the poor man's dyal· with an instrument to set it. made applicable to any place in england, scotland, ireland, &c. by sir samuel morland knight and baronet. 1689. morland, samuel, sir, 1625-1695. 1689 approx. 9 kb of xml-encoded text transcribed from 4 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2006-06 (eebo-tcp phase 1). a51385 wing m2781b estc r221912 99833157 99833157 37632 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a51385) transcribed from: (early english books online ; image set 37632) images scanned from microfilm: (early english books, 1641-1700 ; 2191:21) the poor man's dyal· with an instrument to set it. made applicable to any place in england, scotland, ireland, &c. by sir samuel morland knight and baronet. 1689. morland, samuel, sir, 1625-1695. [2], 5, [1] p. ; ill. and are to be sold at all the button-sellers, cutlers and toyshops about the town. and will be shortly publisht in several other ... and dimensions, for the good of the publick, and for the con... of the manufacture of our nation, [s.l.] : [1689] date of publication from title. copy stained and torn. reproduction of the original in the lambeth palace library, london. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between 1473 and 1700 available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the 25,363 texts created during phase 1 of the project have been released into the public domain as of 1 january 2015. anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng dialing -early works to 1800. 2005-12 tcp assigned for keying and markup 2006-02 spi global keyed and coded from proquest page images 2006-03 andrew kuster sampled and proofread 2006-03 andrew kuster text and markup reviewed and edited 2006-04 pfs batch review (qc) and xml conversion the poor mans dyal . with an instrument to set it. made applicable to any place in england , scotland , ireland , &c. by sir samvel morland knight and baronet . 1689. 〈◊〉 are to be sold at all the button-sellers , cutlers , and 〈◊〉 about the town . and will be shortly publisht in several other 〈◊〉 and dimensions , for the good of the publick , and for the 〈◊〉 of the manufacture of our nation . the poor man's dyal , with an instrument to set it . to set the dyal . first , set the small instrument upon any level place , where the sun comes two or three hours before , and as many after noon , and by the 4 marks at the bottom , make the 4 points ( a , b , c , d ) and by those marks make two lines , ( a , c ) and ( b , d ) crossing each other in the point ( o. ) then mark the point ( e ) where the shadow of the pin terminates in the forenoon . and having from the distance ( o e ) described a circle , watch in the afternoon , when the shadow of the pin cuts the circle in the point ( f ) for the line ( e f ) is a true east and west line ; and the hours of ( vi ) and ( vi ) upon the dyal , being placed upon the said line , the dyal is truly set . directions . for london , or any place within 20 miles , the dyal must be placed exactly level ; but for the following place , the north-side of the dyal ( where is the hour of ( xii ) must be elevated higher than the opposit , or south-side , as is hereafter exprest , which every carpenter and joiner knows how to perform . in england . bedford , about 1 twentieth part of an inch. berwick , 1 tenth . buckingham , 3 hundreths . cambridge , 1 twentieth . carlisle , 1 fourth , or a quarter . chester , 1 tenth . colchester , 1 twentieth . darby , 1 tenth . durham , 1 fourth . glocester , 3 hundreths . hartford , 1 fiftieth . hereford , 1 twentieth . huntington , 1 sixteenth . ipswich , 1 twentieth . kendal , 3 sixteenths . lancaster , 1 fifth . leicester , 1 sixteenth . lincoln , 1 tenth . northampton , 1 sixteenth . norwich , 1 tenth . oxford , 1 fiftieth . stafford , 1 tenth . shrewsbury , 1 tenth . stanford , 1 tenth . warwick , 1 sixteenth . worcester , 1 sixteenth . york , 1 fifth . wales . anglesey , 1 eighth . bermouth , 1 tenth . brecnock , 1 twentieth . cardigan , 1 sixteenth . caermarthen , 1 twentieth . carnarvan , 1 twentieth . denbigh , 1 tenth . flint , 1 tenth . landaff , 1 hundreth . monmouth , 1 fiftieth . montgomery , 1 twentieth . pembroke , 1 fiftieth . radnor , 1 twentieth . st. david , 3 hundreths . the isle of man , 3 sixteenths . scotland . aberdeen , 43 hundreths . dunblain , 1 third . dunkel , 3 eights . edinburgh , 3 tenths . glascow , 3 tenths . kinsale , 45 hundreths . orkney , 59 hundreths . st. andrews , 1 third . skyrassin , 1 half . sterlin , 3 tenths . ireland . antrim , 1 fifth . arglas , 15 hundreths . armagh , 17 hundreths . caterbergh , 17 hundreths . clare , 8 hundreths . cork , 3 hundreths . drogheda , 1 eighth . dublin , 1 tenth . dundalk , 1 eighth . galloway , 1 tenth . kenny , 7 hundreths . kildare , 1 tenth . kingstown , 1 eighth . knockfergus , 1 fifth . kinsale , 1 hundreth . limmerick , 7 hundreths . queenstown , 1 tenth . waterford , 1 twentieth . wexford , 1 sixteenth . youghall , 3 hundreths . for the places hereafter mentioned , the north-side of the dyal must be lower than the south-side , viz. in england . bristol , 1 hundreth . canterbury , 1 tenth . chichester , 1 sixteenth . dorchester , 1 twentieth . exeter , 1 sixteenth . gilford , 1 fiftieth . reading , 1 hundreth . salisbury , 3 hundreths . truro , 7 hundreths . the islands . gvernsey , 1 eighth . jersey , 1 eighth . lindy , 1 hundreth . portland , 7 hundreths . wight , 1 twentieth . if it be required to fix any dyal in any place of england , scotland , &c. not mentioned in this catalogue , it must be set according to the nearest of the places that are mentioned , and it will serve without any sensible error , and much better than those ordinary brass dyals , which are usually made by ignorant apprentices and journey-men : or that cobbled pewter dyal , which was lately made by a brazen-fac'd founder , in imitation of this porr man's dyal , and deserves at least , by way of transposition , the name of that man 's poor dyal ; for tho' he had the wit to make a circle of the same diameter , and to set off all the divisions of the hours , halfs , and quarters from the other ; yet the style is so false and defective in all its parts , that it is not to be mended by one , who knows nothing of a dyal : yet , notwithstanding , in one thing he is to be commended , that he has made the hour-lines so short , that the shadow , of the false gnomon , will not so easily discover his errors . besides , he has carefully fill'd up the vacancies in the middle of he dyal , with the points of the compass , which are only proper for an upright style , and clogg'd the tail of it with five leaden bells , within a pair of rams-horns , to ring aloud the praises of his sheeps-head , for attempting to imitate or counterfeit another man's contrivance , without being able to perform it like an artist : forasmuch as any person who has the least knowledge of those matters , will soon distinguish between the true original , ( which was first calculated , and afterwards exactly delineated by sir samuel morland's own hands , and the molds of the style , which is the principal part of the dyal , carefully contrived and corrected ) and that ignorant founder's counterfeit , and ill-contrived dyal , to which he has put a date of 1690 , to let the world understand that it was none of his own contrivance , but that he did counterfeit one that was made before , viz. in 1689. and of which the style comes false out of the mold , and by cutting and scraping is made ten times worse . if by accident in sending about , or otherwise , any of these dyals , or the stiles of them being but block tin , shall happen to be bent ; then with the help of a square , or for want of that , of a quarter of a sheet of any paper folded up in a double fold , ( which makes an exact square ) it may be bended back , and set right again . finis . a plain declaration of the vulgar new heavens flatform serving not onely fore this age, but also fore the future age of 100 years. halley, edmond, 1656-1742. 1679 approx. 18 kb of xml-encoded text transcribed from 3 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2009-03 (eebo-tcp phase 1). a45349 wing h452 estc r39228 18283100 ocm 18283100 107300 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a45349) transcribed from: (early english books online ; image set 107300) images scanned from microfilm: (early english books, 1641-1700 ; 1635:8) a plain declaration of the vulgar new heavens flatform serving not onely fore this age, but also fore the future age of 100 years. halley, edmond, 1656-1742. 4 p. s.n., [london? : 1679] attributed by wing to halley. date of publication suggested by wing. reproduction of original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between 1473 and 1700 available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the 25,363 texts created during phase 1 of the project have been released into the public domain as of 1 january 2015. anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng astronomy -observations -early works to 1800. sun -observations -early works to 1800. dialing -early works to 1800. 2008-04 tcp assigned for keying and markup 2008-06 spi global keyed and coded from proquest page images 2008-07 paul schaffner sampled and proofread 2008-07 paul schaffner text and markup reviewed and edited 2008-09 pfs batch review (qc) and xml conversion a plain declaration of the vulgar new heavens flatform . serving not onely fore this age , but also fore the future age of 100 years . here you are at first to knowe , that the motion of the sun and the time do alwayes concur , and therefore is the one the measure of the other . fore by the time is to be knowne the place of the sun , and again by the suns place you may knowe the time : therefore you must either knowe the time or the place of the sun , both of them you may easily find in the heavens-flatform , do but lay the dial a or b on the desired day of the yeare and see then what degree the dial doth touch upon the sodiack , and you wil find the thing desired . i. exemple . how to knowe at anny time of the yeare in what degree of the sodiack the sun is . ●y exemple on the first day of may , lay the dial a upon the suid day , and see what degree the dial doth cut upon the sodiack , you wil finde it to be the 12th of taurus being the place of the sun. ii. exemple . how at anny time to knowe the rising and going under of the sun. suppose it be the 24th . of may , then you must lay the dial on the said day , and see where it cuts the sodiac , there you must make upon the dial a signe of chalck or anny thing else that may easily be rubbed out , which signe demonstrates the sun , turn then the dial first so long to the east-side of heaven , til the said signe comes to touch the crombe horizon , see then without upon what houre and minute the dial doth lye on the houre cirkle , you wil find the suns rising at 4 a clock in the morning , and if you turn the said point to the west-side of heaven upon the horizon , then you wil see the dial to lye upon 8 a clock in the evening , being the going under of the sun , then you wil also see that the sun riseth then 36 degr . 30 minut . from the east to the northward , and by consequence so manny degrees and min. lesse from the west to the northward . iii. exemple . hou you may see in the night by the starrs what time it is , suppose you doe but see anny acquainted starre to rise or stand in the s. n. e. or westward , let it be the three kings arising the first day of october , in the night , and that by it you would knove how late it is ; then you must lay the dial a on the said day , and the dial b over the 3 kings , and give then a signe upon the dial b over the 3 kings , then you must firmly turn both dials alike to the eastward , til the said signe upon the dial b doth cut or touch the horizon , and see then upon what houre and minute the dial a lieth , you 'l find it to be 11 a clock in the evening , beingh the right time of the night . i. probleme . how you may upon every poles higth find the riseing and going under of the heavens lights after you have taken the poles higth , and the declination or anny aquainted starres . you must place one foot of the compassis in the centrum of the heavens-mirrour , upon the dial a , and the other downewards , as far as the degree of the declination of the sun or starres ; with this opening of the compassis you must place the one foot upon the edge-side of the dial a upon the complement of the poles higth which you have taken ; and slide the dial a towards 6 a clock , ( or to the dial b which may be laid along by 6 a clock ( or the edge-side of the dial b. ) see then how in anny degrees the dial a lieth upon the houre-circkle from 6 a clock , which wil be the true breadth of the rising and going under of the sun or starres . i. exemple . desiring to knowe on the 21th . of june , where the sun riseth and goeth under , being beginning of cancer , on the poles heigth of 52 degrees . then you must place one foot of the compassis in the centrum of the dial a , and the other downewards as far as on 23 degr . 30 min. being then the declination of the sun ; with this opening of the compassis you must place one foot on the edge-side of the dial a upon the complement of the poles heigth , being 38 degr . then you must turn the dial a towards 6 a clock , til the other foot of the compassis comes triangularly to touch the lin . of 6 a clock . see then how manny degrees the dial a lieth off from 6 a clock . you 'l find it to be very neare 40 degr . 20 min. and so far doth the sun then rise from the e. to the n. upon each northern breadth of 52 degr . and goeth likewise 40 degr . 20 min. under from the w. to the northward . and if the sun be in the first degr . of capricornus , then it is just the same . ii. exemple . desiring to know on the northern breadth of 50 degr . how manny degrees the southern ey of the bul called ●●debaran riseth from the e. to the n. fore the doing of it , you must place one foot of the compassis in the centrum , 〈◊〉 the dial a , and the other as far as the declination of aldebaran being 16 degrees , with this opening of the compassis you must place one foot upon the complement of the poles heigth of 40 degrees , on the dial a , then you mus● turn the said dial til the other foot of the compassis comes triangularly to touch te line of 6 a clock . see then how manny degrees the dial a lieth off from 6 a clock , you 'l find verry neare 25 deg . 20. min. and so manny d●g . doth aldebaran then rise from the e. to the n. it doth also go under 25 deg . 20 min. lesse from the w. to the n. iii. exemple . desiring to knowe on the southern breadth of 20 degr . how many degrees the spica virginis riseth from the e. 〈◊〉 the s. and that , because the declination of spica virginis is sourhly . you must place one foot of the compassis in the center of the dial a , and the other downewards as far as on 9 degr . being the declination of spica virginis , with this opening you must place one foot of the dial a upon the complement of the poles higth of 70 degrees , then you must turn the dial a from the e. to the s. till the other foot of the compassis comes triangularly to touch the east line or the line of 6 a clock . see then how manny degr . the dial a lieth off from 6 a clock , you find verry neare 10 degr . and so far doth s. virginis from the e. to the s. and goeth like wise so far unter from the w. to the s. ii. probleme . how to find the rising and going under of the sun , or of anny acquainted starrs , and that upon every poles higth . place one foot of the compassis in the center of the dial b , and the other downeward , along by the edge-side as far as on the degree of the declination of the sun or starrs , with this opening of the compassis you must place one foot on the edge-side of the dial b upon the complement of the poles heigth , add slide the dial b from the e. to the n. or s. til the other foot comes triangularly to touch the line of 6 a clock . then you must see o● what houre and min. the dial b lieth , which is the true time of the suns rising , which you may also being to the going under . i. exemple . desiring to knowe the rising of the sun , on the n. breadth of 25 degr . being the 21th . of june , when the suns declination is northly 23 degr . 32 min. you must place one foot of the compassis in the centrum of the dial b , and open the other foot douwnewards as far as on 23 degr . 32 min. with this opening of the compassis you must place one foot on the edge-side of the dial b upon the complement of the poles hitgh of 38 degrees , sliding the dial b from the e. to the n. til the other foot of the compassis comes triangulary to touch the line of 6 a clock . see then upon what houre and min. the dial b lieth on the houre circkle , you 'l find it to be in the morning at 5 a clock 15 min. being the right ti●● of the suns rising , the same is in the evening at 8 a clock 15 min. the suns going under . ii. exemple . desiring to knowe on the southern breadth of 40 degr . being on the 21 of june , at what time the sun doth there rise . then you must place one foot of the compassis in the centrum of the dial b , and the other foot downewards , as far as on 23 deg . 30 min. being at the said time the declination of the sun , with this opening you must place one foot of the compassis on the dial b upon the complement of the poles higth of 50 degr . and turn the dial b from the e. to the n. til the other foot of the compassis comes triangularly to touch the east line of 6 a clock , see then upon what houre and min. the dial b lieth , you 'l find neare enough in the morning 17 houres 30 min. being there the rising of the sun , the same is its going under in the evening at 4 a clock 35 min. nota. you must knowe that if you wil , use the heavens flatform over the south-side of the equinoctial line , then you must take the house contrary to that as they are signed upon the heavens mirrour , fore that which is over the north-line 4 a clock in the morning , the same is southly from the line 8 a clock in the morning , and so is the rest accordingly . iii. exemple . desiring to knowe on the northern breadth of 40 degrees , being the first of august , what time the great dog syrius shal rise . lay the dial b over syrius and the dial a upon the first of august , then you must place one foot of the compassis upon the center of the dial b , and open the other as far as on 16 degr . 15 min. being the declination of syrius , with this opening you must place one foot the compassis on the dial b , upon the complement of the poles higth of degrees . then you must firmly turn both dials alike from the e. to the s. , till the other foot of the compassis comes triangularly to touch the east line of 6 a clock . see then upon what houre and minutes the dial lieth , you 'l find neare enough 4 a clock 42 minutes . fore to find its going under , you must firmly turn both dials alike from the west to the southward til the other foot of the compassis comes triangularly to touch the west line of 6 a clock , see then upon what houre and min. the dial a lieth , you 'l find 2 a clock 48 minutes . iii probleme . how to find at al set times the declination of the sun upon the heavens mirrorr . which is indeed verry proffitable for al sea men , fore it serves not onely fore this present age , but also fore the future age of 100 years ; when al books that are made fore that purposse shal be of no worth . fore to find the declination of the sun upon the heavens-mirrour , you must knowe that the suns place is there set according to the two jears , before and after the leape-yeare , and that especially upon the future age , which doth almost differ a whole degree in the sodiack with this present age to the yeare 1700 , and in the suns declination in march and september about 24 minutes . so that al tables of the suns declination which are reckened out with such a difficult calculation , shal after the yeare of 1700 be of no use or worth to a●ny seaman ; because the yeare of 1700 must be a common yeare . desiring then to knowe the declination of the ●un upon some certain or set day in this present age , then you must alwayes lay the dial a one daye farther then the set day , and in the second yeare after the leap-yeare you must lay the dial upon the midst of the day , but being the third yeare after the leape-yeare , then you must lay the dial a on the first fourth part , being in the sodiack about 15 min. backward . but being the first yeare after the leap-yeare , then you must lay the dial upon the third fourth part of the day , being verry neare 20 min. farther in the sodiack , then in the third yeare . and when it is a leap-yeare , then you must lay the dial a upon the beginning of the day , til to the 28 of february ; but being after the 28 of february , then you must al the yeare along lay the dial a upon the end of the day . and if you do truely understand and perform this , then you wil at al times knowe the declination of the sun so perfect and exactly , as the navigation requires . this is the head thing i have to say of my heavens-mirrour . i. exemple . desiring to knowe the suns declination on the 30th . day of april 1691 or 95 being the third yeare after the leap-yeare , then you must lay the dial a upon the first of may , to wit upon the fourth part of the parck of that day . see then where the dial dath cut or touch the sodiack , you find it to be verry neare the 10th . degr . 15 min. ( being at the ●ame time the true place of the sun ) that is 40 degr . 15 min. of ♈ . farther you must lay the dial a over the 40 degr . 15 min. off from ♈ . , in the degrees of the equinoctial or houre cirkle , then you must place one foot of the compassis on the dial a upon the greatest declination of the sun , being 23 degrees 32 min. then you must open the other foot of the compassis towards the line of 6 a clock , to come triangularly with this opening of the compassis . then you must place one foot of it in the center of the dial a , and turn the other downewards , and see where it falls , you find it to be verry neare 14 degrees 55 minutes , being at the said time the suns declination . ii. exemple . desiring to knowe the suns declination on the 30th . of april 1688 or 92. being the first yeare after the leap-yeare . then you must lay the dial a on the first of may , upon the utmost of the days parck , see then where the dial a doth cut the sodiack , you find it to be verry neare the eleventh degree of taurus , being the true place of the set time , that is 41 degrees of ♈ . then you must further lay the dial a on 41 degrees of ♈ . as before , and place one foot of the compassis on the dial a upon the greatest declination of the sun , being 23 degr . 32 min. then you must open the other foot of the compassis , til it comes rect-angularly to touch the line of 6 a clock , with this opening of the compassis , you must place one foot of it in the center of the dial a , and the other you must turn downewards , seeing upon what degree and minute it falls , you 'l find it to be verry neare 15 degr . and 10 min. being at the set time the true declination of the sun , differing litle or nothing with the wise and artificial calculation , do so at al other times , til to the yeare of 1700 , but after that time you must lay the dial upon the set day , considering that this use is principally ( as i told you ) practised and formed upon the future age , where upon we shal also give some exemples , which also wil serve fore the better understanding of the former . iii. exemple . desiring to knowe the suns declination on the 30th day of april 1706. being the second yeare after the leape-yeare . then you must at the same time lay the dial upon the midle of the parck of the said day , and see where the dial a doth toutch the sodiack , you 'l find it to be verry neare the 9th degree and 30th minute of taurus , being at the said time the true-place of the sun , differing almost a whole degree with the yeare of 1686 , fore the sun is 39 degr . 30 minut . of ♈ then you must further place the dial a on 39 degrees 30 minutes of ♈ . upon the dial , in the same manner as i told you before , and then you must place one foot of the compassis on the dial a upon the greatest declination of the sun , being 23 degrees 32 min. , then you must open the other foot of the compassis , till it comes triangularly to toutch the line of 6 a clock : with this opening you must place one foot of compassis in the centrum of the dial a , and the other downewards , see then on what degre and minute the foot stands , you 'l find it to be verry neare 14 degrees 42 minutes , which is on the said 30 day of april about noon , the declination of the sun , differing also in the suns declination with the yeare of 1686 almost 18 min. so that by this exemple you may see that al boocks and tables ●●●●●ning this matter which are formerly made , wil altogether be in vain , and of no vallue , as soon as ever the yeare of 1700 begins . iv. exemple . desiring to knowe the suns declination in the yeare of 1710. whe the sun is in the 18th . degr . of ♌ . then you must lay the dial a upon the 18 degr . you 'l also see that the dial doth then lie on the eleventh day of august , about on the third fourth part of the parck of that day , being in the evening about 6 a clock . then you must further lay the dial a upon the 42th . degree , from ♎ . to cancer upon the houre circkle , being on the 18th . degree of ♌ , then you must place one foot of the compassis on the dial a upon 23 degr . 32 min. as before , en then you must open the other foot , rectangularly unto the line of 6 a clock , with this opening you must place one foot of the compassis in the centrum of the dial a , and then you must turn the other foot downewards , and see on what degree and minute it falls , you 'l find it to be verry neare 15 degr . 30 min. being at the said time the declination of the sun. finis . a nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant framed according to the horizontall projection of the sphere, with the uses thereof. by c.b. maker of mathematic instruments in metall. brookes, christopher, fl. 1649-1651. this text is an enriched version of the tcp digital transcription a69643 of text r4412 in the english short title catalog (wing b4917a). textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. the text has been tokenized and linguistically annotated with morphadorner. the annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). textual changes aim at restoring the text the author or stationer meant to publish. this text has not been fully proofread approx. 26 kb of xml-encoded text transcribed from 12 1-bit group-iv tiff page images. earlyprint project evanston,il, notre dame, in, st. louis, mo 2017 a69643 wing b4917a estc r4412 99834885 99834885 39505 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a69643) transcribed from: (early english books online ; image set 39505) images scanned from microfilm: (early english books, 1641-1700 ; 806:10, 1944:10) a nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant framed according to the horizontall projection of the sphere, with the uses thereof. by c.b. maker of mathematic instruments in metall. brookes, christopher, fl. 1649-1651. [4], 18 p. [s.n.], london : printed in the yeare 1649. dedication signed c.b. (christopher brookes). identified on umi microfilm (early english books, 1641-1700) reel 806 as wing b43 (entry cancelled in wing 2nd ed.). reproductions of the originals in the henry e. huntington library and art gallery (reel 806), and the bodleian library (reel 1944). eng dialing -early works to 1800. quadrant -early works to 1800. a69643 r4412 (wing b4917a). civilwar no a nevv quadrant, of more naturall, easie, and manifold performance, than any other heretofore extant; framed according to the horizontall pr brookes, christopher 1649 4680 5 0 0 0 0 0 11 c the rate of 11 defects per 10,000 words puts this text in the c category of texts with between 10 and 35 defects per 10,000 words. 2006-05 tcp assigned for keying and markup 2006-05 aptara keyed and coded from proquest page images 2006-09 judith siefring sampled and proofread 2006-09 judith siefring text and markup reviewed and edited 2007-02 pfs batch review (qc) and xml conversion a nevv quadrant , of more naturall , easie , and manifold performance , than any other heretofore extant ; framed according to the horizontall projection of the sphere , with the uses thereof . by c. b. maker of mathematic instruments in metall . london , printed in the yeare 1649. to my singular good friend mr. william badiley , mariner , and a lover of the mathematics . worthy sir , having diligently inquired ▪ the reason of the projection of the sphere into plaine , as the ground of all mathematic instruments ( the making whereof in metall is my trade and livelihood ) and compared the severall manners ; i found none so genuine , simple , easie , and manifoldly usefull , as is the horizontall ; which lively representeth the globe rectified to some certaine elevation , and naturally performeth the uses thereof . and having likewise compared the severall quadrants , and pocket instruments hitherto made , and finding them all pieced up with many unnaturall and forced lines and divisions , presupposing an exact diligence both in the calculator , and in the workman ; and yet the performance difficult , troublesome , and tedious : i bethought my self whether out of the horizontall projection i might not by some smal alteration frame a quadrant , that might remedy the defects of all the former instruments , and that with greater ease and certainty . and having by the help of god happily attained my desired intent , my many respects represented you under whose approbation and patronage i might send out into public view this my new quadrant , with the many uses thereof ; as being one to whom i stand obliged for your love and manifold favours to mee both at sea in divers voiages , and at land ; and who through your skill in the mathematicall sciences are able to judge and patronize the first attempts of your affectionately devoted servant , c. b. the description of the quadrant , and the parts thereof . the limbe of the quadrant divided into 90 degr. representeth the horizon . that side of the quadrant where the sights are , is the meridian , or xii a clocke line , unto which is joyned the scale of moneths with every fift day , untill they grow so little toward the solsticeas , that they cannot bee distinguished . this scale ▪ hath five rowes , the midst whereof hath the very same divisions which are on the meridian line : the two next on both sides are for the parts of the moneths , which in the two outermost rowes are noted by their first letters . the other side of the quadrant hath on it the scale of altitudes above the horizon . the short arching lines within the quadrant beside the meridian , are houre-lines , noted by their figures , both for the forenoone , and afternoon ; and halfe houre lines : each halfe houre containing 30 min. of an hour , or deg. 7. 30. of these horary lines , those which serve in the morning before the sunne is full east , or in the evening past the west , ( which is onely in summer halfe yeare ) are reversed . and all the hour lines are noted with two figures ; whereof the upper next the center and scale of altitudes , serve for the afternoon ; and the lower for the forenoon . the two arches which crosse the houre lines , and meet at the beginning of the horizon and scale of altitudes , are two quarters of the ecliptic , and are divided into 90 degr. a piece , in which are noted the xii signes by their proper characters , namely on the upper next the center are ♈ ♉ ♊ & ♋ ♌ ♍ , the summer or northerne signes : and on the lower next the horizon are ♎ ♏ ♐ & ♑ ♒ ♓ , the winter or southerne signes , and contain 30 degr. a piece . this is the circle of the sunnes annuall motion . the long arches , which beginning at the scale of moneths in the meridian betwixt the two quarters of the ecliptic , crosse all the houre lines , are the parallels of declination , or the semidiurnall arches of the sunne ; the middlemost of which is the equinoctiall , the outermost above is the tropick of ♋ , and the outermost below is the tropick of ♑ : although between the equinoctiall and each tropick innumerable parallels are understood to be contained , yet those which are in the instrument drawne , at every second degree of declination , may be sufficient to direct the eye in tracing out an imaginary parallel from every point given in the scale of moneths . the equinoctiall and every tenth parallel are for distinction sake made somewhat grosser than the rest , and all the summer parallels at the east and west line are continued reversedly back unto the horizon . note that upon the right estimation of that imaginary parallel , the manifold use of this instrument doth especially rely ; because the true place of the sunne all that day is in some part or point of the same circle . and note that in this instrument , the direct horary lines , and parallels before their reversion , shew the houre of the day like a direct south upright diall : and the arches of them reversed serve like a direct north upright diall . use i. to finde the declination of the sun every day . seek the day proposed in the scale of moneths very exactly , & mark upon what point it falleth in the middle row of that scale , or ( which is all one ) in the meridian , for there is the declination of the sunne from the equinoctiall , either north or south : which if it fall not directly upon a parallel , but in the space between two , supposing each halfe of that space to containe 60 minutes , estimate with your eye proportionally what minute the point giveth . example 1. what is the sunnes declinaclition upon novemb. 13 ? the day will fall in the space after 20 degrees , from the equinoctiall southward , about 30 minutes : wherefore the sunnes declination is 20° . 30′ south . example 2. what is the sunnes declination upon august 19 ? the day wil fal in the space after 8 degrees , from the equinoctiall northward , one degree and about 40 minutes : wherefore the suns declination is 9° . 40′ . north . note that the declination thus found is to be kept in minde all the day . use ii. to finde the semidinunall arch , or parallel circle in which the sunne moveth every day . seeke out the true point of the sunnes declination upon the meridian by vse i : then from that point by the estimation of your eye , trace out an imaginary parallel : which when it commeth to the east and west line ( as in all northerne parallels it doth ) is to bee reversed unto the horizon or limbe at the same proportionable distance as before . this operation requireth exact diligence . use iii. to finde the time of the sunes rising and setting every day . seek out the imaginary parallel , or semidiurnal arch of the sun for that day by vse ii , and marke where it meeteth with the horizon ; for that is the very point of the sunnes rising and setting , and the hour-lines on both sides of it , ( by proportioning the distance reasonably , according to 30 minutes for halfe an houre ) will shew the time of the sunnes rising and setting . thus at london , novem. 13. the sun will be found to rise at 9 min. before 8 , and to set at 9 min. after 4. also august 19 , the sunne will be found to rise 12 min. after 5. and to set 12 min. before 7. use iv. to finde the suns amplitude , ortive and occasive : that is , how many degrees of the horizon the sunne riseth and setteth from the true east and west points every day . the imaginary parallel of the sunne , together with the time of the sunnes rising , and setting , sheweth upon the horizon the degree of his amplitude from east and west , which in all the northerne parallels is on the north side , and in the southerne on the south side . thus at london , novem. 13. the ampl. ort. will be found 34 degreees . also aug. 19. the ampl. ort. will be found 15° . 10′ . use v. to find the length of every day and night . double the houre of the sunnes-setting , and you shal have the length of the day : or double the houre of the sunnes-rising , and you shall have the length of the night . use 6. to know the reason and manner of the increasing and decreasing of the dayes and nights throughout the whole yeare . when the sunne is in the equinoctial , it riseth and setteth at 6 a clocke : but if the sunne be out of the equinoctial , declining toward the north , the intersection of the parallel of the sunne with the horizon is before 6 in the morning , and after 6 in the evening ; and the diurnall arch greater than 12 houres , and so much more great , the greater the northerne declination is . againe , if the sunne be declining toward the south , the intersection of the parallel of the sunne with the horizon is after 6 in the morning , and before 6 in the evening ; and the diurnall arch lesser than 12 hours , and by so much lesser , the greater the southern declination is . and in those places of the ecliptic in which the sunne most speedily changeth his declination , the length also of the day is most altered ; and where the ecliptic goeth most parallell to the equinoctiall , changing the declination slowly , the length of the day is but little altered . as for example ; when the sun is neare unto the equinoctiall on both sides , the dayes increase and also decrease suddenly and apace ; because in those places the ecliptic inclineth to the equinoctiall in a manner like a straight line , making sensible declination . againe , when the sunne is neare his greatest declination , as in the height of summer , and the depth of winter , the dayes keep for a good time , as it were , at one stay ; because in these places the ecliptic is in a manner parallel to the equinoctiall , the length of the day differeth but little , the declination scarce altering ; and because in those two times of the yeare , the sunne standeth as it were still at one declination , they are called the summer solstice , and winter solstice . wherefore wee may hereby plainely see , that the common received opinion , that in every moneth the dayes doe equally increase , is erroneous . also wee may see , that in parallels equally distant from the equinoctiall , the day on the one side is equall to the night on the other side . use vii . to take the height of the sunne above the horizon . hold the edge of the quadrant against the sunne , so that the sunnes ray or beam may at once passe through the hole of both the sights ; then shall the thread with the plummet shew the sunnes altitude . use viii . to finde the houre of the day , or what a clock it is . having the imaginary parallel or semidiurnall arch of the sunne , already found and conceived in your minde by vse ii , take the sunnes height above the horizon , then stretching the thread over the scale of altitudes , set the bead to the altitude found , move your thread untill the bead exactly falleth upon the imaginary parallel , for there is the houre fought ; and that is the true place of the sun in the quadrant at that time ; to bee estimated upon the horary lines , either direct , or reversed , according as the parallel is . use ix . to finde the sunnes azumith or horizontall distance from the foure cardinall points . the bead being set to the houre of the day , as was shewed in the vse next before , the thread shall in the limbe cut the east or west azumith ; that is , how many degrees of the horizon the verticall circle in which the sunne is , is distant from the east and west points : the complement of which number giveth the azumith from the south meridian , if the bead fell in the right parallels : but if the bead fall upon the reversed parts , the azumith is to be accounted from the north meridian . use x. to finde the meridian altitude of the sunne every day . stretch the thread over the meridian , and set the bead to the true declination of the sunne therein ; then apply the thread to the scale of altitudes ; and the bead shall give the meridian altitude sought . use xi . to finde at what time the sunne commeth to bee full east or west every day in summer . this is shewen by observing at what houre the imaginary parallel meeteth with the east and west line , at which it beginneth to reverse . use xii . to finde how high the sunne is above the horizon at any houre , every day . set the bead to the point in which the imaginary parallel of that day crosseth the houre given : then applying the thread to the scale of altitudes , mark upon what degree the bead falleth ; the same shall bee the altitude of the sun required . use xiii . to finde how high the sunne is being in any azumith assigned every day : and also at what houre . set the bead to the point in which the imaginary parallel of that day crosseth the azumith assigned ; there also shall bee the houre sought : then applying the thread to the scale of altitudes , marke upon what degree the bead falleth ; the same shall be the altitude of the sun required . these two last uses serve for the delineation of the ordinary quadrants , as that of gemma frisius , munster , clavius , master gunter , &c. and also of rings , cylinders , and other topicall instruments ; and for the finding out of the houre by a mans shadow , or by the shadow of any gnomon , set either perpendicular , or else parallel to the horizon . use xiv . to finde the sunnes longitude , or place in the ecliptic . the imaginary parallel of the day being exactly traced will cut in the ecliptick the signe and degree wherein the sunne is : and note , that each semicircle of the ecliptic is doubly noted with characters of the signes ; the first and third quarters goe forward from the equinoctial point unto the meridian , containing ♈ ♉ ♊ & ♎ ♏ ♐ : the second and fourth quarters goe backe from the meridian unto the equinoctiall point , containing ♋ ♌ ♍ & ♑ ♒ ♓ . but because neare unto both tropicks ( namely from may 11 , to july 10 , in the height of summer , and from november 13 , to januarie 12 in the depth of winter ) the declination altereth so slowly , that the true place of the sunne in the ecliptic cannot be distinguished with any certainty , worke according to this foure-fold rule following . 1. before june 10 , out of the number of dayes from may 0 , subduct 11 : the remains shall be the degrees of ♊ : thus for june 3 , ( because there is all may and three dayes of june ) say 34 — 11 = 23 ♊ , the place of the sunne . 2. after june 10 , out of the number of dayes from june 0 , subduct 10 : the remains shall bee degrees of ♋ : thus for july 3 , say 33 — 10 = 23 ♋ , the place of the sun . 3. before december 13 , out of the number of dayes from november 0 , subduct 13 : the reamines shall be degrees of ♐ : thus for december 3 , say 33 — 13 = 20 ♐ , the place of the sun . 4. after december 13 , out of the number of dayes from december o , subduct 13 : the remaines shall be degrees of ♑ : thus for january 3 , say 34 — 13 = 21 ♑ , the place of the sunne . use xv . to find the suns right ascension every day . having by use xiv . found the place of the sunne in the ecliptic , mark diligently upon what houre , and as neare as you can estimate what minute it falleth , counting the houres in the first and third quarters of the ecliptic , from the equinoctiall point ; but in the second and fourth quarters , from the meridian : and adde thereto in the second quarter six hours , in the third twelve houres , and in the fourth eighteen houres : so shall you have the sunnes right ascension , not in degrees , but in time , which is more proper for use . example , in ♌ 6. the sunnes right ascension will bee eight houres , one halfe , and about three minutes ; that is h : 8 : 33. min. reckoning 30′ for halfe an houre . use xvi . to find the houre of the night by the starres . for this , i have set a little table of five knowne stars dispersed round about the heavens , with their declination and right ascension for anno dom. 1650. namely the left shoulder of orion , noted o. the heart of the lion , noted ♌ . arcturus noted a : the vulture volant , noted v. the end of the wing of pegasus , noted p. the table .   declinat . rec. as . o 5° 59′ n h5 6,5′ ♌ 13 39 n 9 50 a 21 4 n 14 00 v 8 1 n 19 34 p 13 15 n 23 55 , 5 the operation is thus ; first by the height of the starre taken , and the parallel of its declination exactly traced , seek out the houre of the starre from the meridian , as before was taught for the houre of the day by the sunne . secondly , out of the right ascension of the starre , subduct the right ascension of the sun ; the remain● sheweth how long time from the noone before the same starre commeth into the meridian . lastly , if the starre be not yet come to the meridian , out of the houre of the starres comming into the meridian , subduct the houre of the starre : but if the star be past the meridian , adde both the houres together ; so shall you have the true houre of the night . note , that if the hours out of which you are to subduct bee lesser than the other , you must adde unto them 24. use xvii . to finde out the meridianline upon any horizontall plaine . about the middle of your plaine describe a circle ; and in the center thereof erect a straight piece of wire perpendicularly . when the sunne shineth , note the point of the circle which the shadow of the wire cutteth , which i therefore call the shadow point ; and instantly by vse ix . seeke the sunnes azumith from the south or north : keepe it in minde . then from the shadow point , if your observation be in the foore-noon , reckon upon the circle an arch equall to the azumith kept in minde , that way the sunne moveth , if the azumith bee south : or the contrary way if it bee north . but if your observation bee in the afternoone , reckon the north azumith that way the sunne moveth ; or the south azumith the contrary way . lastly , through the end of the azumith and the center , protract a diameter for the meridian line sought : which you may note with s. at the south end , and with n. at the north end . you may also note the point of the circle diametrally opposite to the shadow point with sun ; , because it is the azumith place of the sun , at the moment of your observation . use xviii . to finde the declination of any wall or plaine . the safest way ( because the magneticall needle is apt to be drawne awry ) will be by an instrument made in this manner . provide a rectangular board about ten inches long , and five broad : in the midst whereof , crosse the breadth , strike a line perpendicular to the sides ; and taking upon it a center , describe a circle intersecting the same line , in two opposite points , to be noted with the letters t. and a : divide each semicircle into two quadrants , and every quadrant into 90 degrees , beginning at the points t and a , both wayes ; the first quad. beginning on the left hand of t. the second quadrant on the right hand : the third quadrant above it toward a : and lastly , the fourth quadrant . and in the center erect a wier at right angles . the use of this instrument . apply the long side of the board next t to the wall when the sunne shineth upon it , holding it parallel to the horizon , that it may represent an horizontall plaine . marke what degree the shadow of the wyer cutteth in the circle ; and instantly seek the sunnes azumith , either south or north : reckon it on the circle from the shadow to the meridian , as was taught in the use next before , noting that end with the letter contrary to that of the azumith : as if the azumith bee south , note it n. and the opposite end s ; if the azumith bee north , note it s , and the opposite end n : whereby also you have the east ▪ and west sides : so shall the arch s a. or n a. give the declination of the plaine , and the point a , the coast or quarter into which it is . example , june 2 in the forenoone , applying the instrument to a wall , i found the shadow in 23 degr. of quadr. 2. and the height of the sunne was 26 degrees , whereby i found the azumith to be north 84 degr. which reckoned from the shadow against the sunne , fell upon 61 degr. in quad. 1. for one end of the meridian ; and the opposite end ▪ which is n. upon 61 degr. in quad. 3. and a was on the east side of n. wherefore the declination of that wall is 61 deg. from the north eastward . use xix . to finde the declination of an upright wall by knowing the time of the sunnes comming to it , or leaving it . and contrariwise , the declination of an upright wall being known to finde at what time the sunne will come into it . because the declination of a plain is an arch of the equinoctiall intercepted between the horizontall section of the plaine : and the east or west points : or else ( which is all one ) between the meridian , and a , the axis of that horizontall sexion . watch till you see the center of the sunnejust even with the edge of the wall : then instantly take the sunnes azumith from east or west , by use ix . the same is the declination of the wall . likewise if the declination be given , reckon it upon the limbe of your quadrant from the east and west point ; and the thread being applyed to the end of that arch , shal in the suns imaginary parallel for that day , cut the houre and time desired . use xx . certaine advertisements necessary for the use of the quadrant in the night . in which questions as concerne the night , or the time before sunne-rising , and after sunnesetting , the instrument representeth the lower hemisphere , wherein the southern pole is elevated . and therefore the parellels which are above the equinoctiall toward the center , shall be for the southerne or winter parellels : and those beneath the equinoctiall , for the northerne or summer parallels : and the east shall be counted for west , and the west for east ; altogether contrary to that which was before , when the instrument represented the upper hemisphere . use xxi . to finde how many degrees the sunne is under the horizon at any time of the night . seek the declination of the sunne for the day proposed by use i. and at the same declination on the contrary side of the equinoctiall imagine a parallel for the sunne that night ; and marke what point of it is in the very houre and minute proposed : set the bead to that poynt ; then applying the thread to the scale of altitudes , marke upon what degree the bead falleth : for the same shall shew how many degrees the sunne is under the horizon at that time . use xxii . to finde out the length of the crepusculum , or twi-light . it is commonly held that twilight is so long as the sunne is not more then 18 degrees , under the horizon , the question therefore is , at what time the sunne cometh to be 18 degrees under the horizon any night . seek the sunnes declination for the time proposed , and at the same declination , on the contrary side of the aequinoctiall , imagine a paralsel for the sunne that night : then set the bead at 18 degrees in the scale of altitudes ; and carry the thread about till the bead fall upon the imagined parallell : for there shall be the houre or time sought . and in this very manner you may find the time or houre of the night at any other depression of the sunne under the horizon . finis . the triangular quadrant, or, the quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by john brown ... brown, john, philomath. 1662 approx. 38 kb of xml-encoded text transcribed from 14 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2004-08 (eebo-tcp phase 1). a29764 wing b5043 estc r33264 13117453 ocm 13117453 97765 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a29764) transcribed from: (early english books online ; image set 97765) images scanned from microfilm: (early english books, 1641-1700 ; 1545:11) the triangular quadrant, or, the quadrant on a sector being a general instrument for land or sea observations : performing all the uses of the ordinary sea instruments, as davis quadrant, forestaff, crosstaff, bow, with more ease, profitableness, and conveniency, and as much exactness as any or all of them : moreover, it may be made a particular and a general quadrant for all latitudes, and have the sector lines also : to which is added a rectifying table to find the suns true declination to a minute or two, any day or hour of the 4 years : whereby to find the latitude of a place by meridian, or any two other altitudes of the sun or stars / first thus contrived and made by john brown ... brown, john, philomath. [2], 24, [1] p. : ill. to be sold at [his, i.e. brown's] house, or at hen. sutton's ..., [london] : 1662. added illustrated t.p. place of publication suggested by wing. reproduction of original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between 1473 and 1700 available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the 25,363 texts created during phase 1 of the project have been released into the public domain as of 1 january 2015. anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng quadrant. dialing. mathematical instruments. 2004-01 tcp assigned for keying and markup 2004-03 apex covantage keyed and coded from proquest page images 2004-04 mona logarbo sampled and proofread 2004-04 mona logarbo text and markup reviewed and edited 2004-07 pfs batch review (qc) and xml conversion the triangular quadrant : or the quadrant on a sector . being a general instrument for land or sea observations . performing all the uses of the ordinary sea instruments ; as davis quadrant , forestaff , crossstaff , bow , with more ease , profitableness , and conveniency , and as much exactness as any or all of them . moreover , it may be made a particular , and a general quadrant for all latitudes , and have the sector lines also . to which is added a rectifying table , to find the suns true declination to a minute or two , any day or hour of the 4 years : whereby to find the latitude of a place by a meridian , or any two other altitudes of the sun or stars . first thus contrived and made by iohn browne at the sphere and dial in the minories , and to be sold at 〈◊〉 house , or at hen. sutton's in thredneedle-street behind the exchange . 1662. triangular quadrant the triangular quadrant : being a general instrument for observations at land or sea , performing all the uses of all ordinary sea instruments for observations , with more speed , ease and conveniencie than any of them all will do . contrived and made by iohn browne at the spheare and sun-dial in the minories , and sold there or at mr h. suttons behind the exchange . 1662. the description , and some uses of the triangular quadrant , or the sector made a quadrant , or the use of an excellent instrument for observations at land or sea , performing all the uses of the forestaff , davis quadrant , bow , gunter's crosstaff , gunter's quadrant , and sector ; with far more convenience and as much exactness as any of them will do . the description . first , it is a jointed rule ( or sector ) made to what radius you please , but for the present purpose it is best between 24. and 36 inches radius , and a third peice of the same length , with a tennon at each end , to make it an equilateral triangle ; from whence it is properly called a triangular quadrant . ●●secondly , as to the lines graduated thereon , they may be more or lesse as your use and cost will please to command , but to make it compleat for the promised premises , these that follow are necessary thereunto . 1. a line of degrees , of twice 90. degrees on the moveable leg , and outer edge of the cross piece : for quadrantal and back observations . 2. such another line of 60. degrees , for forward observations on the inside of the cross piece . 3. on the moveable leg ▪ a kallender of months , and dayes in 2 lines . 4. next to them , a line of the suns place in degrees . 5. next to that , a line of the suns right assention , in degrees or hours . 6. next above the months on the same leg , an hour and azimuth line , fitted to a particular latitude , as london , or any other place , for all the uses of gunter's quadrant as you may find in the former discourse ( called a joynt-rule . ) 7. on the head leg , and same side , a particular scale of altitudes , for the particular latitude . 8. next to that a general scale of altitudes , for all latitudes . 9. a line of 360 degrees , divided so as to serve for 360 degrees of 12 signs ; and 24 hours , ( and in foot and 2 foot rules for inches also . ) 10. a line of 29½ laid next to the former , serving to find the moons coming to south , and her age , and place , and the time of the night , by the fixed stars . 11. a perpetual almanack , and the right assension , and declination of several fixed stars . 12. a line of lines next the inside , may be put on without trouble , or incumbering one another , all these on one side . secondly , on the other side may be put the lines of natural signs , tangents and secants , to a single , double , and treble radius , and by this means more then a gunters sector , ( the particular lines being inscribed between the general lines . ) thirdly on the other edge there may be artificial numbers , tangents , signs , and versed signs , and by this means it is a gunters rule or a crosstaff . fourthly , on the insides , inches , foot-measure , and a line of 112. parts , and a large meridian line , or the like : as you please . lastly , two sliding nuts with points in them , fitted to the cross piece , makes it a proper beam compass to use in working by the numbers signs and tangents on the edge , or flat side , also it must have four or five sights , a thred , and plummet , and compasses , as other instruments have , thus much for the description , the uses follow . an advertisement . first , for the better understanding and brevity sake , there are ten things to be named and described , as followeth ; 1. the head leg , in which the brass revet is fixed , and about which the other turn ; 2. the moveable leg , on which the months and days must always be ; 3. the cross piece , that is fitted to the head , and moveable leg , by the two tennons at the end ; 4. the quadrant side where the degrees and moneths are , for observation . 5. the other or sector side for operation , 6thly the head center , being center to the degrees on the inside of the crosse peece , for a forward observation as with the forestaffe . 7 the other center , near the end of the head leg , being the center to the moveable leg , for backward observations , ( as the davis quadrant is used , and the bow ) which you may call the foot center , or leg center for backward observations . 8thly the sights , as first the turning or eye sight , which is alwayes , set on one of the centers , with a screw to make it fast there , which i call the turning sight , 9thly the horizon sight that cuts the degrees of altitude , and sometimes is next to the eye , and sometimes remote from the eye , yet called the horizon ( slideing ) sight , 10thly . the object or shaddow sight of which there may be 3 for convenience sake , as two fixed and one moveable to slide as the horizon sight doth : the other two do serve also to pin the crosse peece , and the two legs together , through the two tennons , all whose names in short take thus : 1. the head leg : 2. the moveable leg : 3. the crosse peece : 4. the quadrant side : 5. the sector side , 6. the head ( or forward ) center . 7 the leg ( or backward ) center . 8. the turning sight . 9. the ( slideing ) horzon sight . 10. the object ( or shaddow ) sight . of which there be 3. all differing according to your use and occasions : one to slide to any place , the other 2. to be put into certain holes . nigher , or further off : as will afterwards largely appear . the uses : i. to find the suns declination , true place , right assention , and rising , the day of the moneth being given . first open the rule to an angle of 60. degrees , which is alwayes done when the cross peece is fitted into the mortesse holes , and the pins of the object sights put in the holes through the tennons , or else by the second chapter of the joynt-rule : then extend a thred from the center pin in the head leg , to the day of the month , & on the degrees it cuts the suns declination , in the line of right assention his right assention , in the line of true place his true place , and in the hour line his true rising and setting , in that latitude the line is m●de for : example , on the first of may i would know the former questions , the rule being set by the crosse peece , and the thred on the leg center pin ; and drawn straight and laid over may 1. it cuts in the degrees 18. 4. north declination , and 20. 58. in ♉ taurus for his place , and 3 hours 14. minutes right assention in time , or 48. 32. in degrees : and the rising of the sun that day is at 4. 23 , and sets at 5. 37 , in 51. 32 latitude . the finding of hour and azimuth , either particularly , or generally , with other astronomical propositions , are spoken enough of before in the joynt-rule , and in all other authors that write of the sector , or gunter's rule , so that all i shall speak of now , shall be onely what was forgot in the first part , and what is new as to the using the instrument in sea observations . ii. to find the suns or a stars altitude , by a forward observation . skrew the turning sight to the head center , and set that object sight , whose holes answer to the sliding horizon sight , in the hole at the end of the head leg , and put the horizon sight on the crosse peece next the inside ; then holding the crosse peece with your right hand , and the turning sight close to your eye , and the moveable leg against your body , with your thumb on the right hand thrust upwards , or pull downwards , the horizon sight : till you see the sun through the object sight , and the horizon through the horizon sight , then the degrees cut by the middle of the horizon sight , on the crosse peece shall be the true altitude required : iii. to perform the same another way . if your instrument be parted , that is to say the crosse peece from the other , and an altitude be required to be had quickly , then set the two object sights , in two holes at the end of the line of naturall signs , then set the head of the rule to your eye , so as the sight of the eye may be just over the center , then open or close the joynt , till you see the horizon through one sight , and the sun or star through the other , then is the sector set to the angle required , to find which angle do thus , take the parallel sign of 30 and 30 , and measure it from the center , and it shall reach to the sign of half ●he angle required . example . suppose i had observed an altitude , and the distance between 30 and 30 , should reach from the center to 10. degrees on the signs , then is the altitude of the sun 20. degrees for 10 doubled is 20. iiii. to find the suns altitude by a back observation , skrew the turning sight to the leg center , ( or center to the degrees on the moveable leg ) and put one of the object sights , in the hole by 00. on the outer edge of the crosse peece , and set the edge of it just against the stroke of 00 , or you may use the sliding object sight and set the edge or the middle of that , to the stroke of 00 , as you shall judge most convenient ; and the horizon sight to the moveable leg , then observe in all respects as with a davis quadrant , till looking through the small hole of the horizon sight , you see the crosse bar and button , in the turning sight , cut the horizon : and at the same instant the shadow of the edge or middle of the object or shadow sight , fall on the middle of the turning sight , by sliding the horizon sight higher or lower , then the middle stroke of the horizon sight , shall cut on the moveable leg , the suns true altitude required . as f 〈…〉 t stay at 50 degrees , then is the sun 50 degrees above the horizon . v. but if the sun be near to the zenith or 90 degrees high , then it will be convenient to move the object sight , to a hole or two further as suppose at 10 , 20 , 30 degrees more , toward the further end of the crosse peece and then observe as you did before in all respects , as with a davis quadrant , and then whatsoever degrees the horizon sight cuts , you must ad so much to it , as you set the object sight forwards , as suppose 30 , and the horizon sight stay at 60 , then i say 60 , and 30 , makes 90 : the true altitude required . note that by this contrivance , let the altitude be what it will , you shall alwayes have a most steady observation : with the instrument leaning against your brest , a considerable thing , in a windy day , when you may have a need of an observation in southern voyages , when the sun is near to the zenith at a meridian observation . vi. to find the suns distance from the zenith , by observing the other way , the sun being not above 60 degrees high , or 30 from the zenith . set the turning sight as before on the leg center , then set an object sight in one of the holes in the line on the head leg , nigher or further of , the turning sight : as the the brightnesse or dimnesse of the sun will allow to see a shadow , then looking through the small hole on the horizon sight , till you see the horizon cut by the crosse bar of the great hole , in the turning sight , turning the foreside of that sight , till it be fit to receive the shadow of the middle of the object sight ; then the degrees cut by the horizon sight , shall be the suns true distance from the zenith , or the complement of the altitude . vii . note that by adding of a short peece about 9 inches long on the head leg , whereon to set the slideing shadow sight , you may obtain the former convenience of all angles , this way also , at a most steady and easie manner of observation ; but note whatsoever you set forwards on that peece , must be substracted from that the sight sheweth , and the remainder shall be the suns distance from the zenith required . as suppose you set forward 30 degrees , and the horizon sight should stay at 40 , then 30 from 40 rest 10 , the suns distance from the zenith required ; thus you see , that by one and the same line , at one manner of figuring , is the suns altitude , or coalitude acquired and that at a most certain steady manner of observation . viii . to find an observation by thred and plummet , without having any respect to the horizon , being of good stead in a misty or cloudy day at land or sea . set the rule to his angle of 60 degrees by putting in the crosse peece , then skrew the turning sight to the head center , then if the sun or star be under 30 degrees high , set the object sight in the moveable leg , then looking through the small hole in the turning sight , through the object sight , to the middle of the star or sun , as the button in the crosse bar will neatly shew ; then the thread and plummet , hanging on the leg center pin , and playing evenly by the moveable leg , shall shew the true alti●ude of the sun , or star required counting the degrees as they are numbred , for th : north declinations from 60 toward the head with 10 20 , as if the thred shall play upon 70 10 then is the altitude 10 degrees . ix but if the sun or star be above 30 degrees high , then the object sight must be set to the hole in the end of the head leg : then looking as before , and the thred playing evenly by the moveable leg , shall shew the true altitude required , as the degrees are numbred . note that if the brightnesse of the sun should offend the eye , you may have a peice of green , blew , or red glasse , fixed on the turning sight , or else remove the object sight nearer to the turning sight , and then let the sun beams pierce through both the small holes , according to the usuall manner and the thred shall shew the true altitude required . note also if the thred be apt to slip away from his observed place , as between 25 and 40 it may : note a dexterious handling thereof will naturally shew you how to prevent it : x. to find a latitude at sea by forward meridian observation or altitude . set the moving object sight to the suns declination , shewed by the day of the month , and rectifying table , and skrew the turning sight to the leg center , and the horizon sight to the moveable leg , or the outside of the crosse piece , according as the sun is high or low ( but note all forward observations respecting the horizon , ought to be under 45 degrees high , for if it be more it is very uncertain , by any instrument whatsoever , except you have a plummet and then the horizon is uselesse ) then observe just as you do in a forward observation , moving the horizon sight till you see the sun through the horizon sight , and the horizon through the object sight , or the contrary . ( moving not that sight that is set to the day of the month or declination , ) then whatsoever the moving sight shall shew , if you add 30 to it , it shall be the latitude of the place required ; observing the difference in north and south latitudes ; that is , setting the sight to the proper declination , either like , or unlike , to the latitude example . suppose on the 10. of march when the suns declination is 0 — 10. north , as in the first year after leap year it will be , set the stroke in the middle of the moving object sight to 10 of north declination , and the horizon sight on the moveable leg , then move it higher or lower , till you see the horizon through one , and the sun through the other , then the degrees between , is the suns meridian altitude , if it be at noon , as suppose it stayed at 21 30 ▪ then by counting the degrees between , you shall find them come to 38. 40. then if you add 30. to 21. 30. it makes 51. 30. the latitude required , for if you do take 0 10′ minutes from 38. 40. there remains 38. 30. the complement of the latitude . note , that this way you may take a forward observation , and so save the removing of the ●urning sight . note also , that when the horizon sight shall stay about the corner , you may move the object sight 10. or 20. degrees towards the head , and then you must add but 20. or 10 degrees to what the sight stayed at ; or if you shall set the sight the other way 10 or 20. degr . then you must add more then 30 so much . as suppose in this last observation , it had been the latitude of 45 or 50 degrees , then you shall find the sight to play so neer the corner , that it will prove inconvenient , then suppose instead of 0 10. i set it to 20 degrees 10′ north declination , which is 20. degrees added to the declination , then the suns height being the same as before , the sight will stay at 41. 30. to which if you add 10 degrees , it doth make 51. 30. as before ; here you must add but 10 degrees , because you increased the declination 20. degrees ; but note by the same reason , had you set it to 19. 50. south declination , then it had been diminished 20 degrees , and then instead of 30 you must add 50 ▪ to 1 30. the place where the sight would have stayed . thus you see you may very neatly avoid this inconvenience , and set the sights to proper and steady observations , at all times of observation . xi . to find the latitude by a backward meridian observation at sea. this is but just the converse of the former , for if you set one sight to the declination , either directly , ( or augmented or diminish'd as before , when the moving sight shall stay , about the corner of the triangular quadrant ) then the other being slipt to and fro , on the outside of the crosse peece , till the shadow of the outer edge , shall fall on the middle of the turning sight , then 30 just , or more or lesse added , to that number the moving sight stayed at , ( according as you set the first horizon sight to the declination ) shall be the true latitude required . example . suppose on the same day and year as before , at the same noon time , i set my horizon sight to just 10′ of north declination , you shall find the moving sight to stay at 21. 30. neer to the corner , now if the sun shine bright , and will cast the shadow to the turning sight , then set the horison sight at the declination , forward 10 or 20. degrees , then the moving sight coming lower you , add but 20. or to that it shall stay at , and the summe shall be the latitude . but it is most likely that it will be better to diminish it 20. degrees , then the moving sight will stay about 2 30. on the crosse piece , and so much the better to cast a shadow ; for if you look through the horizon and turning sight to the horizon , you shall find the shadow of the former edge of the moving shadow sight , to stay at 2 30. to which if you add 20. the degrees diminished , and 30. it makes 51 30. the latitude required as before . note also for better convenience of the shadow sight , when you have found the true declination , as before is taught , set the moving object sight to the same , on the crosse piece , counted from 00. towards the head leg , for like latitudes and declinations ; and the other way for unlike latitudes and declinations , then observing as in a back observation , wheresoever the sight shall stay , shall be the complement of the latitude required . if you add or diminish consider accordingly . note likewise , when the declination is nere the solstice , and the same way as the latitude is , and by diminishing , or otherwise the moving sight shall fall beyond 00 on the crosse piece ; then having added 30. and the degrees diminished together , whatsoever the sight shall stay at beyond 00 must be taken out of the added sum , and the remainder shall be the latitude required . example . suppose on the 11. of iune , in the latitude of 51 30 north , for the better holding sake i diminish the declination 30 degrees , that is in stead of setting it to 23. 32. north declination , i set it to 6 : 28 south then the sun being 62 degrees high will stay at 08. 30. beyond 00. the other way now 30 to be added , and 30 diminished , makes 60 , from which take 8. 30 , rest 51 30 the latitude required . xii . to find a latitude with thred and plummet , or by an observation made without respecting the horizon . count the declination on the cross peece ( and let 00 be the equinoctiall and let the declination which is the same with the latitude be counted toward the moveable leg , and the contrary the other way , as with us in north latitude , north declination is toward the moveable leg , and south declination the contrary and contrarily in south latitudes ) and thereunto set the middest of the sliding object or horizon sight , then is the small hole on the turning sight and the small hole on the horizon sight , two holes whereby the sun beams are to pierce to shine one on the other : then shall the thred shew you the true latitude of the place required . example . suppose on the 11 of decem. 1663 , at noon i observe the noon altitudes set the middle of the horizon sight to 23 ▪ 32 counted from 00. toward the head leg end , then making the sun beams to peirce through the hole of this , and the turning sight , you shall find the plummet to play on 51. 30 , the latitude required , holding the turning sight toward the sun . note also that here also you may avoid the inconvenience of the corner , or the great distance between the sights , by the remedy before cited , in the back and forward observation . for if you move it toward the head leg , then the thred will fall short of the latitude , if toward the moveable leg then it falls beyond the latitude , as is very easie to conceive of : thus you see all the uses of the forestaff , and quadrant , and mr. gunter's bow are plainly and properly applyed to this triangular quadrant , that the same will be a sector is easie to perswade you to believe , and that all the uses of a gunter's quadrant , are performed by it , is fully shewed in the use of the joynt-rule , to which this may be annexed , the numbers signs and tangents and versed signs makes it an excellent large gunter's rule , and the cross peece is a good pair of large compasses to operate therewith ; lastly , being it may lie in so little roome it is much more convenient for them , with whom stowage is very precious , so i shall say no more as to the use of it , all the rest being fully spoke to in other authors , to whom i refer you : only one usefull proposition to inure you to the use of this most excellent instrument , which i call the triangular quadrant . note that in finding the latitude , it is necessary to have a table of the suns declination for every of the four years , viz. for the leap year and the 1 , 2 and 3d. after , now the table of the suns declination whereby the moneths are laid down , is a table that is calculated as a mean between all the 4 years , and you may very well distinguish a minute on the rule ; now to make it to be exact i have fitted this rectifying table for every week in the year , and the use is thus : hang the thred on the center pin , and extend the thred to the day of the moneth , and on the degrees is the suns declination , as near as can be for a common year , then if you look in the rectifying table for that moneth , and week you seek for you shall find the number of minutes you must add to or substract from the declination found for that day and year : example , suppose for april 10. 1662. the second after leap year , the rule sheweth me 11. 45 , from which the rectifying table saith i must substract 3′ then is the true declination 11 42 , the like for any other year . note further that the space of a day in the suns swiftest motion being so much , you may consider the hour of the day also , in the finding of a latitude , by an observation taken of the meridian , as anon you shall see that as the instrument is exact , so let your arithmetical calculation be also : by laying a sure foundation to begin to work upon , then will your latitude be very true also . a rectifying table for the suns declination .   d 1 year 2 year 3 year leap year ianuary 7 sub 5 sub 2 add 1 add 4 15 s 6 s 2 a 2 a 5 22 s 7 s 3 a 1 a 6 30 s 7 s 3 a 2 a 7 februar 7 s 8 s 3 a 2 a 7 15 s 8 s 4 a 2 a 8 22 s 9 s 3 a 2 a 7 march 1 s 4 s 1 a 7 sub 11 7 s 3 ad 3 add 9 sub 9 15 add 3 ad 1 sub 9 add 9 22 a 2 sub 4 sub 9 ad 8 30 a 1 sub 4 sub 9 ad 8 april 7 a 2 sub 3 s 8 a 8 15 a 2 s 3 s 8 a 7 22 a 1 s 3 s 8 a 6 30 a 1 s 3 s 6 a 5 may 7 a 0 s 2 s 6 a 5 15 a 0 s 2 s 5 a 3 22 a 0 s 1 s 3 a 3 30 a 0 s 1 s 2 a 1 iune 7 a 0 s 1 s 1 ad 0 15 a 0 s 0 s 0 sub 0 22 s 1 s 0 ad 2 s 2 30 s 1 ad 1 ad 3 s 3 iuly 7 sub 2 add 1 add 3 sub 4 15 s 1 a 1 a 5 s 5 22 s 2 a 1 a 5 s 6 30 s 2 a 2 a 6 s 7 august 7 s 3 a 2 a 7 s 8 15 s 3 a 2 a 7 s 8 22 s 3 a 3 a 8 s 9 30 s 3 a 3 a 9 s 9 september 7 s 3 a 3 add 9 sub 9 15 add 3 sub 3 sub 9 add 9 22 a 2 s 4 s 9 a 8 30 a 2 s 4 s 9 a 8 october 7 a 3 s 3 s 9 a 8 15 a 2 s 3 s 8 a 7 22 a 2 s 2 s 7 a 8 30 a 2 s 2 s 7 a 8 november 7 a 1 s 1 s 6 a 7 15 a 1 s 2 s 5 a 5 22 a 1 s 3 s 4 a 3 30 a 0 s 2 s 3 a 2 december 7 0 s 1 s 1 add 1 15 sub 1 s 0 s 0 sub 1 22 s 1 s 0 add 1 s 3 30 s 3 s 0 add 2 s 5 the declination of the sun being given , or rather the suns distance from the pole , and the complement of two altitudes of the sun , taken at any time of the day , knowing the time between : to find the latitude . suppose on the 11. of iune , the sun being 66 degrees , 29′ distant from the north pole , and the complement of one altitude be 80. 30. and the complement of another altitude 44. 13. and the time between the two observations just four hours ; then say , as the sine of 90 00 to sine of suns dist . f. the pole 66 28 so is the sine of ½ time betw . 30 00 to the sine of ½ the 3d. side 27 17½   27 17½ of a triangle as a b. 54 35 the side a p 66 28 the side p a 66 28 the side a b 54 35 whole sum 187 31 half sum 93 46 the differ . betw ½ sum and ap side op . to inqu . triangle 27 18 then say , as s. of 90 90 00 to s. of suns dist . from pole 66 28 so is the sine of a b 54 35 to the sine of a fourth sine 48 23 then as that 4 to sine of ½ sum 93 46 so is s. of the difference 27 18 to a seventh sine 37 44½ or the versed sine of p a b 77 02 then to find z a b z b is 80 30 and z a is 44 13 the former side a b is 54 35 sum 179 18 half sum 89 39 difference 09 09 as s. of 90 90 00 to sine of a b 54 35 so is s. of z a 44 13 to a 4th . sine 34 38 as s. 4th . 34 38 to s. of ½ sum 89 39 so is s. diff . 09 39 to a 7th . sine 16 15¼ or to the vers . sine of z a b 116 07 then if you take p a b from z a b there will remain z a p 39 05 then say again by the rule as before , as the sine of 90 00 to co-sine of z a p 50 55 so is the tang of a z 44 13 to the tangent of a c 37 04 which taken from a p 66 28 remaineth p c 29 24 then lastly say ,   as the co-sine of a c 52 56 to the co-sine of c p 60 36 so is the co-sine of z a 45 47 to the co-sine of z p 51 30 the latitude required to be found . this question or any other may be wrought by the sines and tangents and versed sines on the rule , but if you would know more as concerning this or any other , you may be fully satisfied by mr. euclid spidal at his chamber at a virginal makers house in thred-needle street , and at the kings head neer broadstreet end . vale. finis . the triangle explained . s p z n a meridian circl . ae ae the equinoctial . ♋ b a ♋ the , tropick of cancer . p b 5 p the hour circle of five , ante , mer. p a 9 p the hour circle of nine , ante , mer. z b the suns coaltitude at 5 — 80 — 30 z a the suns coaltitude at 9 — 44 — 13 p a & p b the suns distance from the pole on the 2 h. li. of 5 & 9 66 — 28 5-9 the equinoctial time between the two observations — 60 — 00 b a in proper measure is as found by the first measure working — 54 — 35 z c a perpendicular on a p from z 29 — 24 p z the complement of the latitude that was to be found — 38-30 a short treatise of dialling shewing, the making of all sorts of sun-dials, horizontal, erect, direct, declining, inclining, reclining; vpon any flat or plaine superficies, howsoeuer placed, with ruler and compasse onely, without any arithmeticall calculation. by edvvard wright. arte of dialing wright, edward, 1558?-1615. 1614 approx. 47 kb of xml-encoded text transcribed from 27 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-10 (eebo-tcp phase 1). a15752 stc 26023 estc s111551 99846866 99846866 11861 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a15752) transcribed from: (early english books online ; image set 11861) images scanned from microfilm: (early english books, 1475-1640 ; 946:04) a short treatise of dialling shewing, the making of all sorts of sun-dials, horizontal, erect, direct, declining, inclining, reclining; vpon any flat or plaine superficies, howsoeuer placed, with ruler and compasse onely, without any arithmeticall calculation. by edvvard wright. arte of dialing wright, edward, 1558?-1615. [52] p. : ill. printed by iohn beale for william welby, london : 1614. another issue, with 3 figures numbered 16-18 on a bifolium signed g1,g2, and with new prelims., of: wright, edward. the arte of dialing. running title reads: a treatise of dialling. signatures: a² b-f⁴ g² ² g² . reproduction of the original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one 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proquest page images 2005-01 emma (leeson) huber sampled and proofread 2005-01 emma (leeson) huber text and markup reviewed and edited 2005-04 pfs batch review (qc) and xml conversion a short treatise of dialling : shewing , the making of all sorts of sun-dials , horizontal , erect , direct , declining , jnclining , reclining ; vpon any flat or plaine superficies , how soeuer placed , with ruler and compasse onely , without any arithmaticall calculation . by edvvard wright . london , printed by iohn beale for william weley . 1614. the contents of this booke . chap. 1. the making of the clinatory . chap. 2. the first diuision of dials into horizontall , erect inclining , and reclining . chap. 3. the second diuision of dials into direct & declining . chap. 4. the third diuision of dials , either agreeing with the plaine of the meridian , or disagreeing from the same . chap. 5. to find the eleuation of the meridian line aboue the horizon . chap. 6. the describing of the figure of the dial first on paper , pasteboard . chap. 7. the making of aequinoctial dials . chap. 8. the finding of the substilar line , and stile , in grounds not equinoctial direct , aud polar . chap. 9. the finding of the distance of the stile from the meridian line in dials , that be neither equinoctial nor polar . chap. 10. the finding of the substilar line & the distance of the stile frō it , when the stile maketh a right angle with the merid ▪ line . chap. 11. from which end of the meridian line the eleuation of the stile is to be counted . chap. 12. the finding of the substilar line & stile in dials that be not polar nor equinoctial the stile making oblique angles with the meridian line . chap. 13. the drawing of the line of contingence , and of the equinoctial circle , and how it must be diuided . chap. 14. the drawing of the houre lines in all dials that be not aequinoctiall . chap. 15. what number must be set to the houre lines . chap. 16. what houre lines are to be expressed in all sorts of dials . chap. 17. how to translate the dial drawne on paper , or paste-board vnto the dial ground . line . errat . correct . b2 b 28 neither nether b3 b 2 cliniatorie clinatory b3 b 2 liue line b3 b 10 made make b4 a 12 alother all other b4 b 14 horozontal horizontal c1 b 21 l●aueth leaneth c3 a 25 ground grounds c3 a 29 drawe drewe d1 a 25 neather nether d3 a 20 contingency contingence d4 b 2 gen●y gence e1 a 8 u●ust must e1 b 23 night from thence : night : from thence e2 a 17 zersection tersection e2 b 4 continue conteine e3 a 8 as the in the e3 a 9 in the as the e3 a 15 nto into e3 b 12 on the one on the e3 b 14 bastbord pastborde . it is also to be remembred that there should haue been numbers set to euery one of the figures , or pictures of the dials , adioined to the end of this treatise , beginning at e4 b , as there is to the three last , viz. the 16. 17. and 18. figure . as in the 7. chap. where i shewe the making of aequinoctial dials sect 4. when i referre you to the first and second figure , i meane the figues or pictures of the aequinoctial dials in e4 b. to the first of these therfore there should be set [ 1. figure , to the next [ 2. figure , and to the first of the polar dials in f1 b , there should be set [ 3. figure , and to the next in f2 a [ 4 figure &c. also in the 18. figure the line f. e. should haue beene continued vpwards so much longer , and it is between r. and e , and at the end thereof being so continued the letter ▪ should be set . a treatise of dialling cap. i. the making of the clinatory . 1 dials are diuersly made according as they are diuersly placed . therefore their situation must first be knowne : which may be done by an instrument not vnfitly called a clinatory . 2 let this instrument be made iust foure square , and let the thicknesse bee about halfe a quarter of the breadth of it , vpon one side therof describe a quadrant , whose two semidiameters or sides must be paralel to the side of the quadrate . 3 the quadrant must bee diuided into 90. degrees , with figures set to euery fifth or tenth degree ( as the manner is ) both forward and backward , and without the peripherie thereof , a groofe or furrowe must be made so deepe that a plummet hanging by a thrid from the center of the quadrant may fall into it , in such sort that the thrid may come close to the degrees of the quadiant . 4 close within the limb of this quadrant make a great round hole or box , for the placing of a magneticall needle within the same , whose true meridian line must bee perpendicular to one of the sides of the quadrant , which shall be called the north side , and the other side of the quadrant shall bee called the east-side , to the which the westside of the clinatory is opposite , as the south side of the clinatorie is opposite to the northside of the quadrant : and the magneticall meridian , must bee drawne in the bottome of the box according to the variation of the place where you are . the figure of the clinatory . cap. ii. the first diuision of dials into horizontall , erect inclining , and reclining . every flat whereupon a diall is to bee made ( which is also called the diall ground ) either lieth leuel with the horizon , or els is eleuated the one side higher then the other . 2 the first kinde is thus known : take the clinatorie and hold it so that the plummet fall vpon on of the semidiameters , or sides of the quadrant : then if the nether side of the clinatory , which way so euer you turne the instrument , will touch the flat , it lieth leuel with the horizon , and dials made vpon such flats , are called horizontal dials . 3 those flats which are eleuated the one side higher then the other , stand either vpright ( which are called erect ) or else , they stand leaning : if they leane to you ward , when you stand right against them , they are called inclining : otherwise reclining , if they leane from you ward . 4 all these flats are thus knowne : holding the clinatory as before ; if either the right , or the left side thereof ( whereto the plumbline is aequidistant ) will touch and lie close to the flat , it is erect : but if either of the neither corners onely touch it , it is reclining : if either of the vpper corners onely touch it , it is inclining . 5 and how much the reclination or inclination is , you shall know after this manner . 6 set one of the sides of the clinatory to the flat , in such sort , that the plumbline hanging at liberty , may fall vpon the circumference of the quadrant : for then the arke of the quadrant , betwixt the plumbline and that side of the quadrant that is parallell , or aequidistant to the slatte , is the reclination thereof , if the center of the quadrant be from the flat , or else the inclination , if it bee towards the same . cap. iii. the second diuision of dials into direct and declining . all flats are either direct , or declining . 2 all flats lying leuel with the horizon are direct . 3 but if the flat lie not leuel with the horizon : you shall thus know whether it be direct or declining . first , draw therein a line parallel to the horizon , after this manner : holde the clinatory to the flat in such sort , that the plumbline may fall vpon one of the sides of the quadrant ; then draw a line by the nether side of the clinatory in recliners ; or by the vpper side in incliners , or by either of those sides in erect flats , for that line shall be parallel or aequidistant to the horizon , and may be called the horizontall line . set the north side of the cliniatorie to this liue , if the north end of the needle looke towards the flat : then if the magnetical meridian be right vnder the needle , it is a direct flat : but if it differ from it , it is declining , and that so much as that difference is , and that way which the north end of the needle declineth from the northend of the meridian line in the clinatory . 4 if the southend of the needle looke towards the flat , made your account contrary wise . chap. iv. the third diuision of dials , either agreeing with the plaine of the meridian , or disagreeing from the same . all flats doe either agree with the plaine of the meridian circle ( which may therefore bee called meridian flats ) or else they disagree from the same . 2 they are knowne thus : if the flat bee erect and declining 90. degrees , it is a meridian flat , otherwise it is no meridian flat ; and then you must first draw therin the meridian line , after this maner . 3 if the flat be horizontal , take the clinatory and lay it flat downe thereupon ; and turning it about till the needle hang precisely ouer the magneticall meridian , by that side thereof that is parallel to the true meridian line of the clinatory , drawe a right line , for that shal be the meridian line desired . 4 in erect flats the meridian line is perpendicular , and therefore laying the clinatory close to such a flat in such sort that the plumbline hang precisely on either side of the quadrant , a line drawne by the side of the clinatory , parallel to that side of the quadrant , shall be the meridian line . 5 in direct flats , a line perpendicular to the line aequidistant from the horizon , is the meridian line we seeke for . 6 in flats reclining or inclining , declining also 90. degrees ( which are commonly called , east , or west reclining or inclining ) the meridian line is parallel to the horizon . 7 for alother inclining or reclining , & withall declining flats , drawe a line vpon some pastbord or paper which shall bee called the horizontal meridian ab . wherein settingone foot of your cōpasses , with the other draw an arch of a circle ; & therin reckon the complement of the declination fc . drawing a right line bc. by the end thereof out of the center b. this right line you shall crosse squire-wise with another as ac . which may be called the base of inclination or reclination , and must also meet with the horizontall meridian at a. and setting one foot of your compasses in the crossing at c. with the other foote draw an arke , counting therein the complement of the reclination or inclination ag. drawing a right line by the end therof , out of the center of the foresaid ark cgd . & from a erect ad. perpendicular to ac . which may meet with cgd . the line of reclination or inclination at d. also from a. draw the line af. perpendicular to the horozontal meridian , ab . in the point a. and equal to the former perpendicular ad. and from the end therof draw a line to ( the center of the arke of declination ) b. then continuing foorth ac . to n. ( that cn . be equall to cd ) from n. you shall draw a line to b. which ( if you haue wrought truely ) must be equall to be. now the angle contained betweene the lines nb. and bc. sheweth how much the meridian line in your diall ground should be distant from the line which you drew aequidistant to the horizon heere represented by bc. in this line therefore ( in the dial ground ) set one foote of your compasses , and extending the other that way which the diall declineth , drawe an arke of a circle , vpwards in recliners , but downewards in incliners : and therein count the said angle from the line parallel to the horizon , and drawe by the end thereof a line , which shall bee the true meridian in the dial ground . 8 from a draw ah perpendicular to eb . make bi . equal to bh . from i. let ik be drawn perpendicular to bn . make cl. equal to ck and drawe a line from l to a. of these three lines ah . ik . and la. make the triangle ahm. for then the angle ahm. is the angle which the dial ground maketh with the plain of the meridian . chap. v. to finde the eleuation of the meridian line aboue the horizon . the meridian line is either parallel to the horizon , or else eleuated the one end higher then the other . 2 if the flat bee either horizontal , or east , or west , and inclining , or reclining , the meridian is parallel to the horizon . 3 in all other flats that disagree from the plaine of the meridian circle , the meridian line is eleuated the one end higher then the other . 4 this eleuation is either vpright , as in all erect dials not declining 90. degrees , or else leaning , as in all inclining , and reclining flats not declining 90 degrees , which if they be direct , is equall to the complement of reclination , or inclination . 5 but if they decline , then the angle abe . in the former figure , is the eleuation of the meridian line . 6 if the meridian line bee not erect , it leaueth either northwards , when the eleuated end thereof looketh towards the north , or else southwards when the eleuated end looketh towards the south . 7 all flats are either polar ( which being continued ▪ would goe by the poles of the world ) as all leaning flats , wherein the eleuation of the meridian line is northwards , and equall to the poles eleuation ▪ and all erect decliners 90. degrees . otherwise they are no polar flats . chap. vi. the describing of the figure of the diall first on paper or pasteboard . now it shall bee best to take a sheet of paper , or rather a pastebord , that you may therein describe the figure of your dial , before you draw the diall it selfe vpon his ground : that is , vpon the truncke , stone , wall , &c. 2 this paper , or pasteboord therefore , you shall place , or vnderstand to be placed so as your dial ground is or must be placed , and therein write the names of the parts of the world , as they lie in respect of your dial ground , as east , west , north , south , zenith , nadir , vpper part , nether part , &c. which you may do by helpe of the magnetical needle : for the north end thereof ( hanging at liberty ) sheweth the north , whereto the south is diametrally opposite ; and your face being turned towards the north , your right hand sheweth the east , your left hand the west , the zenith , or verticall point is aboue your head , the nadir vnder your feete . note also , which end of the meridian line must be higher , and which lower ; if the meridian be not parallel to the horizon . chap. vii . the making of aequinoctiall dials . all diall grounds are aequinoctiall , or not aequinoctiall . 2 an aequinoctiall ground is that which agreeth euen with the plaine of the aequinoctiall circle : which is thus knowne . if the diall ground be direct , and the meridian line eleuated southwards , equally to the complement of the poles eleuation , it is an equinoctiall diall ground , otherwise not . 3 in an equinoctial dial you shall describe the houre lines after this manner . 4 set one foot of your compasses in the meridian line ab . and with the other , drawe a circle dbc . and deuide it into 24. equall parts , as d. e. f. g. &c. beginning at b. the crossing therof , with the meridian line ; for then right lines , as ad. ae . af. ag. &c. in the 1. and 2. figure drawne out of the center , by those diuisions shall bee the houre lines . 5 the stile must stand vpright out of the center of the diall . 6 of equinoctiall dials there be two sorts , the vpper and the nether . 7 the vpper equinoctial diall looketh vpwards to the eleuated pole of the world : and it sheweth the houre of the day , onely in the spring and summer time , as in the first figure . 8 the nether , or lower equinoctial dial , is that which looketh downewards to that pole of the world which is beneath the horizons and sheweth the houres onely in autumne , and winter , as in the second figure . chap. viii . the finding of the substilar line , and stile , in grounds not equinoctiall direct , and polar . in all dial grounds that are not equinoctiall , the substilar line , and the distance of the stile from the substilar must bee found . 2 the substilar line is that right ouer which the stile must be set . 3 the distance of the stile from the substilar , is the angle , or space contained betweene the stile , and the substilar line . 4 the finding out of these is diuers , in diuers kinds , and therefore must bee specially shewed in each kinde . 5 in direct dial grounds not equinoctiall , and polars not meridian , the substilar line is the same with the meridian line , or else parallel thereto , in declining polars . 6 in polar ground ▪ agreeing with the plaine of the meridian , the substilar line may thus be found . 7 set one foot of the compasses in the south-end of the line that you drawe equidistant from the horizon and extending the other foot towards the north end of the same line , draw an arke of a circle : therein reckon the eleuation of the pole beginning at the foresaid line : for a right line drawne thereby out of the center , shall be the substilar line ab . figure . 3. 8 in al polar grounds draw a parallel cd . ( figu . 3. 4. 5. 6. 7. 8. ) to the substilar line at a conuenient distance from the same ; for that shall be the line representing the stile . chap. ix . the finding of the distance of the stile from the meridian line in dials that be neither equinoctial nor polar . in all dial grounds that be not aequinoctiall nor polar , before the substilar line , and distance of the stile from it can be found , first the distance of the stile from the meridian line must be found after this maner . 2 if the meridian line be parallel to the horizon , as bc. the distance of the stile from the meridian line , is equal to the height of the pole , as br . 3 but if the eleuation of the meridian be either vpright , as ag. or leaning towards the north , and withall greater then the poles eleuation , as ah . the height of the pole br . taken out of the height of the meridian line bh . or bg . shal leaue the distance of the stile from the meridian line rh . or rg . 4 if the eleuation of the meridian line be northwards , and lesse then the height of the pole , as bi . take the eleuation of the meridian line bi . out of the height of the pole br . and there shall remaine the distance of the stile from the meridian line ri. 5 if the eleuation of the meridian line be southwards , and either greater , or equal to the complement of the poles eleuation , as af. and ae . then the complement of the meridian lines eleuation , fg. or eg . added to the complement of the poles eleuation gr. shall make the distance of the stile from the meridian line . 6 if the eleuation of the meridian line be southward and lesse then the complement of the poles eleuation as cd . the eleuation of the meridian line cd and the height of the pole c● . put together shall make the distance of the stile from the meridian line . chap. x. the finding of the substilar line and the distance of the stile from it , when the stile maketh a right angle with the meridian line . secondly , in a ground not equinoctial nor polar we must consider whether the stile make a right angle , or an oblique angle with the meridian line . 2 the stile shall make a right angle with the meridian line , if the eleuation of the meridian line be southwards and equall to the complement of the poles eleuation , as in the 9. 10. 11. and 12. figure herein a right line drawne squirewise ouerthwart the meridian line , towards that part of the world , which is opposite to that whereto the dial ground declineth , shall be the substilar line , as ba . in the 9. 10. 11. and 12. figu . and the distance of the stile from the substilar line shal bee equall to the angle which the dial ground maketh with the plaine of the meridian circle as the angle bad . fig. 9. 10. 11. 12. which angle is found by the third chap. chap. xi . from which end of the meridian line , the eleuation of the stile is to be counted . if the stile make an oblique angle with the meridian line , we must first finde out from whether end of the meridian line , the eleuation of the stile must be reckoned , thus : 2 if the meridian line be parallel to the horizon as in the 13. figure , the eleuation of the stile shal be reckoned from the north end of the meridian line in reclining , and horizontal flats looking vpwards , as br . from b in the former figure , but contrariwise in incliners as pc . from c. in the same figure . 3 if the meridian line be eleuated the one end higher then the other from the horizon , and the dial ground looke towards the south , the eleuation of the meridian being also northwards , and lesse then the eleuation of the pole : the eleuation of the stile shal be counted from the vpper end of the meridian line : as ir. from i. 4 but if the eleuation of the meridian be greater then the eleuation of the pole , or vpright , or southwards and greater then the complement of the poles eleuation ; the eleuation of the stile shall bee counted from the neather end of the meridian line , as pm , pn , po , from mno . 5 if the eleuation of the meridian line be southwards and lesse then the complement of the poles eleuation , the eleuation of the stile shal be counted from the vpper ende of the meridian line as dp . from d. 6 if the dial ground looke toward the north , the eleuation of the stile from the meridian line shal be reckoned contrariwise in euery kinde . chap. xii . the finding of the substilar line and stile in dials that be not polar nor equinoctiall , the stile making oblique angles with the meridian line . hauing thus found out from whether end of the meridian line the eleuation of the stile is to be reckoned , set one foot of your compasses in the meridian line as in a. and stretching foorth the other foot towards that end of the meridian line , from which the eleuation of the stile is to bee reckned as towards l. draw an arch of a circle mdln. and ( beginning at the merîdian line ) reckon and marke therein the eleuation of the stile from the meridian line , ld . figure 13. 14. 15. in the rest lo . either eastwards or westwards in direct dials , as in the 13. 14. 15. fig. but in decliners towards that part of the world which is opposite to the part whereunto the dial declineth , as in the 16. 17. 18. fig. 2 then in direct dials , a right line acd . fig. 13. 14. 15. drawne out of the center of the said arke by the marke of the stiles eleuation from the meridian line shall be the line representing the stile , and therefore the distance of the stile from the substilar line shall be the distance of the stile , from the meridian line . 3 but in decliners you shall thus finde the substilar line : from o the point of the stiles eleuation from the meridian line in the foresaid arke drawe op . a perpendicular to the meridian line al. and taking the length of this perpendicular with your compasses , leaue one foote in p. the concurse therof with the meridian line , and with the other describe a quadrant of a circle qro. beginning from the meridian line , and so proceeding vnto o the other end of the perpendicular line : and in that quadrant beginning at the meridian alq. reckon and marke qr . the complement of the angle conteined betweene the plaines of the diall ground and of the meridian circle , and take with your compasses rs. the distance of that marke from the meridian line , and setting one foote of the compasses in p. the meeting of that perpendicular with the meridian line , with the other make a prick t , in the same perpendicular line : for then ab . a right line drawn by this prick t. out of the center of the foresaid arke mdln. shall bee the substilar line . 4 then take with your compasses tr. the distance of the foresaid marke in the quadrant , qro. and this pricke , and leauing one foote of your compasses in the same pricke t. with the other make another pricke v. in the arke you first described ; for then a right line av. drawne thereby out of the arch you first described shall bee the stilar line , or line representing the stile . 5 in dials not polar nor aequinoctiall , if the distance of the stile from the substilar line be but smal as in the fig. 10. 12. 17. it may bee increased by drawing a paralel cd . to the stile already found , which for distinctions sake may bee called , the stile augmented . chap. xiii . the drawing of the line of contingence , and of the equinoctiall circle , and how it must be diuided . now in all dials that be not aequinoctiall , draw a right line , ehf . so long as you can , making right angles with the substilar line , which is called the line of contingence , or touchline . 2 then describe the equinoctiall circle ghi . after this manner : take with your compasses the shortest distance betweene h. the intersection of the line of contingence with the substilar line , and the stilar line , and leauing one foot in that intersection , with the other make a pricke b. in the substilar line , whereupon describe a circle ghi . which shall be called the equinoctiall circle . 3 if the distance of the stile from the substilar be augmented , you must draw two touch lines and two aequinoctial circles : as in 10. 12. 17. figures . 4 the halfe of the aequinoctiall circle next the line of contingence must be deuided into 12. equal parts , beginning at h , the intersection thereof with the substilar line in all direct dials , and erect or meridian polars which are commonly called east or west dials erect , as in the 3. 4. 5. 6. 13. 14. 15. figures . 5 in polars not meridian nor direct , let hk , in fig. 7. & 8. ( the complement of the angle which the dial ground maketh with the plaine of the meridian ) be numbred and marked in the aequinoctial circle , beginning at the substilar line , and proceeding that way which the diall ground declineth as from h. to k. for at that marke k you must begin to diuide . 6 in decliners not polars , if the stile make a right angle with the meridian line , as in the 9. 10. 11. 12. figu . a paralel to the line of contingence , drawne by the center of the aequinoctiall , shall shew the beginning of the diuision , as bk in figu . 9. 10. 11. 12. 7 but if the stile make an oblique angle with the meridian line , and the line of contingencye , cut the meridian line , as in the 16. figu . your ruler laid to that cutting at x and the center of the equinoctial b. shal shew in the peripherie thereof , the beginning of the diuision k if the distance of the stile from the substilar be not augmented . 8 but if it be augmented ( as in the 17. figure ) the shortest distance hx betweene h the intersection of the touch line , with the substilar line , and the stile not augmented av must bee taken with the compasses , and resting one foot in that intersection h , with the other make a pricke y in the substilar line , towards b the center of the equinoctiall ; by which pricke y & z the mutuall intersectiō of the next touch line with the meridian line , let a right line yz be drawne , for bk . and bk . paralels to it drawne out of the cenrers of both the equinoctials , towards the meridian line , at their crossings with the equinoctials k & k shall shew the beginnings of their diuisions . 9 but if the touch line cut not the meridian line as in the figure 18. let a paralel thereto xy bee drawne , which may cut the meridian line in y and take with the compasses the shortest distance za betwixt the intersection thereof with the substilar line and the stile not augmented ; and leauing one foote in that intersection z , with the other make a pricke b in the substilar line towards the center of the equinoctiall ; from this pricke drawe a right line by from b to y the intersection of the said paralel with the meridian line ; for bka paralel to this line drawne out of the center of the equinoctiall b. shall shew the beginning of the diuision k. chap. xiiii . the drawing of the houre lines in all dials that bee not equinoctiall . hauing thus , deuided the equinoctial circle , lay your ruler to the center thereof b. and to euery one of those prickes 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 by which it is deuided , and make marks 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 &c. in euery place where it crosseth the line of contingence for then 2 in all polar dials paralels to the substilar line , drawne by those markes , shall bee the houre lines , as in the 3. 4. 6. 7. 8. figu . 3 in dials not polar , in which the height of the stile is not augmented , right lines drawne out of the center of the dial by those markes shal be the houre lines as in the 9. 11. 13. 14. 15. 16. 18. figu . and if any of the diuisions of the aequinoctiall circle doe fall in to the substilar line , a paralel to the line of contingence drawne by the center of the said diall , shall shew two opposit houres , distant by the space of six houres from the substilar line ; as for example in direct dials , six in the forenoone and six in the afternoone , as in the 13. 14. 15. figure . also if the ruler laid to b. the center of the equinoctial circle , and some diuision thereof , as v in the 16. and 18. figu . cannot crosse the line of contingence , and yet draweth neerer to it , : draw by. a right line from the center of the equinoctial by that diuision , and draw af a paralel to that line , which may crosse the substilar and line of contingency in f. then let ha. the other part of the substilar that is betweene the line of contingence and the center of the dial a. be cut in such sort that the segments , of the substilar line concurring at the line of contingence ah and hb . may keepe the same proportion which the greater segments bh . and ha. haue , which are contained betweene the center of the dial and line of contingence , and betwixt the center of the equinoctiall and the line of contingence . and let aright line bf . bee drawne by that section b and the section of the line of contingence f. for ai a paralel to this right line drawne out of the center of the diall shall be the houre line that wee seeke for . 4 in those dials wherein the distance of the stile from the substilar is augmented , right lines drawne by those markes in both lines of contingence which are proportionately distant from the substilar line shall be the houre lines . chap. xv. what number must be set to the houre lines . in meridian dials , the substilar line is the line of the sixth houre : but for the rest , we must consider whether it be an oriental or an occidental dial . 2 an oriental dial looketh to the east , and the forenoone houres onely must bee set in this dial , and therefore the substilar line sheweth six of the clocke in the morning ; from which towards the south are the morning houres before sixe , viz. 5. 4. 3. &c. but towards the north after six , 7. 8. 9. 10. 11. as in the 3 figure . 3 an occidental dial looketh directly westwards : and onely the houres after noone can bee set into this dial . therfore the substilar line sheweth the sixt houre after noone : from which toward the north are the houres before six in this order . 5. 4. 3. 2. 1. but towards the south after six thus . 7. 8. 9. &c. as in the 4. figure . 4 in dials not meridian , if a ruler laid to the center of the aequinoctial and the beginning of the diuision thereof doe crosse the touchline ; then the houre line drawne by that crossing is the line of twelue a clocke . but if it cannot crosse the touch line , imagine notwithstanding , that crossing and the twelue a clocke line , drawne thereby without the bounds of your dial , whereabouts you thinke it would bee , if the ruler and touch line were continued foorth long inough . 5 then in al dials not meridian , imagine the stile to be fastned in his place , in aequinoctial dials perpendicularly erected out of the center . in dials that be not equinoctial , conceiue it to be placed exactly ouer the substilar line , so much raised from the same as the stilar line in your paper or pastbord , is distant from the substilar line . 6 after this , place your paper or pastebord ( wheron the figure of your dial is described ) in the same site or position that the dial ground is , or must be placed ▪ so that the quarters of the world written thereupon , may answer in like position to the quarters of the world as they lie in respect of your dial ground : for then if the 12. a clocke line be towards the north , from the stile it is the line of the 12. houre of the day . from hence therefore towards the west are the forenoone houres , 11. 10. 9. 8. 7. &c. and toward the east , the afternoone houres , 1. 2. 3. 4. 5. 6. &c. 7 but if the 12. a clocke line bee southward from the stile , it is the line of the twelfth houre in the night from thence : therefore on both sides are the night houres : toward the west , after midnight , 1. 2. 3. 4. &c. towards the east before midnight , 11. 10. 9. 8. &c. chap. xvi . what houre lines are to be expressed in all sorts of dials . in al dials , those houre lines onely are to be expressed , vpon which the shadow of the stile shal fall . therefore the houres of the day onely are to bee expressed . 2 in dials not polar , wherein the height of the stile is not augmented , if the stile point vpwards , and the eleuation thereof from the substilar line , bee not lesse then the complement of the sunnes greatest declination , all the houre lines seruing for the longest day , are to be expressed therein . 3 but if the cleuation of the stile from the substilar be lesse then the complement of the sunnes greatest declination , draw a right line out of the intersection of the line of contingence , and substilar perpendicularly ouerthwart the stilar line : and setting one foot of your compasses in the center of the dial , and extending the other towards the other end of the stilar line , draw an arke there from equal to the complement of the sunnes greatest declination : and thereby draw a line out of the center of the dial , and setting one foot of your compasses inthe intersection of this line with the foresaid perpendicular , extend the other foot to the stilar line : then keeping this distance , set one foote of your compasses in the center of the aequinoctial circle and with the other crosse the line of contingence on both sides the substilar : now if you lay your ruler to these crosses and the center of the dial : right lines drawne thereby beyond the center of the dial shal continue betweene them the space wherein no houre lines are to be expressed . 4 this rule holdeth also in meridional dials inclining , when the eleuation of the stile is counted from the vpper end of the meridian line , and the eleuation of the stile from the substilar is lesse then the complement of the sunnes greatest declination . 5 if the stile point downwards , no houre lines are to be expressed aboue a line parallel to the horizon drawne by the center of the dial. 6 and if the crosse in the line of contingence ( made as before was shewed ) be aboue the line aequidistant to the horizon , drawne by the center of the dial ; no houre lines are to be expressed aboue a right line drawne from the crosse and continued beyond the center of the dial. 7 if any part of the dial whereupon the shadow of the stile may fal , bee void of houre lines : let the houre lines before described bee continued foorth into that part of the dial , as in the 13 and 15 figure . chap. xvii . how to translate the diall drawne on paper or paste-board vnto the dial ground . the figure of your dial being thus described , you shall translate the same into the dial ground , after this manner . 2 place the paper or pastebord whereas the figure of your dial is described in such sort , in the dial ground is placed , so as the quarters of the world written on the paper or pastebord may answer in like position to the quarters of the world as they lye in respect of the dial ground . 3 then as the houre lines and substilar line are described in your pastebord , so in like manner , and in like position , let them be inscribed nto your dial ground that so little part of the ground as may be , be left voide of houre lines seruing for vse , and that the spaces on both sides from the substilar line drawne on the dial ground bee proportionable to the number of houre lines that are to bee expressed in the dial. 4 in polar dials draw a right line squire-wise ouerthwart the substilar in the dial ground ; then take with your compasses the distances of the houre lines from the substilar in the pastebord , and set them into that line drawne squire-wise in the dial ground , setting alwaies on foot in the intersection thereof with the substilar line , and with the other foote making pricks in the said line drawne squire-wise : and let paralels to the substiar line be drawn by those prickes , for they shall bee the houre lines we seeke for , set into the dial ground . 5 the stile must be paralel to the substilar line , and must be placed directly ouer it , so much distant from the same , as the stilar line is distant from the substilar in the figure of your dial drawne on the pastbord or paper . 6 in dials that be not polars , wherein the eleuation of the stile from the substilar is not augmented , describe two peripheries of equal bignesse on the dial ground , the center thereof being placed in the meridian line , the other vpon the center of the dial in the bastbord : then in this peripherie take the distances of the substilar and the houre lines from the meridian with your compasses , out of the figure of your dial in the paper or pastbord , & set those distāces likewise into the dial grounds , and by them draw the houre lines and the substilar from the center of the dial . 7 the stile must bee fastned in the center , and must hang directly ouer the substilar , eleuated so much from the same , as the stilar line in the figure of your dial is distant from the substilar . 8 but in dials that be not polars , wherein the eleuation of the stile from the substilar is augmented , let the substilar line bee described in the dial ground so much distant from the meridian , which you first described therein , as the substilar is distāt from the meridian in the figure of your dial . and let two lines of contingence be drawne squire-wise ouerth wart that substilar in the dial ground , so much distant each from other , as the lines of contingence in the paper are . and let the distances of the houre lines from the substilar line bee taken in both lines of contingence in the figure of the dial , and be set in like manner in to the lines of contingence , answering to them in the diall ground , setting one foot of your compasses alwaies in the substilar line , which is in the diall ground , and with the other making markes in the lines of contingence drawne therein : for then right lines drawne by those markes , differing alike from the substilar line , shall bee the houre lines . the stile must hang perpendicularly ouer the substilar line , so much distant from the same , and from the sections thereof with the lines of contingence , as the stile augmented in the figure of your diall is distant from the substilar . equinoctiall dials . north direct reclining 51. degrees , ●0 . minutes , or the vpper equinoctiall diall . south direct , inclining 51. degrees , 30. minutes . the manner of finding the substilar line in meridian polar dials . this example serueth for the oriental dial . meridian polar dials . a south direct dial reclining 38 degrees 30. minutes , or a south direct polar dial . a north direct dial inclining 38 degrees 30. minutes , or a north direct polar dial . south declining eastward 27. degrees , reclining 34. degrees 40. minutes . north declining westward 36 degrees inclining 32. degrees 15 minutes . north declining eastward 43. degrees reclining 42. degrees 20. minutes wherein the stile and meridian line make right angles . north declining 84. degrees westward , reclining 7 degrees 20 minutes , the stile pendicular to the meridian line . south declining eastward 31 degrees , inclining 48 degrees . ●0 min. the stile perpendicular to the meridian . south declining westward 86 ▪ degrees 40. min. inclining 4. degrees stile perpendicular to the meridian . in all the figures following , the stile maketh a sharpe angle with the meridian : a direct dial lying leuel with the horizon , commonly called an horizontal dial . a south dial erect direct . a north dial erect , direct . a south dial erect declining eastward 30. degr . a south erect dial declining westward 80. degr . north declining westward 41. de . 40. min. reclin . 4● . de . 30. min. notes, typically marginal, from the original text notes for div a15752-e450 also in the diagramme placed there , and in the leafe following , let k & ● be chāged each into others place . notes for div a15752-e1580 the dial ground . horizontal flats : and how they are known . eleuated flats ; erect , inclining reclining : and how they are to be known to know how much the reclination or inclination is . notes for div a15752-e1780 how to drawe the meridian line , in horizontall flats . in erect flats . in direct flats . in reclining or inclining flats , declining 90. degrees . in incl●ning or reclining flats declining lesse then 90. degrees . to finde the angle which the dial ground ( or flat ) maketh with the plaine of the meridian . notes for div a15752-e1970 how to finde the eleuation of the meridian line . the eleuation of the meridian line in erect dials . in reclining or inclining direct flats ; in reclining or inclining declining flats . the eleuation of the meridian whether north or south . flats polar . not polar . notes for div a15752-e2170 how to make equinoctial dials . placing of the stile . vpper equinoctial dial . neather equinoctial dial . notes for div a15752-e2320 substilar line . distance of the stile from the substilar . the finding of the substilar line . in direct flats not aquinoctial in polars not meridian . in meridian polars . the stilar line in all polar dials . notes for div a15752-e2730 the stile augmented . the art of dialling by a new, easie, and most speedy way. shewing, how to describe the houre-lines upon all sorts of plaines, howsoever, or in what latitude soever scituated: as also, to find the suns azimuth, whereby the sight of any plaine is examined. performed by a quadrant, fitted with lines necessary to the purpose. invented and published by samuel foster, professor of astronomie in gresham colledge. foster, samuel, d. 1652. 1638 approx. 60 kb of xml-encoded text transcribed from 30 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-10 (eebo-tcp phase 1). a01089 stc 11201 estc s102472 99838255 99838255 2628 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a01089) transcribed from: (early english books online ; image set 2628) images scanned from microfilm: (early english books, 1475-1640 ; 1097:07) the art of dialling by a new, easie, and most speedy way. shewing, how to describe the houre-lines upon all sorts of plaines, howsoever, or in what latitude soever scituated: as also, to find the suns azimuth, whereby the sight of any plaine is examined. performed by a quadrant, fitted with lines necessary to the purpose. invented and published by samuel foster, professor of astronomie in gresham colledge. foster, samuel, d. 1652. [6], 39, [1] p., [2] folded plates : ill. printed by iohn dawson for francis eglesfield, and are to be sold at the signe of the marigold in pauls church-yard, london : 1638. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between 1473 and 1700 available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the 25,363 texts created during phase 1 of the project have been released into the public domain as of 1 january 2015. anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng dialing -early works to 1800. quadrant -early works to 1800. 2003-06 tcp assigned for keying and markup 2003-06 aptara keyed and coded from proquest page images 2005-02 andrew kuster sampled and proofread 2005-02 andrew kuster text and markup reviewed and edited 2005-04 pfs batch review (qc) and xml conversion the art of dialling ; by a new , easie , and most speedy way . shewing , how to describe the houre-lines upon all sorts of plaines , howsoever , or in what latitude soever scituated : as also , to find the suns azimuth , whereby the sight of any plaine is examined . performed by a quadrant , fitted with lines necessary to the purpose . invented and published by samvel foster , professor of astronomie in grosham colledge . london , printed by iohn dawson for francis eglesfield , and are to be sold at the signe of the marigold in pauls church-yard . 1638. to the reader . reader , here is presented to thy view a short and plaine treatise ; it was written for mine owne use , it may become thine if thou like it ; the subject indeed is old ; but the manner of the worke is all new . if any be delighted with recreation of this nature , and yet have not much time to spend , they are here fitted , the instrument will dispatch presently . if they feare to lose themselves in a wildernesse of lynes , or to out-runne the limits of a plaine , by infinite excursions ( two inconveniences unto which the common wayes are subject ) they are here acquitted of both , having nothing to draw but the diall it selfe , contracted within a limited equicrurall triangle . if want of skill in the mathematicks should deterre any from this subject , let them know that here is little or none at all required , but what the most ignorant may attaine . if others shall thinke the canons more exact ; so doe i , but not so easie to bee understood , not so ready for use , not so speedy in performance , nor so well fitting all sorts of men : and withall an instrument in part must bee used , this will doe all , and is accurate enough . if it must needs be disliked , let a better be shewed and i will dislike it too ; it is new , plaine , briefe , exact , of quicke dispatch . accept it , and use it , till i present thee with some other thing , which will bee shortly . imprimatur . decemb. 1. 1637. sa . baker . the description of the qvadrant , and the manner how the lines are inscribed and divided . chap. i. 1. the description of the fore-side . the limbe is divided into 90 degrees , and subdivided into as many parts as quantity will give leave . the manner of division , and distinction of the subdivided parts is such as is usuall in all other quadrants . to describe the other worke in the superficies ; take from the upper edge of the limbe about 3 degrees , and set off that space from the center r to a. then divide ae into seven parts , whereof let eb containe two . or in greater instruments , if ae be 1000. let eb containe 285. make sc equall to eb , and drawe the line bc. from c , draw cd parallel , and of equall length to ab . upon ab and cd , and be also ( as farre as it is capable ) insert the 90 sines , from b towards a and e , and from c towards d , but let them be numbred from a unto b to 90 , and so to e 113 degrees 30 minutes , from d to c unto 90 degrees . againe : draw es cutting cd at f ; so shall bcfeb containe a parallelogram , whose opposite sides , being parallel , are divided alike , and in this manner . be and cf as whole sines , doe containe the 90 sines , or as many of them as can distinctly bee put in : and from the divisions are drawne parallel lines , having every tenth , or fifth , distinguished from the rest . these serve for the 12 signes and their degrees , and therefore you see upon every 30th degree , the characters of the 12 signes inserted , in such manner as the figure sheweth . and these lines may bee called , the parallels of the suns place . in like manner , the lines bc , ef , being first bisected at x and z , shall make 4 lines of equall length . these 4 lines xb , xc , ze , and zf , are each of them divided as a scale of sines , beginning at x and z , and from each others like parts are parallel lines protracted , having every tenth and fifth distinguished from the rest . they are numbred ; upon bc , from b to x 90 , to c 180 ; upon fe , from f to z 90 , to e 180. these lines are called , the lines of the sunnes azimuth . this done ; upon the center r describe the two quadrants vt , and bc , let their distance vc bee one sixth part of rc , or more if you will. divide them each into 6 equall parts , at e , o , y , n , s ; and a , i , u , m , r , drawing slope-lines from each others parts , as va , ei , on , ym , nr , sb : and these lines so drawne are to bee accounted as houres . then dividing each space into two equall parts , draw other slope-lines standing for halfe houres , which may be distinguished from the other , as they are in the figure . then from the points v and t draw the right line vt . lastly , having a decimall scale equall to tr , you must divide the same tr into such parts as this table here following alloweth , the numbers beginning at t , and rising upto 90 at r. vpon your instrument ( for memory and directions sake ) neere to the line ab , write , the summe of the latitude and sunnes altitude in summer ; the difference in winter . over vt , write , the line of houres . neere to cd write , the summe of the latitude and sunnes altitude in winter ; the difference in summer . by tr , write , the line of latitudes for the delineation of dialls . a table to divide the line of latitudes . 90 10000 62 9360 46 8259 30 6325 14 3325 85 9982 61 9311 45 8165 29 6169 13 3104 80 9924 60 9258 44 8068 28 6010 12 2879 78 9888 59 9203 43 7968 27 5846 11 2650 76 9849 58 9147 42 7865 26 5678 10 2419 75 9825 57 9088 41 7738 25 5505 9 2186 74 9801 56 9026 40 7647 24 5328 8 1949 72 9745 55 8962 39 7532 23 5146 7 1711 70 9685 54 8895 38 7414 22 4961 6 1470 69 9651 53 8825 37 7292 21 4772 5 1228 68 9615 52 8753 36 7166 20 4577 4 984 67 9378 51 8678 35 7036 19 4378 3 739 66 9519 50 86●● 34 6902 18 4176 2 493 65 9496 49 8519 33 6764 17 3969 1 247 64 9454 48 8436 32 6622 16 3758 0 ● 63 9408 47 8348 31 6475 15 3543   s●… 2. the description of the backe-side . upon the backe-side is a circle only described , of as large extent as the quadrant will give leave , noted with abcd , divided into two equall parts by the diameter ac . the semicircle abc is divided into 90 equall parts or degrees , every fifth and tenth being distinguished from the rest by the longer line ; they are numbred by 10 , 20 , 30 , &c. unto 90. the same parts are also projected upon the diameter ac , by a ruler applyed to them from the point d. these are numbred also from a to c by 10 , 20 , &c. unto 90. the other semicircle adc , is first divided into two quadrants at d. and then upon these two quadrants are inscribed 90 such parts as this table insuing doth allow . the inscription is made by helpe of a quadrant of a circle equall to ad or cd , being divided into 45 equall degrees , out of which you may take such parts as the table giveth , and so pricke them downe , as the figure sheweth . every fifth and tenth of these parts is distinguished from the rest by a longer line ; they are numbred from a and c , by 10 , 20 , &c. unto 90 ending in d. a table to divide the upper and nether quadrants of the circle . 1 1.00 14 13.36 27 24.25 40 32.44 53 38.37 66 42.25 2 2.00 15 14.31 28 25.09 41 33.16 54 38.59 67 42.38 3 3.00 16 15.25 29 25.52 42 33.47 55 39.19 68 42.50 4 3.59 17 16 18 30 26.34 43 34.18 56 39.40 69 43.02 5 4.59 18 17.10 31 27.15 44 34.47 57 39.59 70 43.13 6 5.58 19 18.02 32 27.55 45 35.16 58 40.18 72 43.34 7 6.57 20 18.53 33 28.35 46 35.44 59 40.36 74 43.52 8 7.55 21 19.43 34 29.13 47 36.11 60 4054 75 44.00 9 8.53 22 20.32 35 29.50 48 36.37 61 41.10 76 44.08 10 9.51 23 21.21 36 30.27 49 37.03 62 41.27 78 44.22 11 10.48 24 22.08 37 31.02 50 37.27 63 41.43 80 44.34 12 11.45 25 22.55 38 31.37 51 37.51 64 41.57 85 44.53 13 12.41 26 22 40 39 32 . 1● 52 38.15 65 42.11 90 45.00 thus have you both sides decribed . besides all this , there are two sights added , with a threed and plummet like as in other instruments . the threed hath a moovable bead upon it for speciall use . the same threed passeth through the center r. quite behind the quadrant , and is hung upon a pinne at the bottome of the quadrant , noted with w. the reason of the threeds length will be seene when wee come to the uses of the instrument . chap. ii. the use of the quadrant in generall . first upon the fore-side . the limbe serveth especially for observation of all necessary angles . the lines ae , cd , with the parallelogram bcef , are to find out the suns azimuth in any latitude whatsoever . the slope-lines within the arkes vt , cb , by helpe of the threed and bead , doe serve artificially to divide the line of houres tv , into its requisite parts ; which together with tr the line of latitudes , doe serve to protract all plaine dialls howsoever scituated . secondly upon the back-side . note that abc is called the semicircle : ac is called the diameter : ad the vpper quadrant : cd the nether quadrant . the uses of these are to find out the necessary arkes and angles , either for preparation to the dialls description , or serving after for the dialls scituation upon the plaine . in all these uses the threed bearing part , and therefore having asufficient extent of length , that being loosed it may with facility reach over either side of the quadrant . chap. iii. to find the azimuth of the sunne in any latitude whatsoever . before you can make any draught of your diall , you must know the scituation of your plaine , both for declination and inclination . the best way to come to the plaines declination is by helpe of the sunnes azimuth . by having the latitude of the place ; the place of the sunne in the eclipticke , and the altitude of the sunne above the horizon , you may find out the azimuth thereof in this manner . adde the sunnes altitude , and your latitude together , and substract the lesser of them from the greater ; so shall you have the summe of them , and the difference of them . with this summe and difference , come to your quadrant , and according to the time of the yeare ( as the lines will direct you ) count the said summe and difference respectively , and applying the threed unto them , find out the sunnes place in the parallels serving thereto , and where the threed cuts this parallel , observe the azimuth there intersecting , for that is the azimuth from the south , if you number it from the line whereon the summe was numbred . example 1. in the north latitude of 52 gr . 30 min. in the summer-time the sunne entring into 8 , and the altitude being observed 30 gr : 45 min. i adde the latitude 52 gr . 30 min. and the sunnes altitude 30 gr . 45 min so i find the summe of them 83 gr . 15 min. and substracting the lesser of them from the greater , i find the difference of them 21 gr . 45 min. the summe i number in the line ae , and the difference in dc ( because it is in summer ) and to the termes i apply the threed , and where it crosseth the parallel of the beginning of 8 , there i meet with 66 gr . 43 min. which is the azimuth from the south , being reckoned from the line ae whereon the summe was counted . example 2. the latitude and sunnes place being the same if the altitude had beene 9 gr . 15 min. the summe of the latitude and altitude would bee 61 gr . 45 min. the difference 43 gr . 15 min. and so the threed applyed to these termes would have shewed 96 gr . 52 min. for the azimuth from the south . a third example . in the same latitude of 52 gr . 30 min. in the winter-time , the sunne entring the tenth degree of ♏ , and the altitude being 9 gr . 30 min. i would know the azimuth of the sun from the south . i adde the altitude 9 gr . 30 min. to the latitude 52 gr . 30 min. and so find the summe of them 62 gr . 0 min. and substracting the altitude out of the latitude , i find the difference of them 43 gr . 0 min. the summe ( because it is in winter ) i count upon the line dc in the quadrant , and the difference upon ae . so the threed applyed to these tearmes cutteth the tenth of ♏ , at 49 gr . 50 min. which is the azimuth numbred from dc the south . the amplitude . note here by the way , that the threed applyed to the latitude of your place numbred upon both lines ae , dc , will shew you , for any place of the sunne , the due amplitude of his rising or setting , or the azimuth whereon hee riseth or setteth , if you number the same from the middle line noted with xz which here representeth the east and west azimuths . chap. iiii. to find out the declination of a plaine . the declination of a plaine is numbred from the south or north points towards either east or west . and it is the arke of the horizon comprehended betweene the south-north , and a line infinitely extended upon the horizon perpendicular to the horizontall line of the plaine ; which line may be called the axis , and the extremity of it , the pole of the plaines horizontall line . to find out this declination you must make two observations by the sunne : the first is of the distance or angle made betweene the axis of the horizontall line of the plaine , and the azimuth wherein the sunne is at the time of observation . the second is of the suns altitude . both these observations should bee made at one instant , which may bee done by two observers , but if they bee made by one , the lesse distance of time betweene them , will make the worke to agree together the better . 1. for the distance . upon your plaine draw a line parallel to the horizon , to this line apply the side of your quadrant , holding it parallel to the horizon . then holding up a threed and plummer , which must hang at full liberty , so as the shadow of the threed may passe through the center of the quadrant , observe the angle made upon the quadrant by the shadow of the threed , and that side that lyeth perpendicular to the horizontall line , for that angle is the distance required . 2. at the same instant as neere as may be , take the sunnes altitude ; these two being heedfully done , will helpe you to the plaines declination by these rules following . when you have taken the altitude , you may find the sunnes azimuth by the former chapter . then observe , whether the sunne bee betweene the pole of the horizontall line and the south north point or not . if the sunne be betweene them , adde the azimuth and distance together , and the summe of them is the declination of the plaine . if the sunne be not betweene them , subduct the lesser of them from the greater , and the difference shall be the declination of the plaine . ¶ by your observation you may know to what coast a plaine declineth . for if the south north point bee in the midst betweene the sunnes azimuth and the pole of the plaines horizontall line , then doth the plaine decline to the coast contrary to that wherein the ☉ is : if otherwise , the declination is upon the same coast with the sunne . chap. v. to find the inclination of a plaine . the inclination of a plaine is the angle that it maketh with the horizon . when you have described your horizontall line upon a plaine , as in this figure ef , crosse it with a perpendicular gh , for the verticall line . and because the inclinations of the upper and under faces of the plaine , are both of one quantitie in themselves , if therefore you apply the side of the quadrant noted with ab unto the verticall line of the under face , or to the under side of a ruler applyed to the verticall line of the upper face , as is here shewed in this figure ; then shall the degrees of the quadrant give you cad the angle of inclination required . chap. vi. of upright declining plaines . those plaines are upright , which point up directly into the zenith or verticall point of the horizon , and may be tryed by a perpendicular or plumb-line . in these , as in the rest that follow , before the houres can be drawne , two things must bee found ; 1. the rectifying arke ; 2. the elevation of the pole above the plaine . 1. to find the rectifying arke . extend the threed from your latitude counted in the upper quadrant of the circle on the backeside , to the complement of the plaines declination numbred in the semi-circle ; so shall the threed shew you on the diameter the arke required . 2. to find the elevation of the pole above the plaine . extend the threed from the rectifying arke numbred in he upper quadrant , to your latitudes complement taken in the semicircle ; so shall the threed shew upon the diameter , the elevation of the pole above the plaine . according to these rules , in the latitude of 52 gr . 30 min. supposing an upright plaine to decline 55. gr . 30 min. i find the rectifying arke to bee 28 gr . 36 min. and the elevation of the pole above the plaine to be 20 gr . 10 minutes . chap. vii . in east and west incliners . those plaines are called east and west incliners , whose horizontall line lyeth full north and south , and their inclination is directly towards either east or west . 1. to find the rectifying arke . extend the threed from your latitudes complement taken in the upper quadrant of the circle on the backside , to the complement of the plains inclination counted in the semicircle ; so shall the threed shew upon the diameter the arke required . 2. to find the elevation of the pole above the plaine . extend the threed from the rectifying-arke counted in the upper quadrant , to your latitude taken in the semicircle ; so the threed upon the diameter gives the elevation of the pole above the plaine . thus in the latitude of 52 gr . 30 min. if a plaine incline eastward 40 gr . to the horizon , the rectifying-arke will be 35 gr . 58 min. and the elevation of the pole 37 gr . 26 min. above the plaine . chap. viii . in north and south incliners . such plaines are called north and south incliners , whose horizontall line lyeth full east and west , and their inclination is directly upon either north or south . 1. for the rectifying-arke . there is no use of it in these plaines , because they all lye directly under the meridian of the place . 2. to find the elevation of the pole above the plaine . if the inclination be toward the south , adde the inclination to your latitude ; for the summe is the elevation of the pole above the plaine . if the summe exceed 90 degrees , take it out of 180 , and the supplement gives you the poles elevation . if the inclination bee northward , compare the inclination with your latitude , and subduct the lesser out of the greater : the difference is the elevation of the pole above the plaine , if there bee no difference , it is a direct polar plaine . chap. ix . in declining incliners . those plaines are called declining incliners , whose horizontall line declineth from the east or west , towards either north or south , and their inclination also deflecteth from the coasts of north and south towards either east or west . the best way to find the rectifying-arke , and the poles elevation for these plaines , will be first , to referre them to a new latitude , wherein they may lye as east or west incliners . for which purpose you are first to find out an arke , which in respect of its use may fitly be called , the prosthaphaereticall arke , it is found by this rule : ¶ extend the threed from the complement of the plaines declination counted in the upper quadrant , to the inclination numbred in the semicircle ; so the threed shall give you upon the diameter the prosthaphaereticall-arke required . this prosthaphaereticall-arke is to be used as the inclination was in the precedent chapter . for , if the plaine doe incline towards the south , it must be added to your latitude : and so the summe ( if lesse then 90 degrees ) gives you the new latitude : but if the summe bee greater than 90 , then the residue , or supplement of it to 180 degrees will be the new latitude required . if the plaine incline toward the north , compare this prosthaphaereticall-arke with your latitude , and subduct the lesser of them out of the greater ; so the difference shall give you the new latitude . if there be no difference , it is a declining polar plaine . secondly , it will be required to know what inclination these plaines shall have in this their new latitude ; and that is done by this rule : ¶ extend the threed from the prosthaphaereticall-arke taken in the upper quadrant to the plaines declination counted in the semicircle : so the threed shewes on the diameter , the new-inclination in their new latitude . being thus prepared , you may now proceed as in east and west incliners you did before . 1. to find the rectifying-arke . extend the threed from the new latitudes complement taken in the upper quadrant , to the new-inclinations complement numbred in the semicircle ; so the threed upon the diameter shewes the arke required . 2. to find the elevation of the pole above the plaine . extend the threed from the rectifying-arke in the vpper-quadrant to the new latitude in the semicircle ; so the threed upon the diameter gives the elevation of the pole above the plaine . according to these rules , supposing a plaine to incline towards the north 30 degrees , and to decline from the south towards the west 60 degrees in the latitude of 52 gr . 30 min. first i find the prosthaphi-arke 60 gr . 6 min. and because the plaine inclineth toward the north ; i compare this arke with the latitude of the place , and taking it out of the latitude there remaineth 36 gr . 24 min. for the new latitude . then i find the new inclination to bee 25 gr . 40 min. and so the rectifying-arke 59 gr . 8 min. and the elevation of the pole above the plaine to be 32 gr . 20 minutes . chap. x. to draw the houre-lines upon the horizontall , the full north or south plaines , whether standing upright or inclining . in the foure last chapters we have seene the uses of the circle on the backe-side of the quadrant : in this and the next chapter we shall shew the use of tr the line of latitudes , and of tv the line of houres ; which two lines with the helpe of the limbe vctb , and of the threed and bead , will serve to pricke downe any diall , by the precepts hereafter delivered . and first we begin with those plaines which have no declination , whose poles lye directly under the meridian of the place ; of which sort are the horizontall , the erect south and north plaines , with all incliners looking directly north or south . having then by the former precepts found the elevation of the pole above your plaine , you may begin your draught in this manner . first , draw the line rat of sufficient length , and out of the line of latitudes in your quadrant , take off the elevation of the pole above the plaine , and pricke it downe from the point a , unto r and t both wayes . 2. take the line of houres tv also out of the quadrant , and with that extent of your compasses upon r and t as upon two centers , draw the arkes bv and cv , crossing each other in v ; and draw the lines rv and tv : then comming to your quadrant againe ; 3. apply the threed to every houre point in the limbe vt or cb , as first to s , or r , so shall it cutte the line of houres tv in 1 ; then take off with your compasses t1 , and pricke it downe here from v to 1 , and from t to 7. again , apply your threed to the next houre in the limbe at n or m , it will cut the line of houres tv in 2 take off t2 , and prick it down here from v to 2 , and from t to 8. so againe , the threed applyed to the third noure at y , or u , cuts the line tv , in 3 ; take off t3 , and pricke it downe here from v to 3 , and from t to 9. in like manner , the threed applyed to the fourth houre at o , or i , will cut the line tv in 4 take off t4 , and pricke it downe here from v to 4 , and from t to 10. so also the threed laid upon the fifth houre at e , or a , cutteth tv in 5 ; take off t5 , and prick it downe here from v to 5 , and from t to 11. thus are all the houres pricked downe . an horizontull diall to 52 gr : 30 m : lat : lastly then , laying your ruler to the center a , through each of these points , you shall draw the houre-lines a7 , a8 , a9 , a10 , a11 , av which is 12 , a1 , a2 , a3 , a4 , a5 , rat is the line of the two sixes . so having 12 houres , which is halfe the diall , drawne , you may extend the necessarie lines , as many as you will , beyond this center , as 5a5 , 4a4 , 7a7 , 8a8 , &c. in the same manner may the halfe houres bee pricked downe and drawne , by applying the threed to the halfe houres in the limbe , &c. and note also that in these plaines before mentioned ; as the extent from v to 1 , is the same with that from t to 7 , so likewise is it the same with v11 , r5 ; and as v2 is the same with t8 , so likewise is it the same with v10 , r4 : so likewise v9 and t9 are all one , and both equall to r3 and v3 . so that the three first houres taken from the quadrant , that is to say , t1 , t2 , t3 , will give all the houres for these dialls . t1 , gives v1 , v11 , r5 , t7 . t2 , gives v2 , v10 , r4 , t8 . t3 , gives v3 or r3 , v9 or t9 . but in other plaines it is not so , for which cause i have rather set downe this way before at length , as a direction for what comes after , for that is generall . here note againe , that if you desire to make your draught greater , you may in your description either double or triple every length which you take in your compasses . and so i proceed to all declining plaines . chap. xi . to draw the houres upon all sorts of declining plaines , whether erect or inclining . by the former precepts you must first get the rectifying-arke , with the elevation of the pole above the plaine . after they are had , you may pricke downe the houre points in this manner following , little differing from the former . a plaine ▪ inclininge eastward 40 gr : the horizointall line , parallel to the line of 12. 1. asbefore ; upon the line rat , set off the elevation of the pole above the plaine , being taken out of the line of latitudes in the quadrant , from a both wayes , to r and t. 2. take the line of houres tv out of the quadrant , and with that extent upon r and t as upon two centers , describe the two arkes bv and cv crossing at v , and draw the lines rv , tv , and av. thus farre we goe along with the last chapter . 3. if we take the example in the seventh chapter , that plaine is the upper face of an east incliner , whose elevation is 37 gr . 26 min. and so much doth this line ta reach unto in the line of latitudes : the rectifying arke is 35 gr . 58 min. this arke i number below in the limbe of the quadrant es , and thereto applying the threed i observe in the upper limbe vcb t which of the houres and where it cutteth , i find it to cut the slope line o u in the point p ; to this point p i set the bead , which by this meanes is rectified and fitted to the description of the diall . here you see the use of the bead , and the reason why this arke counted upon the limbe is called the rectifying arke : and here bee carefull that you stretch not the threed . 4. the threed and bead being thus placed and rectified , you shall see the threed to cut the line tv at a upon the quadrant ; take t a in your compasses , and pricke it downe here from v to 12 , and from r to 6. here by the way observe , that because this plaine is an eeast-incliner , the face of it looketh toward the west , and then if you imagine the true scituation of this diall upon the plaine whereon it must stand , you will easily conceive that the line of 12 is to stand on the right hand from the line av. and so the line of 6 on the left hand , whereas if this plaine had faced toward the east , the line of 12 must have stood on the left hand , and 6 on the right hand . your owne conceit , together with the precepts of the chapter following , must helpe in this , and in other things concerning the right scituating of the lineaments of your diall . to proceed then , in the same manner must you apply the bead to every houre line , as in the next place i remove it to the line y m in the quadrant , and then i see it to cut the line tv in b ; i take 1 b in my compasses , and with it doe pricke downe from v to 1 , and from r to 7. againe , the bead being applyed to the lines nr , sb , the threed will cut the line tv upon the quadrant in c and d ; i take the points tc , td , in my compasses , and pricke them downe from v to 2 and 3 , and from r to 8 and 9. then againe , the bead applyed to the lines ei , va , the threed will cut the line tv in the points e and o ; i take then te and tf , and pricke them downe from u ●o 11 and 10 , and from r to 5 and 4. 5. lastly , lay your rule to a , and draw a10 , a11 , a12 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 . thus have you twelve houres , and if you extend these beyond the center , you shall have the whole 24 houres , of which number you may take those that shall bee fit for the plaine in this scituation . the halfe houres may thus bee pricked on and drawne also , by applying the bead to the halfe houres pricked downe in vcb t the upper limbe of the quadrant , for so the threed will give you the halfe houre points upon the line tv , which may be taken off , and set downe upon the diall as the houres themselves were . chap. xii . how to place the diall in a right scituation upon the plaine . after the houre-lines are drawne by the last chapter , they are to be placed in a right scituation upon their plaine . which to doe , upon some plaines is more difficult than the description of the diall it selfe . to give some directions herein , i have added this chapter , where you have 9 ▪ questions with their answers , giving light sufficient to what is here intended and required : but first be admonished of three things . 1. that the inclination mentioned chap. 8. is the very same in use with the prosthaphaereticall arke mentioned chapter 9. and therefore when i mention the prosthaphaereticall arke , because it is of most frequent use , you must remember i meane both the prosthaph : arke , chap. 9 , and the inclination , chap. 8. 2. that these rules , though given primarily for places of north-latitude , lying within the temperate , torrid , and frigid zones , yet are also as true , and may bee applyed to all places of south-latitude , if we exchange the names of north and south , for south and north. here by the way note , that the north part of the torrid zone extendeth from 0 degrees of latitude to 23 gr . 30 min. the temperate zone reacheth from 23 gr . 30 min. to 66 gr . 30 min. the frigid zone extendeth from 66 gr . 30 min. to 90 gr . of latitude . and so i come to the 9 questions . 1. what pole is elevated above the plaine . upon all upright plaines declining from the north : upon the upper faces of all east or west incliners : upon the upper faces of all north-incliners , whose prosthaph : arke is lesse than the latitude of the place : on the under faces of all north-incliners , whose prosthaph : arke is greater then the latitude of the place ; and on the upper faces of all south-incliners , the north pole is elevated . and therefore contrarily , upon all upright plaines declining from the south : on the under faces of all east and west , and south incliners : on the under faces of all north-incliners , whose prosthaphaereticall arke is lesse than the latitude of the place : on the upper faces of all north-incliners , whose prosthaph : arke is greater than the latitude of the place , the south pole is elevated . 2. what part of the meridian ascendeth or descendeth from the horizontall line of the plaine ? in all upright plaines the meridian lyeth in the verticall line , and if they decline from the south it descendeth , if from the north it ascendeth . upon both faces of east and west incliners the meridian lyeth in the horizontall line . in all north-incliners , the north part of the meridian ascendeth , the south part descendeth : in all south incliners the south part of the meridian ascendeth , the north part descendeth : upon both upper and under faces . and if these north and south incliners be direct , then the meridian lyeth in the verticall line , and so maketh a right angle with the horizontall line : but if they decline , then the meridian on the one side maketh an acute angle with the horizontall line . 3. to which part of the meridian is the style with the substyle to be referred , as making with it an acute angle ? the style is the cocke of the diall ; the substyle is the line whereon it standeth , signed out in the former descriptions by the letters av. in all plaines whereon the north pole is elevated , it is referred to the north part of the meridian , and maketh an acute angle therewith . in all plaines whereon the south pole is elevated , it is referred to the south part of the meridian , and is to make an acute angle therewith . except here only those south-incliners , whose prosthaph : arke is more than the complement of your latitude : for on these plaines the substyle standeth on that part of the meridian , whose denomination is contrary to the pole elevated above the plaine . for on the upper faces the north pole is elevated , but the substyle standeth toward the south end of the meridian : and on the under 〈◊〉 the south pole is elevated , but the substyle lyeth toward the north end of the meridian . note here , that in south-incliners whose prosthaphaereticall arke is equall to the complement of your latitude , the substyle lyeth square to the meridian upon the line of 6 a clocke ; which line in such plaines alwayes lyeth perpendicular to the meridian line . amongst these falleth the equinoctiall plaine . 4. on which side of the meridian lyeth the substyle ? in all direct plaines it lyeth in the meridian . in all decliners it goeth from the meridian toward that coast which is contrary to the coast of the plaines declination . and so doe all houres also goe upon the plaine to that coast which is contrary to the coast whereon they are ; as all the morning or easterne houres goe to the westerne coast of the plaine , and all the evening or westerne houres goe to the easterne coast of the plaine . which being observed will bee a great helpe to place them aright . 5. what plaines have the line of 12 upon them , and which not ? all upright plaines , in all latitudes whatsoever , declining from the south have the line of 12 ; and decliners from the north in the temperate zone have it not , but in the other zones they also have it . the upper faces of east and west incliners in all latitudes have it , the underfaces have it not . the upper faces of all north incliners whatsoever have it ; their under faces in the temperate zone want it , in the frigid zone have it , and in the torrid zone likewise if the prosthaph : arke bee greater than the sunnes least north meridian altitude , but if it be lesse they want it also . for south incliners , consider the sunnes greatest and least meridian altitude upon the south coast . for if the prosthaphaereticall arke bee such as falleth betweene them , that is , if it be greater than the least , or lesse than the greatest , then have bothsides the line of 12 upon them ; but if it be lesse than the least , then doth the underface want it universally , and the upper face alone hath it ▪ if greater than the greatest , then doth the upper face want it , and the under face alone hath it : except in the frigid zone where the upper face hath it also , by reason of the sunnes not setting there for a time . 6. whether the north or south part of the meridian serveth for the line of 12 ? in those plaines that have the line of 12 , where the north pole is elevated , there the north part of the meridian serveth for 12. and where the south pole is elevated , there the south part of the meridian serveth for the line of 12 or mid-day . except , in all latitudes , the under faces of those south incliners , whose prosthaphaereticall arke falleth betweene the sunnes greatest and least meridian altitudes , for in them the south pole is elevated , but the north part of the meridian serveth for the line of 12. except in speciall those upright plaines in the torrid-zone which looke toward the north , and the under faces of north-incliners also , whose prosthaphaereticall arke is greater than the least north-meridian-altitude ; for these have the south or lower part of the meridian serving for 12 , though the north pole be elevated . 7. which way the style pointeth , and how it is to bee placed ? in plaines where the north pole is elevated , it pointeth up towards it ; and where the south pole is elevated , it pointeth downe towards it . the style lyeth perpendicularly over the substyle , noted in the former figures with av , and is to be elevated above it to such an angle as the elevation of the pole above the plaine shall be found to be by the 6 , 7 , 8 , and 9 chapters . 8. when is it that that part of the meridian next the substyle , and the line of twelve doe goe contrary wayes ? in all latitudes , upon the upper faces of south-incliners , whose prosthaphaereticall arke is greater than the complement of the latitude , but lesse than the sunnes greatest south meridian altitude : and on the under faces of those south-incliners also , whos 's prosthaph : is lesse than the complement of the latitude , but greater than the sunnes least south meridian altitude : in the torrid-zone alone you must adde hither also , north upright plaines , and those north-incliners on the under-face , whose prosthaphaereticall-arke is greater than the least north-meridian altitude of the sun ; for these have the line of midday standing on that coast which is contrary to the coast of that part of the meridian next the substyle , and none else . the line of 12. i call herethe line of midday because in the frigid-zone , where the sunne setteth not in many dayes together , there are two twelves , the one answering to our midday , and the other to our midnight : and so all upper faces of south-incliners , whose prosthaphaereticall arke falleth betweene the least and greatest south meridian altitudes , have there two 12 a clockelines upon them . 9. how much the meridian line ascendeth or descendeth from the horizontall line ? the quantity of the angle is to be found upon the circle on the back-side of your quadrant , in this manner ; extend the threed from the complement of the plaines inclination taken in the lower quadrant , to the complement of the plaines declination counted in the semicircle , and the threed will shew you upon the diameter , the degrees and minutes of the meridians ascension or descension . in the example of the 9. chapt. taking the upper face of that plaine , i find the meridian to ascend above the horizontall line 33 gr . 41 minutes . ¶ these directions are sufficient for the bestowing of every line into its proper place and coast . as may bee seene in the example of the ninth chapter . for , first , upon the upper face of that north incliner , because his prosthaph : arke 16 gr . 6 min. is lesse than 52 gr . 30 min. the latitude of the place , therefore the north pole is elevated above it : by the answer to the first quest. 2. because it is a north-incliner , therefore the north part of the meridian ascendeth above the horizontall line , by the answer to the second question . 3. because the north pole is elevated , therefore the style with the substyle maketh an acute angle with the north end of the meridian , by the answer to the third question . 4. because this plaine declineth toward the west , therefore the substyle lyeth on the east-side of the meridian , and so doe the houres of the afternoone : by the answer to the fourth question . 5. this plaine , being the upper face of the north-incliner , will have the line of 12 to bee drawne upon it , by the answer to the fifth question . 6. because the north pole is elevated , therefore the north part of the meridian serveth for the line of 12 : by the answer to the sixt question . 7. because the north pole is elevated , therefore the style pointeth upward toward the north pole ; by the answer to the seventh question . 8. that part of the meridian next the substyle , and the line of 12 are both one , and so therefore goe both one way : by the answer to the eight question . 9. by the second the meridian line ascendeth , and the quantity of the ascent is 33 gr . 41 min. above the horizontall line : by the answer of the ninth question . thus you see every doubt cleared in this example : the like may be done in all others . chap. xiii . the making and placing of polar plaines . place this diagram betweene folio 32. and 33. the horizontall line of the plaine . these plaines may have dialls described upon them by this quadrant , but the better way is the common way , to protract them by an equinoctiall circle , for otherwise the style will be alway of one distance from the plaine , be the diall greater or lesser . the polar plaines that decline , before they can be described , must have their new-inclination known , and then their delineation will be easie , the manner of it may be seene in this example . suppose the upper face of a north-inclining plaine , lying in the latitude of 52 gr . 30 min. to decline from the south toward the east 68 gr . and to incline towards the north 73 gr . 57 min. you shall find by the ninth chapter , the prosthaph : arke to be 52 gr . 30 min. the same with the latitude of the place , and therefore you may conclude this plaine to be polar . by the same chapter you shall find the new inclination to be 63 degrees . when you have these you may draw your semicircle ab4 , and divide it into 12 equall parts for the houres : so signing the new-inclination 63 degrees from a to b , draw cb : and supposing the altitude of your style to be cd , through d draw the perpendicular d 12 ; and where the lines drawne from c through the divisions of the semicircle doe cut the line d 12 , there raise perpendiculars for the houres , and so finish it up as the manner is . the style lyeth directly over and parallel to the substyle cb , & the distance of it from the plain is cd , and in this example the substyle cb standeth from the line of 12 westward , because the plaine declineth eastward , according to the rules in the former chapter , and so doe the morning houres also . for the placing of the diall in a true site upon the plaine , you shall find by the answer to the 9 quest. in the former chapter , that the meridian ascendeth 55 gr . 38 min. for other necessaries , the precepts of the former chapter will direct you . onely observe , that in upright east and west plaine , the line of 6 is alwayes the substyle , and it ascendeth above the north end of the horizontall line , as much as the latitude of the place commeth to . finis . an appendix shewing a ready way to find out the latitude of any place by the sunne . because in the third chapter , and quite through this treatise , the latitude of the place is supposed to bee knowne , when as every one perhaps cannot tell which way to find it out ; i thought good therefore to adde this appendix as a ready helpe to shew how it may bee attained sufficiently for our purpose . know then that for the finding out of the latitude of a place by the sunne , these things are required . 1. to find the meridian line . the readiest way to find the meridian line is by the north-starre . this starre is within 2 degr . 37 min. of the north-pole . the north-pole lyes very neere betweene allioth , or the root of the great beares tayle , and this starre ; you may therefore imagine where the pole is , if you conceive a right line drawne from the pole-starre to allioth , and by your imagination suppose ⅔ parts of the distance of the next starre of the little beares taile from the pole-starre towards allioth , for there is the very pole-point . now then if you set up two poles aslope , and from the tops of them hang two cords with weights at the ends of them , and turne them till you standing on the south-side of them may see them both together with the pole-point , as it were all in one line , then be sure these two cords doe hang in the meridian line , or very neere it , yea so neere it , that though you should erre 3 degrees herein ( wherein you need not to erre one degree ) yet will not the meridian altitude in these climates ( especially more northward ) faile you above 3 minutes , which is neere enough to our purpose . i have here given you the chiefe starres of the great and little beares , that by them you may come to know the starres used in this observation , and so find the very pole-point it selfe . 2. to find the sunnes meridian altitude . observe diligently about noone when the shadow of the south cord shall fall upon the north cord , for then is the sun in the meridian . at that instant observe the suns altitude stedily and carefully , for that is the meridian and greatest altitude of the sun for that day . 3. to find the sunnes declination . for this purpose the limbe hath the characters of the 12 signes fixed to each 30 degree , and a scale of declinations under the limbe noted with mn . the scale is divided by this table ; for looke what degr . and min. of the eclipt . doe answer to the degr . of declination in the table , the same are to be numbred in the limbe , and by a ruler applyed to them , the degrees of declination are drawne upon the scale . a table to make the scale for the declination of every part of the eclipticke . degr. of decl . deg. of the ecl . degr. of decl deg. of the ecl . degr. declin . degr. eclipt . degr. declin . degr. eclipt degr. declin . degr. eclipt . degr. declin . degr. eclipt . 0.0 0.00 4.0 10.04 8.0 20.26 12.0 31.26 16.0 43.44 2.0 59.04 0.15 0.38 4.15 10.43 8.15 21.06 12.15 31.09 16.15 44.34 20.15 60.14 0.30 1.15 4.30 11.21 8.30 21.46 12.30 32.52 16.30 45.25 20.30 61.26 0.45 1.53 4.45 11.59 8.45 22.26 12.45 33.36 16.45 46.17 20.45 62.41 1.0 2.31 5.0 12.37 9.0 23.06 13.0 34.21 17.0 47.09 20.0 46.00 1.15 3.08 5.15 13.16 9.15 23.46 13.15 35.05 17.15 48.03 21.15 65.22 1.30 3.46 5.30 13.54 9.30 24.27 13.30 35.50 17.30 48.57 21.30 66.48 1.45 4.24 5.45 14.33 9.45 25.08 13.45 36.35 17.45 49.52 21.45 68. ●● 2.0 5.01 6.0 15.12 10.0 25.49 14.0 37.21 18.0 50.48 22.0 69.58 2.15 5.39 6.15 15.51 10.15 26.30 14.15 38.07 18.15 51.45 22.15 71.44 2.30 6.16 6.30 16.30 10.30 27.12 14.30 38.54 18.30 52.43 22.30 73.41 2.45 6.55 6.45 17.08 10.45 27.53 14.45 39.41 18.45 53.43 22.45 75.53 3.0 7.33 7.0 17.48 11.0 28.36 15.0 40.28 19.0 54.44 23.0 78.30 3.15 8.10 7.15 18.27 11.15 29.17 15.15 41.16 19.15 55.47 23.15 81.52 3.30 8.48 7.30 19.6 11.30 30.00 15.30 42.05 19.30 56.50 23.30 90.00 3.45 9.26 7.45 19.46 11.45 30.43 15.45 42.54 19.45 57.56 finis . before you can find the declination , you must know the sunnes place , and for such as know not the use of the astronomicall tables , an almanacke will serve , where for every day at noone , you shall find the sunnes place in signes , degrees and minutes . the degr . and min. must bee numbred in their signes upon the limbe , and the threed applyed thereto will shew the declination answerable . as for example . september 21. 1637 in the almanack for this yeare , the sunne is found to be in 8 gr . 23 min. of ♎ . in the quadrants limbe i looke for the signe ♎ and number there , 8 gr . 23 min. whereto apply the threed , i find it to cut in the scale of declinations 3 gr . 20 min. 4. by the meridian altitude , and declination of the sun had ; how to find the latitude of the place , or the elevation of the pole above the horizon . compare the sunnes meridian altitude and declination together , and if the sunne be in a north signe as ♈ ♉ ♊ ♋ ♌ ♍ , then substract the declination out of the meridian altitude , so shall the difference give you the height of the equinoctiall . but if the sun be in the south signes , as ♎ ♏ ♐ ♑ ♒ ♓ , then adde the declination to the meridian altitude , so shall the summe give you the height of the equinoctiall , which being taken out of the quadrant or 90 degrees , leaveth the latitude of your place , or the elevation of the pole above your horizen . for example . upon the 21 of september 1637. i observed the sunnes altitude in the meridian to be 34 gr . 10 min. upon which day i find the sunnes place to be ( as before ) 8 gr . 23 min. of ♎ , and the declination 3 gr . 20 min. and because the sun is in a south signe , i adde this declination and meridian altitude together ; the summe 37 gr . 30 min. is the altitude of the aequator , and this taken out of 90 degrees leaveth 52 gr . 30 min. for the latitude of coventrie . speculum nauticum a looking-glasse for sea-men. wherein they may behold, how by a small instrument, called the plain-scale, all nautical questions, and astronomical propositions, are very easily and demonstratively performed. first set down by john aspley, student in physick, and practitioner of the mathematicks in london. the sixth edition. whereunto are added, many new propositions in navigation and astronomy, and also a third book, shewing a new way of dialling. by h.p. and w.l. aspley, john. 1662 approx. 136 kb of xml-encoded text transcribed from 39 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2008-09 (eebo-tcp phase 1). a75737 wing a4013 estc r229501 99899278 99899278 152809 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a75737) transcribed from: (early english books online ; image set 152809) images scanned from microfilm: (early english books, 1641-1700 ; 2323:2) speculum nauticum a looking-glasse for sea-men. wherein they may behold, how by a small instrument, called the plain-scale, all nautical questions, and astronomical propositions, are very easily and demonstratively performed. first set down by john aspley, student in physick, and practitioner of the mathematicks in london. the sixth edition. whereunto are added, many new propositions in navigation and astronomy, and also a third book, shewing a new way of dialling. by h.p. and w.l. aspley, john. h. p. w. l., 17th cent. [4], 64 [i.e. 72] p. : ill., charts printed by w. leybourn, for george harlock, and are to be sold at his shop at magnus church-corner, in thames street, near london-bridge, london : 1662. page 72 misnumbered 64. reproduction of original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language 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these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng dialing -early works to 1800. navigation -early works to 1800. nautical astronomy -early works to 1800. 2007-12 tcp assigned for keying and markup 2007-12 apex covantage keyed and coded from proquest page images 2008-01 john pas sampled and proofread 2008-01 john pas text and markup reviewed and edited 2008-02 pfs batch review (qc) and xml conversion speculum nauticum . a looking-glasse for sea-men . wherein they may behold , how by a small instrument , called the plain scale , all nautical questions , and astronomical propositions , are very easily and demonstratively performed . first set forth by john aspley , student in physick , and practitioner of the mathematicks in london . the sixth edition . whereunto are added , many new propositions in navigation and astronomy , and also a third book , shewing a new way of dialling . by h. p. and w. l. london , printed by w. leybourn , for george hurlock , and are to be sold at his shop at magnus church-corner , in thames-street , near london-bridge , 1662. to the worshipfvll , the master , wardens , & assistants of the trinity hovse ; john aspley , in testimony of the honour he bears to the governours & practisers of the art of navigation , dedicates these his first labours . the printer to the reader . this little book having been well accepted of among sea-men , being the first fruites of mr. aspley's mathematical studies , hath passed five impressions , without any alteration ; and so i doubt not might have done still : but because since that time there have been severall bookes put out of this nature , i have procured this to be revised , and severall alterations and additions to be made therein , so that here you have both the old , and a new booke intermingled all in one , with a third part added thereto , concerning dialling ▪ by a way not formerly published by any . all which i doubt not you will kindly accept of , and receive much delight and profit thereby . your . g. h. errata . page 34 line 26 read 360. page . 45. l. 8. r. distance i m. page 50. line 13. for 14 &c. 〈…〉 which is just the length of the gnomon . page . 50 line . 28. for increase , read decrease , page 52 line 4. r. h a i. line 18. r. point o. page 57 line 11 r. point l. also for some lite●all faults we shall desire your pardon ▪ speculum nauticum , or the sea-mans glasse . the first book . chap. i. the explanation of certain terms of geometry . being intended in this treatise of the plain scale , to declare the manner of projection of the sphere , in plano , i have thought fitting first , to shew unto you some tearms of geometry which are necessary for the unlearned to know , ( for whose sake chiefly i write this treatise ) before they enter into the definition of the sphere . first therefore i intend to relate unto you , what a point or prick is , and afterward a line both right and crooked , and such sorts thereof as are appertinent unto the operations and use of this scale . punctum , or a point , is the beginning of things , or a prick supposed indivisible , void of length , breadth , and depth : as in the figure following is noted by the point , or prick a. linea , or a line , is a supposed length , or a thing extending it self in length , not having breadth nor thickness , as is set forth unto you by the line bad . parallela , or a parallel line , is a line drawn by the side of another line , in such sort that they may be equidistant in all places . and of such parallels , two only belong unto this work of the plain scale , that is to say , the right lined parallel , and the circular parallel . right lined parallels are two right lines equidistant one from another , which being drawn forth infinitely , would never touch or meet one another , as you may see in the figure , where the line h i is parallel unto the line ce , and the line gf is parallel unto them both . a circular parallel is a circle drawn either within or without another circle upon the same center , as you may plainly see by the two circles bcde , and xvyw . these circles are both drawn upon the center a , and therefore are parallel the one unto the other . there is another kind of parallel also , which is called a serpentine parallel , but because it is not belonging unto the use of this scale , i will omit it , and so proceed unto the rest . perpendiculum , or a perpendicular is a line raised from , or let fall upon , another line , making equal angles on both sides , as you may see declared in the figure , where in the line ac is perpendicular unto the line bad , making equal ●ngles in the point a. diameter circuli , or the diameter of a circle , is a right line drawn thorow the center of any circle , in such sort that it may divide the circle into two equal parts , as you may see the line bad is the diameter of the circle bcde , because it passeth thorow the center a , and the two ends thereof do divide the circle into two equal parts , in the two extreams b and d , making the semicircle bcd equal unto the semicircle deb . semidiameter circuli , or the semidiameter of a circle is half of the diameter , and is contained betwixt the center , and the one side of the circle , as the line ad is the semidiameter of the circle bcde . this semidiameter contains 60 degrees of the line of chords , which we sometimes call the radius . semicirculus , or a semicircle , is the one half of a circle , drawn upon his diameter , and is contained upon the superficies , or surface , of the diameter , as the semicircle bcd which is half of the circle bcde , and is contained above the diameter bad . quadrans circuli , is the fourth part of a circle , and is contained betwixt the semidiameter of the circle , and a line drawn perpendicular , unto the diameter of the same circle , from the center thereof , dividing the semicircle into two equal parts , of the which parts , the one is the quadrant , or fourth part of the same circle . as for example , the diameter of the circle bcde is the line bad , dividing the circle into two equal parts : then from the center a raise the perpendicular ac , dividing the semicircle likewise into two equal parts ; so is abc , or acd , the quadrant of the circle bcde , which was desired . chap. ii. the manner how to raise a perpendicular from the middle of a line given . 〈◊〉 first a ground line whereupon you would have a perpendicular raised , then open your compasses unto any distance ( so it exceed not the end of your line , ) placing one foot of the said compasses in the point from whence the perpendicular is ●o be raised , and with the other foot make a mark in the line on 〈…〉 removing your compasses unto any other distance that 〈…〉 set one foot thereof in one of the marks , and with the 〈◊〉 foot make an arch over the middle point , then with the same distance of your compasses set one foot in the other mark upon the line , and with the other foot make another arch of a circle over the middle point , so that it may cross the first arch , and from the meeting of these two arches , draw a right line unto the middle point , from which the perpendicular was to be raised , which line shall be the perpendicular desired . example , suppose your base or ground line whereupon a perpendicular is to be raised be the line flk , and from l the perpendicular is to be raised , set one foot of your compasses in the point l , and with the other , make the marks g and m on both sides of the point l , the● opening your compasses wider , set one foot in the point m , and with the other draw the arch s over the point l , then with the same distance of your compasses , set one foot in g , and with the other make the arch r , crossing the arch s in the point t , then from t draw the line tl , which line is perpendicular unto the line flk from the point l , which is the perpendicular desired . chap. iii. to let a perpendicular fall from any point assigned , unto the middle of a line . let the line whereupon you would have a perpendicular let fall be the line lfk , and the point assigned to be the point t , from whence you would have a perpendicular let fall upon the line flk , first set one foot of your compasses in t , and open your compasses unto any distance so that it be more than the distance tl , which here we suppose to be the distance tm ; then make in the line flk the marks g and m , then with your compasses take the one half of gm , which is the point l , then from l draw a line unto the point t , so the line tl shall be the perpendicular , which was desired to be let fall from the assigned point t unto the middle of the line flk . chap. iv. to raise a perpendicular upon the end of a line . suppose the line whereupon you would have a perpendicular raised , be the line flk , and from the point f a perpendicular is to be raised : first open your compasses unto any distance , which here we put to be the distance fg , and set one foot of your compasses in the point f , and with the other draw the arch deg , then set one foot of your compasses in the point g , and with the other draw the arch e ; then placing one point of your compasses in e , with the other draw the arch db ; then place your compasses in d , and with the same distance draw the arch a , cutting the arch db in c , then draw a line from c unto the end of the line flk , unto the assigned point f , so shall the line cf be a perpendicular raised from the end of the line flk , and from the assigned point f. chap. v. to let a perpendicular fall from any point assigned unto the end of a line . let the line flk be the base or ground line , and from the point i a perpendicular is to be let fall upon the end of the line at k , first from the assigned point i , draw a line unto any part of the base , which shall be the line ihm , then find the middle of the line im , which is at h ; place therefore one foot of your compasses in the point h , and extend the other unto i , with which distance draw the arch ink upon the center h , cutting the base or ground-line in the point k , then draw the line ki , which line shall be the perpendicular desired . chap. vi. a right line being given , how to draw another parallel there unto at any distance required . let the line given be ab , unto which it is required to draw another right line cd which shall be parallel to the former line ab , and at the distance ac . first open your compasses to the distance ac , then set one foot in the point a , with the other describe the arch c ; again , place one foot in b , and with the other describe the arch d ; then draw the line cd , so that it may only touch the two arches c and d , so shall the line cd , so drawn , be parallel to ab , and at the distance required . chap. vii . a right line being given , how to draw another parallel thereunto , which shall also pass through a point assigned . let ab be a line given , and the point assigned be c : and let it be required to draw another line parallel thereunto , which shall pass through the given point c. now i doubt not but you understand the way to let fal , or to raise any manner of perpendicular line , either from , or upon any part of a line : as also to draw lines parallel one to another at any distance required , therefore now i intend to proceed unto the main point here aimed at , which is , to declare , and make known unto you the several operations performed by the plain scale , which though it be in use with very few , yet it is most necessary for sea-men , because all questions in navigation are thereby easily and plainly wrought . and also all questions in astronomy ( belonging unto the expert , and industrious sea-men ) may both speedily and easily be wrought by the same scale : in regard whereof i have declared in this little book , that knowledge ( which god hath been pleased to bestow upon me ) concerning the necessary use and practice thereof ; hoping that you will as kindly accept it , as it is freely offered unto your courteous considerations . chap. viii . of the description of the scale . the figure of the plane scale . the second part of the scale , is the single chorde of a circle , or the chord of 90 , and is divided into 90 unequal divisions , representing the 90 degrees of the quadrant : and are numbred with 10 , 20 , 30 , 40 , &c. unto 90. this ghord is in use to measure any part or arch of a circle , not surmounting 90 degrees : the number of these degrees from 1 unto 60 is called the radius of the scale , upon which distance all circles are to be drawn , whereupon 60 of th●se degrees are the semidiameter of any circle that is drawn upon that radius . the third part of the scale is divided into eight parts , representing the points or rumbes of the mariners compass ; which in all are 32 points : but upon the scale there are only s reckoned , which is but one quadrant or quarter of them , being to be reckoned from the meridian of north and south both wayes , as you may see more plainly by this figure , representing the order of the points of the compasse . it is usuall also to have another line placed upon your scale , to she● you how many leagues make a degree of longitude in every latitude , concerning which you shall have directions in the 14 chapter following . chap. ix . knowing the course any ship hath made , and the leagues she hath sailed : to find how much she hath raised or depressed the pole ; and likewise how much she is departed from her first meridian . the course is south-west and by south , the leagues sailed are 100 , the difference of the latitude , and the distance of the meridians is required . now you must heedfully observe this point d , for this represents the place where your ship is , and doth shew both the differencefo the latitude of the place you are in , and also your distance or departure from your first meridian . first for the latitude , you see the line df , being paralel to the line ab , cuts the meridian line af in the point f : so that if you take the distance fa with your compasses , and apply it to the scale of equal leagues , you shall find it is just 83 leagues , which counting 20 leagues to a degree , makes 4 degrees 9 min. and so much you have altered your latitude by the said course , which degrees and minutes being added to , or substracted from the latitude of the place you came from , according as your course requires , shews you alwaies the true latitude you are in . likewise from this point d , take with your compasses the distance df , and you shall find it by your scale of equal leagues to be 56 leagues , and so much you are departed from your first meridian to the west-ward ; which when you are near the equinoctial , where the degrees of longitude are equal to the degrees of latitude , would shew the longitude , by taking 20 leagues for one degree , &c. so it would be two degrees , and 48 min. for your difference of longitude , from y●ur first meridian af. but in other places , you must first 〈◊〉 howmany leagues make a degree of longitude about that latitude where you are , and so turn your leagues of distance from the meridian , into degrees and minutes of longitude , of which more hereafter , chap. 14. i have been the larger in these two propositions , because they are ●he first , for the better understanding of all the rest ; and because they are most necessary , for thereupon depends the knowledge of the true traverse point , and the keeping of your dead reckoning . now because this cannot alwaies be kept exactly , it is to be corrected by the observation of the latitude , according to this following proposition . chap. x. knowi g the difference of latitude of two places , and the rumb you have sailed upon , to find the leagues you have sailed , and the difference of meridians . the pole depressed four degrees and the rumb south-west by south or the third from the meridian , to find your true traverse point , viz. how far you have sailed , and how much you are departed from your first meridian . in the first figure draw the lines as in the former chapter , so that akf may represent the meridian line , and acd may represent the third rumb from the meridian ; then because you have altered your latitude 4 degrees , which make 80 leagues , take 80 leagues with your compasses out of your scale , and set them upon the meridian line af , from a to l : then keeping the same distance of your compasses , draw the line lm parallel to ab , ( or else you may erect lm perpendicular to the line af , in the point l ) and mark where the said lm crosseth the rumb line acd , which is in the point m. this point m is the true traverse point , the leagues sailed are shewed by the line am , which being measured in the scale , will be found to be 96 leagues and an half , and the departure from the meridian is lm , which is 54 leagues . now by this proposition ( as i said ) you may correct your dead reckoning ; for suppose by the former proposition you reckon you had sailed 100 leagues upon the ●hird rumb , then as you see there , you should have been at the point d , and have altered your latitude 83 leagues , and departed from your meridian 56 leagues ; but now suppose that by a good observation of the latitude , you find that you have altered the latitude onely 80 leagues , from a to l , by drawing this line lm , which crosseth the rumb or ships way in m , you may conclude your true traverse point to be at m , so that you have sailed only from a to m , which is 96 leagues ½ , and departed from your meridian 54 leagues . so that as you are short of the latitude you reckoned for 3 leagues or 9 min. you are also short of your way you reckoned 3 leagues ½ , and two leagues less in your departure from the meridian . and this you must account for your true reckoning , being thus corrected . chap. xi . by the difference of the latitudes of two places and the distance between their meridians , to find the rumb by which you must sail from the one place to the other , and how far it is from the one place to the other ? the difference of latitude between the two places is 4 deg . 9 min. and the distance between the two meridians is 56 leagues , and it is required to find the rumb from the one place to the other . in the former figure draw the quadrant akcb , then turn your four degrees 9 min. of latitude into leagues , it maketh 83 leagues , which you must place upon the meridian line from a to f. and from the point f draw the line fd parallel to the line ab . then open your compasses to the distance of the meridians which is 56 leagues , and set it on the line fd , from f to d. then lay your ruler by this mark d and the centre a , and draw the line acd . then mark where this line cuts the quadrant , which is in the point c , and setting one foot of your compasses in the point c , open the other to k , and keeping your compasses at that distance ck , measure it upon your scale , either in the line of chords , or in the line of rumbs , you sh ll find it to be in the one 33 deg . 45 min. and in the other just the third rumb from the meridian . so that the rumb from a 〈◊〉 d ●s south-west and by south , and the rumb from d to a is the rumb opposite thereunto , which is north-east and by north. then for the distance between the two places in the rumb , ●et one foot of your compasses in the one place at a , and open the other to the other place at d , and the length of the line a d ineasured in the scale of leagues , shews the distance between them to be just 100 leagues . these three ( or rather these six ) propositions , ( for they are each of them double ) are the most usefull and necessary in the art of navigation . by the first of these , knowing the point of the compass you ●ail upon , and judging howmany leagues you have sailed thereon , you know and are able to give a reasonable account where you are , both in respect of latitude and longitude . by the second having a fair observation of the latitude at any time , you may more perfectly know where you are ; and thereby correct your former account . and by this third you may know how to direct your course from any place to your desired haven . so that in effect you need no more , but yet for your better instruction by variety of cases and examples , i shall proceed . chap. xii . the difference of latitude and the lea●●es sailed being given , to find the distanee from the meridian , and the rumb you have sailed upon . sailing 100 leagues between south and west , untill the pole be depressed 4 deg . 9 min. the distance from the meridian is demanded , and what rumb you have sailed upon ? in the first figure draw the quadrant akcb , as in the former chapters , and then reduce your degrees of latitude into leagues , so 4 deg . 9 min. make 83 leagues , which you must take with your compasses out of your scale of leagues , and set them off in this meridian line from a to f. then from the point f draw the line fd , parallel to the line ab , which you may do with the foresaid distance of your compasses . then open your compasses unto your distance failed , which is 100 leagues , and setting one foot of your compasses in the point a , with the other draw the little arch hg , cutting the line fd in the point d. so the line fd measured in the scale of leagues , shall shew you the distance from the m ridian , which is 56 leagues , and if you draw the line acd , it i● the rumb line upon which you have sailed , and the arch kc 〈◊〉 ed in the scale of rumbs , shews it to be the third rumb from the meridian , or south-west by south . chap. xiii . to find the distance of any island from you , that you may discern at two stati n● , knowing the po●ut of the compasse , the island beareth unto each of the stations . suppose , being at sea you discover an island bearing north-east off you , which place let it be your first station , and then sailing seven leagues full north you observe the island to bear full east off you , which let be the second station ; the aemand is to find the distance of the said island from both the said stations ? in the second figure , or demonstration , let a be the first station , and upon the center a draw the quadrant abde ; then in regard you found the island to bear north-east from you , take 4 of your 8 points of the compass our of the scale , and place them upon your quadrant from b to d , then from the center a by the point d , draw the line adf , representing the visuall line passing between your sight and the island , being at the first station a. then seeng when y●● had sailed 7 leagues north , you observed the island to bear full east off you , set off the said 7 leagues from a to c , ( reckoning every 10 leag● s of your scale to be but on● ) and from this point c , which is the second station , draw the line c f parallel to ae , and it will cut the line adf in the point f : so shall the point f , be the place of the island desired , and the distance af , is the distance of the island from the first station , viz. 9 leagues 90 parts or almost 10 leagues : likewise the distance from c , to f , is the distance of the island from the second station , which is just seven leagues . and by this manner of work , you may find the ▪ distance of any island or head land from you , or you may take the distances of as many places as you will or can see at any two such stations , and by the crossing of their visuall lines , find their position and distances each from other . chap. xiv . to find how many leagues , miles , and parts do make one degree of longitude in every latitude . note , all this while we have been sailing according to the rules of the plain chart , which supposith the degrees of longitude to be equal to the degrees of latitude , in all latitudes , but that is very false and erroneous ; it being true onely in places near the equinoctiall , where every degree of longitude contains 20 leagues , as the degrees of latitude do ; but in places neer the poles it alt●rs very much , so that in the latitude of 60 degrees , 10 leagues make a degree of longitude : and in other latitudes the degrees of longitude alter , as in this little table , which shews at what degree and minute of latitude , any nnmber of leagues make a degree of longitude , by which you may divide a line upon your scale for your ready use . leagues in one degree . 20 00 d 00 m 19 18 11 18 25 50 17 31 47 16 36 52 15 41 25 14 45 34 13 49 27 12 53 08 11 56 38 10 60 d 00 m 9 63 15 8 66 25 7 69 31 6 72 32 5 75 31 4 78 28 3 81 22 2 84 16 1 87 08 now to return to the question , and shew you by demonstration how to find how many leagues , miles , and parts , make a degree of longitude in any degree of latitude ? the larger you make your quadrant , the more exact will the work be , and shew the leagues and miles more exactly , which you may make into a table , as this following . a table shewing how many leagues , miles , and hundred parts of a mile make one degree of longitude in any latitude . latitude leagues miles parts difference   latitude leagues miles parts diffe ence   latitude leagues miles parts diff●rence 0 20 0 0 —   30 17 0 96 —   60 10 0 0 — 1 19 2 99 1   31 17 0 43 53   61 9 2 09 91 2 19 2 96 3   32 16 2 88 55   62 9 1 17 92 3 19 2 92 4   33 16 2 32 56   63 9 0 24 93 4 19 2 85 7   34 16 1 74 58   64 8 2 30 94 5 19 2 77 8   35 16 1 15 59   65 8 1 36 94 6 19 2 67 10   36 16 0 54 61   66 8 0 40 96 7 19 2 55 12   37 15 2 92 62   67 7 2 44 96 8 19 2 42 13   38 15 2 28 64   68 7 1 47 97 9 19 2 26 16   39 15 1 63 65   69 7 0 50 97 10 19 2 09 17   40 15 0 96 67   70 6 2 52 98 11 19 1 90 19   41 15 0 28 68   71 6 1 53 99 12 19 1 69 21   42 14 2 59 69   72 6 0 54 99 13 19 1 46 23   43 14 1 88 71   73 5 2 54 100 14 19 1 22 24   44 14 1 16 72   74 5 1 54 100 15 19 0 96 26   45 14 0 43 73   75 5 0 53 101 16 19 0 68 28   46 13 2 68 75   76 4 2 52 101 17 19 0 38 30   47 13 1 92 76   77 4 1 50 102 18 19 0 06 32   48 13 1 15 77   78 4 0 48 102 19 18 2 73 33   49 13 0 36 79   79 3 2 45 103 20 18 2 38 35   50 12 2 57 79   80 3 1 42 103 21 18 2 1 37   51 12 1 76 81   81 3 0 38 104 22 18 1 63 38   52 12 0 94 82   82 2 2 35 103 23 18 1 23 40   53 12 0 11 83   83 2 1 31 104 24 18 0 81 42   54 11 2 27 84   84 2 0 27 104 25 18 0 38 43   55 11 1 41 86   85 1 2 23 104 26 17 2 93 45   56 11 0 55 86   86 1 1 18 105 27 17 2 46 47   57 10 2 68 87   87 1 0 14 104 28 17 1 98 48   58 10 1 80 88   88 0 2 09 105 29 17 1 48 50   59 10 0 90 90   89 0 1 05 104 30 17 0 96 52   60 10 0 0 90   90 0 0 0 105 chap. xv. the difference of latitude , and the rumb or distance sailed being known , to find the distance of the meridians , and thereby to find the degrees and minutes of the difference of longitude in any latitude . sailing from the north parallel of 56 degrees and 5 min. latitude , 100 leagues upon the third rumb from the meridian ▪ viz. south-west and by south untill i find the pole is depressed 4 deg . 9 m. and the meridional distance 56 leagues ; the longitude is desired thereby ? i● the first figure now to reduce this 56 leagues into degrees of longitude , you must consider from what latitude you have sailed , and to what latitude you are come , viz. from latitude 56 d. 5 m. to 4 deg . 9 min. lesse , which is 51 d. 56 m. and take the middle latitude ( or somewhat more ) between the two places , which in this example falls out to be 54d . 01 m. then by the table in the former chapter , find out howmany leagues and miles in the said middle latitude make one degree of longitude , and you shall find in that table , that in the latitude of 54 d. there is but 11 leagues , and 2 miles , and 27 parts in one degree of longitude ; therefore open your compasses upon your scale of leagues , to this 11 leagues , 2 miles , 27 parts , and keeping your compasses at that distance , set one foot of them at 56 leagues in your scale of leagues , or in the line df in the figure , ( or upon the like line in your chart at any time ) either at f or d , and measure howmany times you find that distance either to the end of your scale coming backward , or in the line df , for so many degrees is the difference o● longitude , and if any odde part remain , you may proportion i● by your eye , judging it to be a quarter , a third , an half , or any part more or lesse of a degree , which you may either reckon by parts , or 15 , 20 , 30 &c. minutes , thus this line df being 56 leagues , opening your compasses to 11 leagues 2 miles 27 parts , you will find this distance in it , 4 times and 3 quarters ; so that the difference of longitude is 4 deg . 45 min. or you may reduce it into miles and work by the rule of proportion , so you shall find as 11 leagues , 2 miles , 27 parts , that is 35 miles 27 parts . 35,27 to one degree of longitude in the latitude of 54 d. 01,00 so is 56 leagues , or 168 miles . 168,00 to 4 degrees , 76 parts . 04 76 but if your scale be large , the other way with your compasses will give you the degrees and parts of longitude as exactly as you need for most uses . also if the latitude fall not out in equal parts , you may find out for your odde minutes by proportion , for which purpose i have set the differences between each degree in the table . so that as one hundred parts or 60 minutes being one degree , to the difference in the table between the two next degrees ; so the odde hundred parts or minutes of latitude , to the parts and minutes proportional to be allowed . chap. xvi . sailing from the south latitude of 60 degrees , 51 min. and from longitude 25 degrees , 24 min. 99 leagues , upon a south-west course : the latitude and longitude of the second place is demanded . in the second demonstration , draw the quadrant abcde , as is formerly taught , then in regard you sail south-west , take 4 points of the compasse from your scale , and place them from b unto d , then by the point d draw the line adf , then place your ninety nine leagues upon the line adf , from a unto f , so shall f be the place of your ship. then from f draw the line fc parallel unto ae , cutting the line abc in c , so shall the distance ca be the leagues you have run south , which is seventy leagues , or 3 deg . 30 minutes , which being added to the latitude from whence you dearted , makes 64 deg . and 21 minutes for the latitude of the second place : then take the distance cf , and apply it unto the line of equal parts , and you shall find it likewise 70 leagues : then finding the middle latitude 62 degrees 36 minutes in the table , chap. 14. you shall find that 9 leagues and 0 miles , and 61 parts , do alter a degree of longitude in that latitude . then opening the feet of your compasses to 9 leagues 0 miles 61 parts , in the scale of equall leagues , and keeping the compasses at that distance , see howmany times that distance is in the line cf , which is seven times and somewhat above an half , the true difference of longitude being 7 deg . 36 m. which being substracted from the longitude from whence you departed , leaves 17 degrees and 48 minutes for the longitudeof the second place . chap. xvii . a ship sayling from the north parallel of fifty degrees , having an hundred leagues to sail south-west , and by west , by the way is enforced by contrary winds to sail upon several points of the compasse , first sailing thirty leagues upon a direct course , then west north-west twenty leagues , then south sixty leagues , the question is to find the latitude of the second place , how far it is to the place whereuuto you are bound , the distance of the rumb that is betwixt them , the distance that you are from your first meridian , and thereby the difference of longitude . in the third demonstration , draw the line ad , and from the point a , raise the perpendicular ab , then open your compass unto the radius of your scale , and place one foot thereof in the center a. and with the other draw the quadrant bcd , then take three points of the compasse & place them upon the quadrant from d. unto c , then from the center a , by the point c , draw the line acl , 100 leagues in length , which is the true course you are to saile , then in regard you sayled thirtie leagues direct , take thirtie leagues from your scale of equall parts , and place them upon the line aec , it extends from a unto e : then in regard you turned your course , west , northwest , from the center e , draw the line eg parallell unto a. d. and again from the center e draw the line eh perpendicular to eg , and parallell to ab , then witn the distance of the radius , set one foot of your compasses in the center e , and with the other draw the quadrant gmh , and in regard you sayled west , northwest , which is two points from the west northward , take from your scale two points of the compass , and place them upon the quadrant gmh , from g unto m , then from the center e unto the point m , draw the line efm , then take 20 leagues with your compasses from the scale of equall parts , and place them upon the line efm , from e unto f , then is your ship in the point f. lastly , in regard you run south 60 leagues from f , draw a line parallell unto the meridian ab , which is the line fi , then take from your scale of equall parts sixtie leagues , and place them from f , unto i , then is your ship in the point i : then last of all is to be found how far it is to the place where unto you were bound , the distance of the rumb that is betwixt you , the degrees and minutes you have raised the pole , the distance of departure from the first meridian , and thereby t●e difference of longitude : and that you may so doe , first draw the line oik , perpendicular unto the line if in the point i , and with your compasses opened unto the distance of the radius , set one foot of your compasses in the center i , and with the other draw the quadrant knf , then in regard your ship is in the point i , and the place whereunto you are bound is the point l , therefore from i , thorow the point l draw the line iln , cutting the arch knf , in the point n , therefore let il , be the leagues you have unto the place whereunto you are bound , which is fortie one leagues and a halfe , and the rumb the distance kn , which is west , and by north , and three degrees unto the northward , so likewise is the line ao , the number of leagues you have run due south , which is sixtie eight leagues and one mile , or three degrees and twenty five minutes , which being taken from fiftie degrees , the parallell from which you departed , leaves fortie six degrees and thirtie five minutes for the parallel you are in . last of all , shall the line io , be the leagues that you have departed your first meridian , which are fortie two leagues and one mile , then take the middle latitude which is fortie eight degrees seventeen minutes and in the table chap. 14 you shall find that thirteen leagues 0. mile , 92 parts , do answer unto a degree of longitude in that parallell ; then setting one foot of your compasses in thirteen leagues , and ninety two parts , extending the other to the beginning of the scale , keeping the compasses at that distance , turn them over the line i o , and you shall find it contains that distance three times and almost a quarter , so the difference of longitude is three degrees eleven minutes . chap. xviii . two ships departing from one parallel , and port , the one in sayling eight leagues betwixt the north , and the west , hath raised the pole two degrees , the other in sailing a hundred leagues betwixt the north , and west , hath raised the pole four degrees , i demand by what rhumbs the said ships have sailed , and the rhumb and distance that is betwixt them ? in the fourth demonstration , draw the quadrant abcde , then in regard the first ship hath raised the pole two degrees , which is fortie leagues , take fortie leagues off your scale , and applie them unto the meridian line agl , from a unto g : then from the point g , draw the line gf , parallel unto ab , then opening your compasses unto 80 leagues , set one foot in the center a , with the other make a marke in the line gf , which will be at f , so shall f be the place of the first ship ; the second ship hath raised the pole four degrees , which is 80 leagues , therefore place 80 leagues upon the meridian line agl , from a unto l , and from the point l draw the line lm , parallel unto ghf , then open your compasses unto the distance of a hundred leagues , which are the leagues the second ship did run , and set the foot of your compasses in the center a , and with the other make a mark in the line lm , which will be at m , then draw the line ma , which is the course of the second ship , and the line fa , is the course of the first ship , then from f let a perpendicular fall , being perpendicular to the line gf , which is the line fk , then opening your compasses unto the radius of your scale , set one foot in the center f , and with the other draw the quadrant hik , likewise from f , the place of the first ship , draw a line by the point m , the place of the second , cutting the quadrant khi , in i , so let ik , be the course that is betwixt them , that is , if you will saile from the first ship unto the second , you must saile north and by east , and one and fortie minutes to the eastward , likewise let f m , be the distance that is betwixt them , which in this demonstration is fortie leagues , two miles , so shall bc , be the course of the first ship from the west northward , wh ch here is found to be thirtie degrees and one minut from the west northward , or northwest by west , and three degrees and fortie four minutes to the west ward . lastly the arch ed , is in the distance of the course that the second ship made from the north westward , which is found by this demonstration to be northwest and by north , and three degrees five minutes to the westward . chap. x●x . two ships departing from one parallell and port in the parallell of 47 deg . 56 min. the first in sayling 80 leag . betwixt the north and west , hath raised the pole two degrees , i demand by what course the second ship must runne , and how much she shall alter in her first meridian or longitude , to bring her selfe 40. leagues and two miles north and by east , and 41. minutes to the eastward of the first ship ? in the fourth demonstration draw the quadrant abcde , then multiplie your two degrees you have altered your latitude by twentie and it maketh fourtie leagues ; which fourtie leagues set upon the line ael , from a unto g , then from the point g draw the line gf , parallel unto ab , then open your compasses unto the distance of 80 leagues , which are the leagues your first ship did runne , and place one foot of your compasses in the center a , and with the other make a marke in the line gf , which will be at the point f , then from the center a unto the point f draw the line af , representing the distance of the course of the first ship 80 leagues : then from f let fall a perpendicular fk , and upon the center f , with the radius of the scale draw the arch hik , then in regard you must bring the second ship north and by east , and 41 minutes eastward of the first ship , take 11 degrees 56 minutes from your scale of chords , and place them from kunto i , upon the quadrant kih . then from f draw the line if , and upon the line , fi , place the distance that you must bring the second ship from the first ( which is fourty leagues and two miles ) from f unto m. so is m the place of your second ship . then from m draw the line ml parallel unto fg , cutting the line agl in l , then draw the line ma , cutting the quadrant bde in d. so shall the arch de be the course that the second ship must run , to bring her self fourty leagues and two miles north and by east , and 41 minutes east of the first ship . then to know what you have altered the latitude , first take the distance la and apply it unto the scale of equall parts , and you shall find it to be 80 leagues , which is just 4 degrees , which you have altered your latitude , or poles elevation : which 4 degrees added unto the latitude you depar ed from , it makes 51 degrees 56 min. for the latitude that your second ship is in , then take the distance lm and apply it to the scale , it gives 60 leagues ; then open your compasses unto the distance of the middle latitude , which is 40 deg . 5● min. of the chord , and apply it unto the table of longitudes , and it gives 12 leagues , and 2 miles , and 62 parts , to alter one degree of longitude in that parallel : then set one foot of your compasses in 12 leagues 2 miles , and 62 parts , and open the other to the beginning of the line , and with that distance measure the line l m , being 60 leagues , and you shall find that it is contained there in four times and two thirds , so the longitude is 4 degrees 40 minutes . chap. xxi . of the ebbing and flowing of the sea , aud of the tides , and how to find them in all places . a generall table for the tides in all places . the moons age . hours and minutes to be added . hours and minutes to be added .   the moons age . hours and minutes to be added : hours and minutes to be added : daies . degrees : minutes :   daies . degrees : minutes : 1 0 48   16 0 48 2 1 36   17 1 36 3 2 24   18 2 24 4 3 21   19 3 12 5 4 0   20 4 0 6 4 48   21 4 48 7 5 36   22 5 36 8 6 24   23 6 24 9 7 12   24 7 12 10 8 0   25 8 0 11 8 48   26 8 48 12 9 36   27 9 36 13 10 24   28 10 24 14 11 12   29 11 12 15 0 0   30 0 0 the use of the table of the tides . first it is to be understood , that by the swift motion of the first mover , the moon and all the rest of the stars and planets , are turned about the world in four and twenty hours , upon which swift motion of the moon , the daily motions of the sea , do depend , which motion of the sea falleth not out alwaies at one hour , the reason thereof is , because of the swift motion of the moon in regard she goeth almost thirteen degrees in four and twenty hours , and the sun moveth scarce one degree , which gives every day twelve degrees , that the moon cometh slower to any point in the heaven than the sun : which twelve degrees makes fourty eight minutes of time for the difference of every full sea , according unto the middle motion of the moon , which difference is here set down in this table for every day of the moons age . therefore if you would know the full sea at any place in the world , first you must know at what hour it is full sea at the new or full moon ; which hours and minutes keep in mind , then seek the age of the moon as is before taught , and with the number of her age enter this table , under the title of the moons age , and having found her age in the table , against it you shall find the hours and minutes which are to be added unto the time that the moon maketh full sea in any place , and the whole number of hours and minutes is the time that the moon maketh full sea in that place upon the day desired . as for example , i desire to know the full sea at london bridge upon the 13 of july 1624. the age of the moon being found as before , is eight daies , then in the table i find eight daies , and against it 6 hours , and 24 minutes , which being added unto 3 hours , the full sea upon the change day gives 9 a clock 24 minutes for the time at the full sea upon the 13 day of july 1624. the sea-mans glasse . the second book . vvherein is declared the definition of the sphear , a description of the six great circles , and also of the four lesser circles , last of all , certain questions astronomicall , performed by the said scale . chap. i. of a sphear , and the circles thereof . the figure of the plaine scale . a sphear according to the description of theodosius , is a certain solid sup● ficies , in whose middle is a point , from which all lines drawn unto the circumference are equall ; which poi●● is called the center of the sphear , by which c●●●er a right line being drawn , and excending himself on either side unto that part of the circumference whereupon the sphear is turned , is called axis spherae , or the axle-tree of the world. a sphear accidentally is divided into two parts , that is to say , in sphaeram rectam & sphaeram obliquam . sphaera recta , or a right sphear , is onely unto those that dwell under the equinoctiall , quibus neuter polorum magis altero elevatur : that is , to whom neither of the poles of the world are seen , but lie hid in the horizon . sphaera obliqua , or an oblique sphear , is unto those that inhabit on either side of the equinoctial , unto whom one of the poles is ever seen , and the other hid under the horizon . the circles whereupon the sphear is composed are divided into two sorts : that is to say , in circulos majores & minores . circuli majores , or the greater circles , are those that divide the sphear into two equall parts : and they are in number six , viz. the equinoctial , the middle of the zodiack , or the ecliptique line , the two colures , the meridian , and the horizon . minores vero circuli , or the lesser circles , are such as divide the sphear into two parts , unequally , and they are four in number ; as the tropick of cancer , the tropick of capricorn , the circle artike and the circle antartike . chap. ii. of the six greater circles . i. the equinoctial is a circle that crosseth the poles of the world at right angles , and divideth the sphear into two equall parts , and is called the equinoctial , because when the sun cometh unto it , ( which is twice in the year , viz. in principio arietis , & librae , that is , in march and september ) the daies and nights are equal thoroughout the whole world , whereupon it is called equator diei & noctis , the equall proportioner of the day and night artificiall : and in the figure is described by the line cae . ii. the meridian is a great circle passing thorow the poles of the world , and the poles of the horizon , or zenith point over our heads ; and is so called , because that in any time of the year , or in any place of the world , when the sun ( by the motion of the heavens ) cometh unto that circle , it is noon , or twelve of the clock . and it is to be understood , that all towns and places that lie east and vvest one of another , have every one a severall meridian : but all places that lie north and south one of another , have one and the same meridian . this circle is declared in the figure following by the circle bcde . iv. the two colures , colurus solstitiorum , or the summer colure , is a circle passing by the poles of the world , and by the poles of the ecliptick , and by the head of cancer and capricorn , whereupon , the first scruple of cancer , where the colure crosseth the ecliptick line , is called punctus solstitiae aestivalis , or the point of the summer solstice : to which place when the sun cometh , he can approach no nearer unto our zenith , but returneth unto the equator again . arcus vero coluri , the ark of the colure contained betwixt the summer solstice and the equator , is called the greatest declination of the sun , which ptolomy found to be 23 degrees , 31 minutes : but by the observation of copernicus it was found to vary , for ●e found the declination sometimes to be 23 degrees 52 minutes , and in the processe of time to be but 23 degrees 28 minutes . and in these our daies ( by the observation of ticho de brahe , and that late famous mathematician , mr. edward right ) it is found distant from the equinoctiall 23 degrees , 31 minutes , 30 seconds . v. the other colure passeth by the poles of the world , & by the first point of aries and libra , whereupon it is called colurus distinguens equinoxia . these two colures do crosse each other at right angles in the poles of the world , whereupon these , verses were made . haec duo solstitia faciunt cancer capricornus , sed noctes aequant aries & libra diebus . chap. iii. of the four lesser circles . the sun having ascended unto his highest solstitial point doth describe a circle , which is the nearest that he can approach unto the north pole , whereupon it is called circulus solstitii aestivalis , the circle of the summer solstice , or the tropick of cancer , and is noted in the figure before , by the line h y i. the sun also approaching unto the first scruple of capricornus , or the winter solstice , describeth another circle , which is the utmost bounds that the sun can depart from the equinoctiall line towards the antartike pole , whereupon it is called circulus solstitii hyemalis , sive tropicus hyemalis , vel capricorni : the circle of the winter solstice , the vvinter tropick , or the tropick of capricorn , and is described in the figure by the line gxf. so much as the ecliptick declineth from the equinoctiall , so much doth the poles of the ecliptick decline from the poles of the vvorld , whereupon the pole of the ecliptick , which is by the north pole of the vvorld , describeth a certain circle as it passeth about the pole of the vvorld , being just so far from the pole as the tropick of cancer is from the equator , and it is the third of the lesser circles , and is called circulus arcticus , or the circle of the north pole , and is described in the diagram , in the second chapter by the line po. the fourth and last of the lesser circles is described in like manner , by the other pole of the ecliptick , about the south pole of the world , and therefore called circulus antarcticus , the antarctick circle , or the circle of the antarctick or south pole , and is demonstrated in the former figure , by the line nm . chap. iv. definitions of some peculiar terms fit to be known by such as intend to practice the art of navigation or astronomy . the zenith is an imaginary point in the heavens over our heads , making right angles with the horizon , as the equinoctiall maketh with the pole. the nadir is a prick in the heavens under our feet , making right angles with the horizon under the earth , as the zenith doth above , and therefore is opposite unto the zenith . the declination of the sun is the ark of a circle contained betwixt the place of the sun in the ecliptick , and the equinoctiall , making right angles with the equinoctiall . but the declination of a star is the ark of a circle let fall from the center of a star , perpendicularly unto the equinoctiall . the latitude is the ark of a circle contained betwixt the center of any star , and the ecliptick line , making right angles with the ecliptick , and counted either northward , or southward , according to the scituation of the star , whether it be nearer unto the north or south pole of the ecliptick . the latitude of a town or countrey , is the height of the pole above the horizon , or the distance betwixt the zenith and the equinoctiall . the longitude of a star is that part of the ecliptick which is contained betwixt the stars place in the ecliptick , and the beginning of aries , counting them from aries according to the succession or order of the signes . the longitude of a town or countrey are the number of degrees , which are contained in the equinoctiall , betwixt the meridian that passeth over the isles of azores , ( from whence the beginning of longitude is accounted ) east wards , and the meridian that passeth over the town or country desired . the altitude of the sun or star is the arch of a circle , contained betwixt the center of the sun , or any star , and the horizon . the amplitude is that part of the horizon which is betwixt the true east or west points , and the point of the compasse that the sun or any star doth rise or set upon . azimuth's are circles , which meet together in the zenith , and crosse the horizon at right angles , and serve to find the point of the compasse , which the sun is upon at any hour of the day , or the azimuth of the sun or star , is a part of the horizon contained betwixt the true east or west point , and that azimuth which passeth by the center of the same star to the horizon . the right ascension of a star is that part of the equinoctiall that riseth or setteth with the star , in a right sphere : or in an oblique sphere , it is that portion of the equinoctiall , contained betwixt the beginning of aries , and that place of the equinoctiall , which passeth by the meridian with the center of the star. the oblique ascension is a part of the equinoctiall , contained betwixt the beginning of aries , and that part of the equinoctiall that riseth with the center of a star , in an oblique sphere . the difference ascensionall , is the difference betwixt the right and oblique ascension : or it is the number of degrees contained betwixt that place of the equinoctiall that riseth with the center of a star , and that place of the equinoctiall that cometh unto the meridian , with the center of the same star. almicanterahs are circles drawn parallel unto the horizon , one over another , untill you come unto the zenith : these are circles that do measure the elevation of the pole , or height of the sun , moon , or stars above the horizon , which is called the almicanter of the sun , moon , or star : the ark of the sun or stars almicanter , is a portion of an azimuth contained betwixt that almicanter which passeth thorow the center of the star , and the horizon . questions astronomical , performed by the plain scale . chap. v. the true place of the sun being given , to find his declination . the sun being in the head of taurus , his declination is desired . by the seventh demonstration , draw the line ad , then upon the center a raise the perpendicular ab , then opening your compasses to the radius of your scale , place one foot in the center a , and with the other draw the quadrant bcd , then opening your compasses unto the greatest declination of the sun , place it upon the quadrant , from d unto k , then from the point k draw the line kh , parallel to da , cutting the line ab in h , then with the distance ah draw the small quadrant geh , and in regard the sun is in the head of taurus , which is 30 degrees from the beginning of aries , let ad be the equator , and d the beginning of aries , dc 30 degrees , or longitude of the sun , then from the point c draw the line ca , cutting the quadrant geh in e , then from e draw the line ei parallel to ad , cutting the quadrant bcd in i , so shall the arch id be the declination of the sun desired , which in this demonstration is found to be eleven degrees , and thirty one minutes . chap. vi. the declination of the sun , and quarter of the ecliptick that he possesseth , being given , it is desired to find his true place . the declination is 10 deg . 31 min. the first quarter that he possesseth , is betwixt the head of aries and cancer . first , by the seventh demonstration , draw the quadrant abcd , as is taught in the former chapter , then set the greatest declination of the sun upon the chord from d unto k , which is 23 deg . and 31 min. then from k draw the line kh parallel unto the equator da , cutting the line ba in the point h. so shall ha be the sign of the suns greatest declination , then with the distance ah draw the quadrant geh , then from d upon the quadrant dbc set the declination of the sun , which is 11 degrees 31 minutes from d unto i , then draw the line ie parallel unto ad , cutting the quadrant geh in e. then from the center a by the point e , draw the line aec , cutting the quadrant bcd in c. so shall the ark cd be the distance of the sun from the head of aries , which is here found to be just 30 degrees , which is in the beginning of taurus . chap. vii . by the elevation of the pole , and declination of the sun , to find the amplitude of the sun , or his distance of rising , or setting from the true east or west point . the elevation of the pole is 51 deg . 32 min. the declination of the sun is 14 deg . 52 min. north. by the eight demonstration , first draw the line bd , then upon the center a draw the circle bcde , then from a raise the perpendicular cae , then is your circle divided into four equall parts : then suppose the elevation of the pole to be 51 degrees , 32 minutes , which must be placed upon the circle , from d unto f , then from the point f , by the center a , draw the line fag , representing the pole of the world , f being the north pole , and g the south pole , then substract 51 deg . 32 min. from 90 deg . and the remainder is the height of the equinoctiall , which is 38 deg . 28 min. which must be placed upon the circle from the horizon b , unto the point i , then from i , by the center a , draw the line iah , representing the equinoctiall circle . then from i unto m set the declination of the sun , being here supposed 14 deg . 52 minutes north , then from the point m draw the line , or parallel of declination mtn , parallel unto the equator i a h , cutting the horizon bd in t , then from t raise the perpendicular tv , cutting the circle bcde in v , so shall the distance cv be the true amplitude of the sun desired , which here is found to be 24 deg , 21 minutes north. chap. viii . by the amplitude of the sun , to find the variation of the compasse . having found the amplitude of the sun by the last chapter , first observe with a compasse , or rather with a semicircle , upon what degree and minute the sun riseth or setteth , beginning to reckon from the east or west , and ending at the north or south at 90 degrees : and when you have diligently observed the magneticall rising or setting , by the semicircle , or by some other like fitting instrument : and also the true amplitude found , as is declared in the last chapter , the difference of these two amplitudes , is the variation of the compasse : but when the sun riseth upon the same degree of the compasse , as is found by the scale , the variation is nothing , but the needle pointeth directly unto the poles of the world , which by m. mulinux was affirmed to be at the westernmost part of s. michaels , one of the islands of the azores , from whence he will have the longitude reckoned . secondly , when the sun is in the equinoctial circle , where he hat● no amplitude , look what distance the compasse maketh the sun to rise from the east or west of the compasse , the same distance is the compasses variation , from the north or south . thirdly , if the sun rise more to the south of the compasse , or setteth more to the north of the compasse , than is shewed by the scale , the difference betwixt the amplitude given by the scale , and the amplitude given by the needle , is the variation of the compasse from the north westward . fourthly , if the compasse sheweth the sun to rise more northward , or set more southward , than is shewed by the scale , the difference is the variation of the compasse , from the north eastward . fifthly , if the scale shew the amplitude of the sun rising southerly , and the compasse shew it to be northerly , adde both the amplitudes together , and they shew you the variation westernly . chap. ix . the place of the sun being given , to find his declination , by a whole circle . the suns place is the tenth degree of taurus . according unto the eighth demonstration , first draw the circle bcde , then draw the horizon bad , and then the equinoctial iah , as is before taught : and then the tropick of cancer kl , twenty three degrees and a half from the equinoctial : then draw the tropick of capricorn po , of like distance from the equinoctial , and after from k to o draw the ecliptick line kao. and when you have thus laid down the sphere , suppose the sun to be in the tenth degree of taurus , at which time his declination is desired . and in regard the sun is more near unto the tropical point cancer , than unto capricorn ; first find how many degrees he is from the tropick of cancer , and you shall find him to be 50 degrees ; therefore take with your compasses 50 degrees from the chord , and apply it from the tropical point cancer at k , unto v , upon one side , and unto p on the other side : then draw the line vp , cutting the ecliptick ko in the point r , then from r draw the line mrn parallel unto the equinoctial iah , and cutting the quadrant bc in the point m. so shall the arke mi be the declination of the sun desired , which being applyed unto your scale , gives you 14 deg . and 52 minutes . chap. x. the elevation of the pole , and declination of the sun given , to find his height in the vertical circle . the pole is elevated 51 degrees 32 minutes , the declination of the sun is 14 degrees 52 minutes north , his height in the verticall circle is found as followeth . first , according unto the former chapter , draw the circle bcde , then the horizon bad , and after the verticall line cae , then the axis of the world fg , and likewise the equator iah , this being done , place the declination of the sun 14 degrees 52 minutes , upon the circle from i unto m , and also from h unto n , then draw the line mn , cutting the line cae in s , then from s draw the line svv , parallel unto the horizon bad , cutting the meridian circle bcde in vv : so shall the distance dvv be the height of the sun in the vertical circle , for the time demanded , which by this proposition is found to be 19 degrees and 8 minutes . chap. xi . the elevation of the pole , and the amplitude of the sun , being given , to find the declination . the elevation of the pole is 51 degrees 32 minutes , the suns amplitude is 24 degrees 21 minutes , the declination is found as followeth . first , as in the eight demonstration , upon the center a , draw the circle bcde , then draw the line bad , representing the horizon : dividing the circle into two equall parts then draw the line cae , perpendicular to bad , representing the east and vvest points of the compasse , then placing the elevation of the pole 51 degrees and 32 minutes , from d unto f , from f , by the center a ▪ draw the line fag , which let be the pole or axletree of the world , then from b unto i , and from d unto h , set the complement of the poles elevation : which shall represent the equinoctiall , in regard it maketh right angles with the pole of the world , in the center a. then from c unto v place the amplitude of the sun , which is 24 degrees and 21 minutes : then from v let fall the perpendicular vt , cutting the horizon bad in the point t , then from the point t , draw the line mtn parallel unto the equinoctiall iah , and cutting the circle bcde in the points , m and n , so shall the distance , m , or hn , be the declination of the sun , which was desired : which being applied unto your scale , gives you fourteen degrees and fifty two minutes . chap. xii . the elevation of the pole , the declination of the sun , and hour of the day being given ▪ to find the almicanter . the elevation of the pole is thirty degrees , the declination of the sun is twenty degrees north , the hour is nine in the morning , at which time the almicanter is found , as followeth . by the ninth demonstration , first upon the center a , draw the circle bcde , then draw the line bd for the horizon , then place your poles elevation , which is thirty degrees , upon the circle from d unto r , then from r by the center a , draw the line ras , representing the axis of the world , then from b unto f place the complement of the poles elevation , which is ●0 degrees , and from the point f , by the center a , draw the line fah , representing the equinoctial line , and then set the declination of the sun from f unto l ▪ and from l draw the line lpo parallel unto the equator fah , cutting the axis of the world in the point p , then set one foot of your compasses in the point p , and extend the other either unto l or unto o , and with the same distance of your compasses , upon the center p , draw the circle lnoq , which is called the hour circle : so shall l be the point of twelve a clock at noon , n the place of six a clock after noon , o the place of twelve a clock or midnight , and q the place of six a clock in the morning : every one of the four quarters must be divided into six equall parts , or hours , making the whole circle to contain twenty four parts , representing the twenty four hours of the day and night , then in regard the hour of the day was nine of the clock , which is three hours before noon , take three of those twenty four hours , and place them upon the circle lnoq , from the meridian point l unto k , the nine a clock point in the morning , and unto m the point of three a clock after noon , then draw the line mk , cutting the parallel of the sun lo in the point i , then from i draw the line ig parallel unto the horizon bad , which shall cut the meridian circle bcde in the point g , so shall the distance of g and b be the almicanter the sun , which was desired , which in this demonstration is found to be fourty eight degrees and eighteen minutes . chap. xiii the elevation of the pole , the almicanter , and declination of the sunne , being given , to finde the houre of the day . the elevation of the pole is thirty degrees , the declination of the sun , is twentie degrees , the almicanter of the sun , is fortie eight degrees , and eighteene minutes , the houre of the day is found as followeth . first , as in the ninth demonstration , upon the center a , draw the circle bcde , then draw the diameter bd , representing the horizon , then from d unto r , set 30 degrees , the elevation of the pole , then from r unto the point a , draw the line ras , representing the pole of the world , then draw the line fah , crossing the pole in a , at right angles , cutting the meridian circle in f , then from f , set twenty degrees , the declination of ●he sun unto l , and then from the point l , draw the line lpo , representing the parallell of the sun , and cutting the pole of the world in p , then placing one foot of your compasses in p , extend the other unto l , with which distance of your compasses , draw the hour circle lnoq , then from the horizon at b , place the suns almicanter : ( which is fortie eight degrees , and eighteen minutes ▪ ) upon the quadrant bgl , from b unto g , then from the point g , draw the line g● parallel unto the horizon bad , cutting the line lo , in i , then from the point i , draw the line kim , parallell to the pole of the world qan , cutting the circle lno , in m , then let ln , be divided into six houres , whereof lm , are there : whereupon i conclude , that is is three houres from noon , that is , at nine a clock in the morning , or three in the after noon . chap. xiv . the latitude of the place , the declination of the sun , and the altitude of the sun being given , to finde the hour of the day : by a n●w way differing from that in the former chapter .   deg . min.   deg . m the suns altitude is 48 18       the lat●ude of the place is 30 00 its comple . 60 00 the suns declination is 20 00 n. 70 00       sum 130 00       difference 10 00 the complement of any arch lesse then 90 degrees , is so much as the arch wants of 90 degrees , as the complement of 20 degrees is 70 degrees , &c. first , finde the sum and difference of the complement of the suns declination , and the complement of the latitude , as above is done , where the sum is 130 deg . and the difference 10 deg . then your compasses being opened to the radius of your line of chords : describe the semicircle abc , and divide it into two quadrants by the perpendic●lar bd , then out of your line of chords ; take 48 deg . 18 min. the suns altitude , and set it from b to e , and draw e f parallel to b d : then from your line of chords take 130 deg . the sum , and set it from a to g , ( or its complement to 180 deg . which is 50 deg . from c to g ) and draw the line gh also parallel to bd. again , out of your line of chords , take 10 deg . ( which is the difference ) and set that distance from a to k , and draw k l parallel to ef or bd. this done , take with your compasses the distance from f to h , and seting one foot in a , with the other describe the arch mp , likewise take the distance from f to l , and seting one foot in c , with the other describe the arch nq . lastly draw the streight line pq ▪ which only touching the two former arkes will cut the line ac in o , upon the point o , therefore , erect the perpendicular or , cutting the semicircle in r , so will cr being measured upon your line of chords , give you the degrees of the sun from the south part of the meridian , which here you will finde to be 45 degrees , which make 3 hours , allowing 15 degrees for an hour , for 15 degrees make one hour , and one degree makes 4 minuts of an hour , so that it is either 9 of the clock in the morning , or 3 in the afternoon . chap. xv. the almicanter , or height of the sun being given , to finde the length of the right shadow . the almicanter is 45 degrees . according unto the tenth diagram , draw the line af , and upon the center a , raise the perpendicular ac , then upon the center a , draw the quadrant cdf , then suppose the height of your gnomon , or substance yelding shadow be the line , ab , which is to be divided into 12 equall parts , which gnomon , i have here made just 12 degrees of the equall leagues of the scale , then from b , to the top of the gnomon draw the line be , parallel unto af , then set the almicanter which is fortie five degrees from f , unto d , and from the point d , draw the line da , cutting the line be in the point g , so shall bg , be the length of the right shadow desired , which here is found to be fourteen degrees and eighteen minutes , which is but just the length of your gnomon , and 2 / 12 and ⅓ of a twelfe over : note that the right shadow , is the shadow of any poste , staffe , or steeple , that standeth at right angles with the horizon , the one end thereof respecting the zenith of the place , and the other the naedir . chap. xvi . the almicanter , or height of the sun being given , to finde the length of the contrary shaddow . the almicanter given is 70 deg . by the verse or contrary shadow , is understood the length of any shadow , that is made by a staffe or gnomon , standing against any perpendicular wall , in such a manner that it may l●e parallel unto the horizon , the length of the contrary shadow , doth increase as the sun riseth in height , whereas contrariwise the right shadow doth increase in length , as the sun doth increase in height : the way to finde the verse shadow is as followeth . first , draw your quadrant as is taught in the last chapter , wherein let ab , be the length of the gnomon , likewise from b , draw the line be , parallel unto af , as before , then set your almicanter from c upon the quadrant which is given to be seventie degrees and it will extend from c unto h , then from the point h draw the line ha , cutting the line be , in the point k , so shall kb , be the length of the contrary shadow , which here is found to be thirtie four degrees and eight minutes , or twice so long as your gnomon , and ●0 / ●2 about ½ part of a twelfth more . chap. xvii . the latitude of the place , the almicanter , and declination of the sun being given , to find the azimuth . the latitude of the place is fiftie one degrees , thirtie minutes , the declination of the sun twenty degrees north , the almicanter thirtie eight degrees thirtie minutes , the true azimuth of the sun is desired . first as in the eleventh demonstration upon the center a , draw the circle bcde , then draw the diameter bad ▪ and from d unto f , set the elevation of the pole , which is one and fiftie degrees , and thirtie minutes , whose complement is eight and thirtie degrees and thirtie minutes , which must be placed from b unto h , then from h , draw the line hal , representing the equinoctial line , and from f , draw the line fag , representing the pole of the world , then from h unto p , and from i unto q , set the declination of the sun , which is twentie degrees , and by those two points draw the line pq , for the parallel of the suns declination ; then upon the circle from b unto h ▪ set the suns almicanter , thirtie eight degrees , and thirtie minutes , then from h , draw the line hr ▪ parallel unto the horizon cutting the suns parallel poq in o , then draw the line tvae perpendicular unto the line bad , in the center a , and cutting the line hvr , in v , then seting one foot of your compasses in the point v , extend the other unto r , and with the same distance draw the semicircle hlr , then draw the concentricke circle upon the radius of the scale mtn , and where the line poq , and the line mon do meet in the point v , raise the perpendicular ol , cutting the semicircle hlr in l , then lay the scale from the center a to the point l , and draw the line lk , cutting the semicircle mtn , in k , so shall m k , be the true distance of the sun from the east , or west point southward , or the suns true azimuth , which is here found to be seventie two degrees , and fortie minutes from the south part of the meridian . chap. xviii . the latitde of the place , the declination of the sun , and the altitude of the sun being given to finde the azimuth : by a new way differing from that in the former chapter .   deg . min. s. deg . m. the suns declination is 20 00       the latitude of the place is 51 30 its comple . 38 30 the suns altitude is 12 00   78 00       sum 116         difference 39 30 having found the sum and the difference of the complement of the suns altitude , and the complement of the latitude as above is expressed where you finde the sum of them to be 116 deg . 30 min. and their difference 39 deg . 30 min. secondly , take 116 deg . 30 min. the sum out of your line of chords , and set it from c to g , and draw the line gk parallell d to b , thirdly take 39 deg . 30 min. the difference , out of your line of chords , and set it from c to h , and draw the line hl parallell also to bd. fourthly take in your compasses the distance from f to k , and setting one foot in a , with the other describe the arch s. fifthly , take the distance from f , to l , and setting one foot in c , with the other describe the arch r. sixthly , lay a rular , that it may only touch these two arches , s , and r , and by it draw a line as sr , cutting the line ac in n. lastly , upon the point n , erect the perpendicular nm , then the distance am , measured upon your line of chords , is the azimuth from the south part of the meridian , which in this example will be found to be 34 deg . mc the azimuth from the north 146 deg . and md , the azimuth from the east or west , 56 deg . chap. xix , the place of the sun being given , to find the right ascension , suppose the sun be in the twentieth degree of taurus , his right ascention is found as followeth . first , as in the 12 demostrastion , draw the line baf , for the pole of the world , the ● upon the center a draw the circle bcde , then from the center a , raise the perpendicular cae , for the equator , then place your greatest declination from c unto q , and from e unto p , then daw the line qap , which doth represent the eclipticke line , then in regard the sunne is in the twentieth degree of taurus , which is forty degrees , from the head of cancer , which forty degrees , place from q unto l , and unto k , then draw the line kl , cuting the eclipticke in i , then from the point i draw the line hi , parallel unto cae , cuting the pole of the world in o then set one foot of your compasses in o , and extend the other unto g , with which distance draw the semicircle hdg , then opening your compasses unto the radius of the scale , and upon the center o , likewise draw the circle hnfg , then draw the line im , parallel unto aod , cutting the semicircle hmdg , in m , then lay your scale from the center o , unto the point m , and draw the line nm , cutting the concentricke circle in n , so shall the distance nf , be the right ascention , which is here found tobe two and fortie degrees , seven and twentieminutes . chap. xx. the elevation of the pole , and declination of the sunne given , to finde the difference of the ascensions . the poles elevation is 51 degrees , 32 minutes , the declination of the sun is 21. degrees . first , as in the 13th . demonstration , draw the line bak , representing the horizon , then upon the center a , draw the circle bcdef , then from k unto d , set the elevation of the pole which is 51 degrees , and thirty two minutes : then from the point d , by the center a , draw the line daf , representing the pole of the world , then from b unto c , set the complement of the poles elevation which is thirty eight degrees , and 28 minutes : then from c by the center a , draw the line cae , representing the equinoctiall line ; then from c unto g ▪ and likewise from e unto h , for the declination of the sunne , which is 21 degrees , then from g unto h , draw the parallel of the sunnes declination , cutting the pole of the world in l , and he horizon in i , then set one foote of your compasses in the point l , and extend the other unto g , then with that distance of your compasses draw the semicircle gmnh , then opening your compasses unto the radius of your scale , upon the same center draw the concentricke circle , gxoh , then from i , where the declination of the sunne doth cut the horizon , draw the line in , parallell unto the pole of the world am , cutting the circle gmh in n , then lay your ruler from the point i unto the point n , and so draw the line no , cutting the concentricke circle gxoh , in o , so shall the distance of o and x , be the difference of the ascentions , which is here found to bee eight and twentie degrees , and foure and fiftie minutes . chap. xxi . the right ascention of the sun or of a star being given , together with the difference of their ascention , to finde the oblique ascention or descention . the sun is in the 4th . degree of sagitarius , his right ascention is 242 degrees , or 16 hours 8 minutes , the difference of ascention is 1 houre 53 min. or 28 deg . 28 min. the oblique ascention or desce●tion is required . the right ascention of any point of the heavens being known , the difference of the ascention is either to bee added thereunto , or else to bee substracted from it , according as the starre is situate in the northern or southerne signes : as for example , if the sunne be in any of these sixe signes , aries , taurus , gemini , cancer , leo , or virgo , then the difference of the ascentions is to bee substracted from the right ascention , and the remainder is the oblique ascention . suppose therefore the sunne to be in the fourth degree of gemini , where the right ascention is found to be foure houres , and 8 minutes , or 62 degrees , and the difference of ascention where the pole is elevated 51 degrees , is found to be one houre 53 minutes , otherwise 28 degrees 50 minutes , which being taken from the right ascention , leaves two houres and 16 minutes , or 33 degrees and 42 minutes , which is the oblique ascention of the sunne in the fourth degree of gemini . but if the sun be upon the south side of the equinoctiall , either in libra , scorpio , sagitarius capricornus , aquarius , or pisces , then the difference of the ascentions is to bee added unto the right ascention , and the product will be the oblique ascention . suppose the fourth degree of sagitarius is given , for which sign and degree the oblique ascention of the sun is desired , his right ascension being then found to be 242 degrees , or 16 hours 8. min. the difference of the ascensions is one hour , 53 minutes , or 28 degrees , 18 minutes : which being added unto the right ascension , makes 18 hours , and one minute ; or in degrees 270 degrees , and 18. minutes : which is the oblique ascention of the sunne , when he is in the fourth degree of sagitarius . and if you would finde the oblique descention , you must adde the difference of the ascentions unto the right ascention , when the sunne is in these six signes . aries , taurus , gemini , cancer , leo , virgo : and contrariwise , when the su●n is in the other six signes , you mnst substract the difference from the right ascention , and you shall have the oblike descention of the sun or any starre , whose right ascention and difference of ascentions is knowne . but it is to be understood , that this manner of operation , doth serve no longer than you are upon the north side of the equinoctiall . for if the south pole be elevated , the worke is contrary : for so long as the sunne is in any of the northerne signes , the difference of the ascentions is to be added unto the right ascention , to find the oblique ascention . and contrariwise , substracted to finde the oblique descention . likewise if the sunne or star be in the south●rn signes , then is the difference of ascentions , substracted from the right ascention , to finde the oblique ascention , and added , to finde the oblique descention . the end of the second book . the sea-mans glasse : the third book . shewing how by the plain-scale , to delineate houre-lines upon all kinde of upright plains , either direct or declining , in any latitude . the figure of the plaine scale . chap. i how to draw hour lines upon an horizontal plain , in any latitude . vvith the radius of your line of chords , upon e as a center , describe the circle abcd , and crosse it with he diameters ab , and cd . this done , out of the line of chords take the complement of the latitude of your place ( which we here suppose to be london , whose latitude is 51 deg . 30 m. and its complement 38. deg . 30 m. ) which set from b to g , from g to n , and from d to m ; then lay a ruler from a to g , and it will cut the line cd in h , and from a to n it will cut c d in o , and from a to m it will cut the same line in f. this done , upon o ( as a center ) place one foot of your compasses , and extend the other foot to f , and with this distance describe an arch of a circle , which ( if the rest of your worke be true ) will fall just in the points a and b , and so constitute the arch afb , representing the equinoctiall circle , and so we shall hereafter call it . having drawn the equinoctiall afb , divide the semicircle adb , into 12 equall parts in the points *** , &c. then laying a ruler to the center e , and every one of these marks *** &c. it will divide the equinoctiall circle into 12 unequall parts in the points ●●●● &c. again , lay a ruler to h , and every of these unequall parts ●●●● , &c. it will cut the semicircle adb in the points 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , 5 and 6. lastly , if you lay a ruler on the center e , and from thence draw right lines to the severall points 7 , 8 , 9 , 10 , &c. they shall be 12 of the true houre-lines belonging to an horizontall diall for the latitude of 51 degrees , 30 minutes . but for the houres before 6 in the morning , and after 6 at night , do thus ; draw the hour liues of 4 and 5 in the evening , quite through the center e , and they shall be the hours of 4 and 5 in the morning ; also , 7 and 8 in the morning drawn through the center , shall give the hours of 7 and 8 at night , as in the figure . chap : ii. concerning direct south dials . a direct south diall is no other then an horizontall diall , the makeing whereof is before described , the difference consisting only in the numbring of the houres , and in the placing of it , the one being to be fixed on a poste or the like , and the other to be fixed to a wall which exactly beholds the south , i say here is no other difference : for   degrees   degrees an horizontall diall for the latitude of 10 will be a direct south dial in the latitude of 80 20 70 30 60 40 50 50 40 60 30 70 20 80 10 and the like in any other latitude , as 15 , 16 , 33 , &c. chap. iii. of driect north dialls . a direct north diall , is the same with a direct south diall ; for , i● you take a south diall and turn it upside down , causing the sc●le or cock to point upwards , as the cock of the south doth down wards ; and leaving out the hours neer the meridian , in these northern latitudes ; as the hours of 9 , 10 , 11 , and 12 at night , and 1 , 2 and 3 in the morning , all which time the sunne is under the horizon . i say a south diall so disposed , and fixed against a direct north wall , shall give you the true houre of the day . chap. iv. how to draw the houre lines on a direct east or west plain . this done , upon the point g , with the radius of your chord , descirbe an occult arch of a circle h i , and set thereon 15 degrees fr om h to i , then from g , through i , draw the line g k , cutting n min k , on k , as a center , with the radius of your chord , describe the quadrant k s t , which divide into 6 equall parts in the points ●●●● , through which points and k , draw the lines , k● , k● , &c. cutting the equinoctiall eb in **** &c. through these points *** , &c. draw right lines quiet through your plain perpendicular to the equinoctial , which will be parallel to your lines of vi , and xi , and will be the true hours of vii , viii , ix , and x , then the like distances of vii and viii , set above vi , on the other side , and drawn parallel thereto , shall be the true hours of iiii. and v. and thus have you all the hours of an east dial truly drawn , which is from four in the morning , till eleven at noon , and is the same with a west diall only naming the hours contrary : for , in the east diall 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , in the morning , are in the west diall 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , h in the evening . the stile of either of these dials , is a t in plate of brasse , made directly of the breadth of the distance between the hours of vi , and ix , and must be placed directly perpendicular upon the line of vi , and so is your diall finished . chap. v. of upright declining plains . before we come to draw the houre lines upon a declining plain , two things are first to be discovered , viz. first . the height of the pole above the plain , which is the height of the cock or stile . secondly , the deflexion , or distance of the substile from the meridian or line of twelve a clock . 1. to finde the height of the pole above a declining plain . vvith the tadius of your line of chords , upon a , as a center , describe the quadrant ab c , then your latitude being 51 deg . 30 min. take it out of your line of chords , and set it from b to f , and draw the line ed parallel to ab , cutting the line ac in d , then with the distance de , on the center a , describe the quadrant ghr . then supposing your plain to decline 30 deg . set 30 deg . from b to f , in the quadrant bec , and draw the line fa cutting the quadrant ghr in h , through which point h , draw the line shn parallel to ca , and cutting the quadrant be c in n , so shall the arch cn be the height of the pole above the plain , and in this example contains 32 deg . 37 min. 2. to finde the deflexion , or the distance of the substile from the meridian . out of this figure , take the distance hs , and set it in the line de , from d to k ; through which point k , draw the line akl , cutting the quad ant bc in l ; so shall the arch cl be the distance of the substile from the meridian : and in thls example will be found to be 21 degrees 42 minutes . chap. vi. how to draw the houre-lines upon an upright plain declining from the meridian towards the east or west . vve will here take for example a south erect plain , declining eastward 30 deg . having ( by the fifth chapter of this book ) found the defl●xion of such a plain to be 21 deg . 42 min. and the height of ●he ●●ile ( by the same chapter ) to be 32 deg . 37 min. we may proceed to draw the diall in manner following . with the radius of your line of chords , on the center c , describe the circle xnsw ; and in it , draw sn through the center c , for the meridian , or line of 12. then the deflexion being found to be 21 deg . 42 min. set that from n to e , and draw the line ●c through the center to g ▪ this line representeth the substilar line of your diall , upon which line the stile or co●k must stand ▪ also , out from your line of chords take 32 deg . 37 min. the height of the s●ile , and set that distance from e to h , and draw the line ch for the stile of your diall ; so shall the triangle ech , be the true pattern for the cock of your diall . the substilar line eg being 〈◊〉 ●●aw the line xw through the center c , and perpendicular to eg . this done , take the distance eh , ( which is equall to the stiles height ) and set that distance from a to b , and from w to d. likewise , take the distance from w to b , and set it from b to i. these three points i , b and d , being found in the circumference of the circle xnsw , lay a ruler from x to i and it will cut the substilar line ec being extended in the point g , which is the center upon which the equinoctiall circle must be described . again , a ruler laid from x to b , will cut the substilar line in f , and a ruler laid from x to d , will cut the substilar in o. now , if you set one foot of your compasses in g , and extend the other to x or w , you may describe the equinoctial circle xow , which ( if you have not erred in your former worke ) will passe exactly through the point o in the substilar line before found . in the next place , if you lay a ruler from f to n , it will cut the equinoctiall circle in p , and a ruler laid from c to p , will cut the diall circle in v. these things being performed , the next thing is to draw the hour lines , which will be easily effected if you 〈◊〉 the former directions . first , from the point v last found , begin to divide your houre circle into 24 equall parts ( or only one halfe of it into 12 parts ) which you may do by taking 15 deg . out of your line of chords and set that distance on both sides of v at the marks ⚹ ⚹ ⚹ &c. so many times as the plain is capable of hours . this done , if you lay a ruler on the center c , and every of these points **** &c. you shall divide the equinoctiall circle into 12 unequall parts in the points ●●●● &c. now a ruler laid from f to every of these unequall points ●●●● , &c. will divide the houre circle into 12 other unequall parts marked with 4. 5. 6. 7. 8. ▪ 9. 10. 11. 12. 1. on the one side of v , and with 2. 3 ▪ ●n the other side of v. lastly , a ruler laid from c to the severall points 4. 5. 6. 7. 8. 9. 10. 11. 12. 1. 2. 3. and lines drawn by the side thereof they shall be the true houre lines belonging to such a declining plain of 30 deg . in the latitude of 51 deg . 30 min. but if you desire more hours then 12 , the equinoctiall may be divided into more unequall parts , being continued beyond x and w , and if you will , quite round the whole circle , but that is needlesse without you would make 4 dialls in the makeing of one as you may easily do . for , the hours that are on the west side of the meridian of a south east diall , being drawn through the center , will make a north west diall of the same declination . and the hours on the east side of the meridian of a south west diall ; being drawn through the center , will produce a north east diall of the same declination . and again , the reall houre lines of a south east diall being drawn on the other side of the paper , and the hours named by their complements to 12 , that is , 10 for 2 , 9 for 3 , 8 for 4 , &c. will make a south west diall of the same declination . chap. vii . how to place any upright diall truly . all upright dialls , in what oblique latitude soever have the meridian perpendicular to the horizon , wherefore to set your diall exact , hang a line with a plummet at the end thereof , and with a nail fixed in the line of 12 towards the top thereof , to hang the plummet upon , apply the diall to the place where it is to be fixed , so that the line and plummet may hang just down upon the line of 12 , neither inclining on one side or the other , the diall thus fixed if the declination were truly taken , and the dial rightly made , by the former directions , shall at all times ( the sun shining upon it ) give you the true hour of the day . chap. viii . how to insert the halve and quarters of hours in all dialls . the halves and quarters of hours are drawn in all plaines by the same rules , and the like reason , that the hours are inserted . therefore take notice that if you would insert the halfe hours into any diall , you must divide your equinoctiall circle into 24 equall parts instead of 12 , and if you would insert the quarters , then you must divide it into 48 parts , and then proceed in all respect , as you did for the whole hours . chap. ix . how to finde the declinatioon of any upright wall. the declination of a plain is an arch of the horizon comprehended between the pole of the plains horizontall line , and the meridian of the place . to finde this declination , two observations must be made , the sun shining , and both at one instant of time ( as neer as may be . ) the first is the horizontall distance of the sun from the pole of the plain . the second is the suns altitude . first , to finde the horizontall distance . apply the side of a quadrant to your plain , holding it ( as neer as may be ) horizontall , that is to say , levell , then holding up a thrid and plummet , which must hang at full liberty , so that the shadow of the thrid may passe directly through the center of the quadrant , then diligently note ● through what degree of the quadrant the shadow passed , and count those degrees from the side of your quadrant which is perpendi●cular to the plain , for those degrees are the horizontall distance . secondly , at the same instant , take the suns a●●itude , these two being heedfully taken , will help you to the plains declination by th rules following . by the 17 or 18 chapters of the second book find the suns azimuth . then observe whether the sun be between the pole of the plains horizontall line and the north or south points , or not . if the sun be between them , adde the azimuth and horizontall distance together , and the sum of them is the declination of the plain . if the sun be not between them , substract the lesser of them from the greater , and the difference shall be the declination of the plain . these rules sh●w you the quantity of your plains declination . but , chap. x. shewing how to know whether your plain declin from the meridian towards either the east or west . you must take notice in your observation , that if the meridian point fall between the azimuth and the pole of the plains horizontall line , then doth the plain decline to the coast contrary to that wherein the sun is , that is to say , if the sun be to the eastward of the meridian , the plain declines to the westward , but if the meridian point be not between the forementioned distance and the pole of the plain , then doth the plain decline to the same coast in which the sun was at the time of observation . chap. xi . concerning polar dials . a polar diall is made in all respects as an east or west diall is made , onely the line of 6 a clock in the east or west diall , is 12 a clock in the polar diall , the houre of 7 is 1 , of 8 is 2 , of 9 is ● , of 10 is 4 , and of 11 is 5. also the houre of 5 in the east or west diall , is 11 in the polar , of 4 is 10 , of 3 is 9 , of 2 8 , of ● is 7 , &c. the cock of this diall is a plate of iron or brasse made of the breadth between 12 and 3 a cloock , and set perpendicular upon the line of 12 , as in the east or west diall it is upon the line of 6. in these dialls the equinoctiall line is to lie parallel to the horizon , and not to be elevated according to the complement of the latitude of the place , as in the east or west diall it is . chap. xii . concerning equinoctiall dialls . an equinoctiall diall is of all other dialls , the most easie to make , for if you describe a circle , and divide it into 24 equall parts , and draw lines from the center through eve●● one of those equall parts , the lines so drawn shall be the true houre lines . for the stile of these dialls , it is no other but a streight wyre of any length set perpendicular in the center of the circle , whose shadow shall give the true houre of the day . chap. xiii . of such plains as decline very far from the east or west towards the meridian as 75 , 80 , or 85 , deg ▪ above which plains the pole hath small elevation . such plains as decline above 60 degrees the houre lines will come very close together , so that if they be ▪ not extended very far from the center , there will be no sensible distance between hour and hour ▪ to remedie this inconvenience , there are severall wayes , i will instance only in one which is familiar and easie , and that is this . when 〈◊〉 have 〈…〉 your diall on a large sheet of paper , fix it on some large table or smooth floor of a room , if the diall you are to make be very large , as 5 , 6 , or 7 ▪ foot square , then by the side of a long rular laid to the center and every hour line , as also to the stile and substile , draw lines to the full extent of the table or flour , and you shall finde them to be of a competent largnesse . then according to the bignesse of your plain , cut off the houres . stile and substile , leaving the center quite ou● , and yout work is finis●ed . chap : xiiii concerning declining reclining and inclining dials . vve should now shew the manner of drawing houre lines upon declining reclining and inclining plains , of which there are severall varieties , and many cautions , which in this place and at this time , would be too many to ennumerate : but if this which hath been already delivered concerning upright decliners shall be kindly accepted , it shall animate me to do the like for all other plains whatsoever . finis . advertisement . note , that this scale and all other instruments for the mathematicks , are made by walter hayes , at the crosse dagers in moore , fields next doore to the popes head tavern , london . mr. de sargues universal way of dyaling, or, plain and easie directions for placing the axeltree and marking the hours in sun-dyals, after the french, italian, babylonian, and jewish manner together with the manner of drawing the lines of the signs, of finding out the height of the sun above the horizon, and the east-rising of the same, the elevation of the pole, and the position of the meridian ... / [edited] by daniel king, gent. maniére universelle pour poser l'essieu. english desargues, gérard, 1591-1661. this text is an enriched version of the tcp digital transcription a35744 of text r17188 in the english short title catalog (wing d1127). textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. the text has been tokenized and linguistically annotated with morphadorner. the annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). textual changes aim at restoring the text the author or stationer meant to publish. this text has not been fully proofread approx. 146 kb of xml-encoded text transcribed from 72 1-bit group-iv tiff page images. earlyprint project evanston,il, notre dame, in, st. louis, mo 2017 a35744 wing d1127 estc r17188 13154958 ocm 13154958 98167 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial 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(eebo-tcp ; phase 1, no. a35744) transcribed from: (early english books online ; image set 98167) images scanned from microfilm: (early english books, 1641-1700 ; 414:7) mr. de sargues universal way of dyaling, or, plain and easie directions for placing the axeltree and marking the hours in sun-dyals, after the french, italian, babylonian, and jewish manner together with the manner of drawing the lines of the signs, of finding out the height of the sun above the horizon, and the east-rising of the same, the elevation of the pole, and the position of the meridian ... / [edited] by daniel king, gent. maniére universelle pour poser l'essieu. english desargues, gérard, 1591-1661. king, daniel, d. 1664? bosse, abraham, 1602-1676. [17], 108 p. : ill. printed by tho. leach, and are to be sold by isaac pridmore ..., london : 1659. translation of: maniére universelle pour poser l'essieu. added illustrated t.p., engraved. the diagrams are reproductions of the engravings by abraham bosse who published the original french edition. advertisement on p. [17]. reproduction of original in cambridge university library. eng dialing. sundials. a35744 r17188 (wing d1127). civilwar no mr. de sargues universal way of dyaling. or plain and easie directions for placing the axeltree, and marking the hours in sun-dyals, after t desargues, gérard 1659 29465 52 0 0 0 1 0 21 c the rate of 21 defects per 10,000 words puts this text in the c category of texts with between 10 and 35 defects per 10,000 words. 2006-06 tcp assigned for keying and markup 2006-06 apex covantage keyed and coded from proquest page images 2007-04 robyn anspach sampled and proofread 2007-04 robyn anspach text and markup reviewed and edited 2008-02 pfs batch review (qc) and xml conversion mr. desargves . vniversall way of makeing all manner of sun dialls ▪ published by daniell king ▪ & sold by isaake pridmore at y , golden falcon in y strand ▪ a● 1659 mr. de sakgves universal way of dyaling . or plain and easie directions for placing the axeltree , and marking the hours in sun-dyals , after the french , italian , babylonian and jewish manner . together with the manner of drawing the lines of the signs , of finding out the heighr of the sun above the horizon , and the east rising of the same , the elevation of the pole , and the position of the meridian . all which may be done in any superficies whatsoever , and in what situation soever it be , without any skill at all in astronomy . by daniel king gent. london , printed by tho. leach and are to be sold by isaac pridmore at the golden faulcon in the strand , near the new exchange , 1659. to the illvstriovs george villiers , duke , and marquess of buckingham , earl of coventry , viscount villiers , baron whaddon , and ros , knight of the most noble order of the garter , &c. sir , having had the honour to observe your graces great affection , and love to sciences and arts , and your own excellency being most eminent therein , together with your unparallel'd love and inclination to the splendour of your native country , in promoting learning and ingenuity . these high merits with my own particular obligations and attendance , encourage my endeavours of the patronage to a new birth never presented to the english nation ; presuming by gods assistance to bring forth something of worth that hathnot yet seen light , and if your grace shall please to pardon my observant presumption , you will hereby more strictly engage him ever to honour your heroick worth , who is , the very humblest of your servants , daniel king . the preface . concerning the particulars of this treatise . whereas the superficies or outsides whereon dyals may be made , may be either flat , bowed , or crooked , plain or rugged , & situated diversly , the most part of the books treating of this matter , contain severally the manner of making flat dyals in all kinds of positions , horizontal , vertical , meridional , septentrional , oriental , occidental ; declining , inclining ; inclining , and declining ; and accordingly in all other kinds of superficies . they may also shew , for those that are ignorant of it , the way to find the elevation of the pole , the meridian line , the declinings , inclinings , and other particularities . but monsieur de sargues intention being to publish nothing , if it be possible , that is to be found in another book , and to give you only the general rule to make , and not to copy out a number of examples all differing one from another ; i will give you but one example only in this volume , by this universal manner , the discourse whereof may be applyed generally to all kinds of superficies , and in what situation soever they be , without having any knowledge of the pole , nor of the height of the sun , nor of the declining or inclining , nor of the meridian line , nor of any other thing in astronomy , and without a needle touch'd , nor of any kind of thing that may give a beginning to that , as you shall see yet better when we shall treat of the practice . and in practising this general rule , you shall find at one and the same time , the elevation of the pole , and the position of the meridian , you shall know how to place the needle of your dyal , and so you shall come to find the equal hours , which are called hours after the french way , alias astronomical . the rest being more curious than necessary , i thought to set down nothing else but those two things ; but i have been perswaded , for the satisfaction os some , to add also the manner of drawing upon the same superficies , the lines of the signs , the hours after the italian and babylonian way , and of the antients , the height of the sun , and the situation thereof , in respect of the horizon . of the practice of sun-dyals . many diverse things are represented in sun-dyals by the shadow of the sun , to divers ends , the hour is shewn by them , and serves only for that purpose every day . other things are represented also by it , as the signs , and other particularities , whereby it may serve sometime for the divertisement of a few in antient time the hours were not counted as they are now a dayes , and in italy at this day they are counted otherwise than in france the manner that they count their hours in france is according to astronomy , and here is at length a generall way of framing and making dyals with houres equal to the sun , according as the hours are now counted in france , after which way one may come to shew if need be , by the shadow of the sun , and also of the moon , whatsoever can be shewn concerning other circumstances to satisfie curiosity . there are two things which together do compose those kinds of dyals of equal hours , after the french way , the one is as piece that shoots out or sallies out of the superficies of the dyal , the shadow whereof falling upon this superficies , shews what a clock it is , the other are the lines drawn upon the superficies of the dyal , each of them representing one of the hours after the french way . they make dyals after the french way wherein there is only the shadow of one portion alone , as might be a button of the piece that shoots out , that shews what a clock it is . but in this general way , there is still the whole shadow of all the length , in a direct line of this piece that shoots out , shewing continually what a clock it is , of which piece or length , you may take if you will a button , and mark the hours with that button only , together with all the other particularities that may be added to such dyals . some call by one and the same name these two kinds of pieces the shadow whereof shews the hour , as well that same whose shadow shews continually the hour at length , as that which hath but the shadow of a button to shew it . but to the end we may distinguish both kinds of pieces one from the other , that same whose shadow shews the hour at length , and is the original spring of all the others , i call it the axeltree of the dyal . this axeltree may be made as well with a straight , round , and smooth rod of yron or brasse , as with a flat piece , and cut of one side in a straight line . there are often other rods in the dyals , which serve to bear up the axeltree that shoots out , and those kind of rods i call the supporters of the axeltree . the lines that are drawn upon the superficies of the dyal , and that shew each of them one of the hours after the french way , i call them lines of the hours after the french way . in the innumerable number of such kinds of dyals as may be made after the french way , it happens that the superficies of the dyal , is either all flat , or is not so altogether . when the superficies of the dyal is all flat , every line of the hours is a straight line . and when the superficies of the dyal is not altogether flat , it may be that every line of hours is not all straight also . to make one of those dyals of equal hours after the french way , by this universal way , there are two things to be done one after another . the first is to place the axeltree as it ought to be , that is , shooting out of the superficies of the dyal ; the second is , to draw the lines of the hours as they must be upon the same superficies . and by means of this general way , you shall do those two things without knowing in what day , nor in what time of the year , nor in what country you are , without knowing what the superficies of the dyal is , whether it is plain or rough , nor which way it looks , without knowing any thing concerning the making and the placing of the parts of the world , or without any skill in astronomy , without any needle touch'd with the loadstone , or any instrument or figure that may serve for a beginning towards the making of a dyal : but by the means only of the beams of the sun , by one general rule you shall place the axeltree , and draw the lines of those hours upon one of those dyals whatsoever the superficies may be , and which way soever it looks , with all the celerity and exactnesse that is possible in art ; and if you are equally exact in every operation , you shall make by this means many dyals upon different superficies , and turned towards several parts of the world , which shall agree plainly among themselves , and if you do not do it , you may be sure that the fault is on your part , and not in the rules , since that others do succeed well in it . there are some pieces that are requisite for the framing of a dyal , and whereof it is composed , such are the axel-tree rod with its supporters . there are other pieces that must be used in the making of a dyal , as rules , compasses , a squirt , a lead with his two frames , one to mark with , and the other to level . there are some other things that you shall use also , as pegs , and rods , either of yron , or of brasse , or of wood , some sharp at both ends , the others sharp at one end only , a table , either of wood , or of slate , or of any other stuff , flat and solid , to draw upon if need be , some straight lines with the rule ; and in case the superficies of the dyal were plain and even , you must use some fine strings , supple and strong , some mastick , cement , or plaster , or such like stuff fit to seal with , &c. all which things you must have in readinesse , whensoever you will go about the making of a dyal . and though you would learn to make but one of these dyals well , it is fit you should have some models of all those pieces , and when you are upon those chapters that concern them , as you shall understand an article , it will be requisite that with the models of those pieces , you work at the same time an actual model of the thing which that article shall teach you to do , and so you must work from one end to another , till you have at last every way compleated an actual model of this kind of dyals , and you shall need to make but few of such models of dyals upon any superficies , turned towards several parts of the world , to bring you acquainted with the practice of making dyals to the life , or after the natural , in what kind or odd situation of superficies soever they may be . lastly , you shall find the precepts and the descriptions to be more troublesome , than the actual making or working , study only to be as exact in every one of these operations of making of these dyals , as in the practice of other arts . the epistle to the reader . courteous reader , this treatise being originally written in french , and generally approved of all those that have any skill in the art of dyalling , i have thought it my duty to lay hold upon this occasion , to shew how desirous i have ever been to procure any good unto my country . therefore i have caused it to be carefully translated into english , and have set it forth for the good and utility of all such , as are curious , and true lovers of that art ; reputing my self most happy to meet with any occasion , whereby i may contribute any thing towards the advancement of learning , and of the publick good , — non enim nobis solum nati sumus ; we are not meant to be wholely and soly for our selves . as for the work it self , i am so confident it will so gain the attentive readers approbation , as that i shall forbear to say any more in commendation of it , than that it is an expedite , and sure way of obtaining the site of the axis , and of other requisites in the framing of all sorts of dyals , of no lesse curiosity , than use , performed without the ordinary rules and presupposals of the spiritual calculations and practice ; i need premise no more , but advise to follow the directions that are set down through all the book , for effecting that which is promised , and thou shalt see the same plainly and readily performed . accept then , courteous reader , this small labour , the undoubted testimony of my love , as kindly ▪ as i offer it cordially unto thee , hoping that god will enable me to give thee hereafter some thing of more consequence , so farewell . vtere & fruere . thine d. k. to all lovers of ingenious practices . the french have excelled all other nations in the art of perspective for this last age , their many books and curious writings so excellently composed do witnesse for them . dyalling i accompt one kind of perspective , for that glorious body the sun , the eye of the world , traceth out the lines and hour-points by his diurnal course , and upon the resubjected plane by the laws of picture , scenographically delineates the dyal . many have writ upon this subject of several countryes , in several ages , many are the rules and practices set down ; but among all those of forein parts , none hath performed the same with more ease , and lesse trouble , than monsieur du sargues the author ; as wholy laying aside those tedious observations of azimuth's , declination , reclination , inclination , meridian ; substile , &c. and performing the operation only by three observations of the suns shadow from a point . it will not be amisse to give the reader a small consideration hereof ; the point b of the pin ab , in all the figures is alwayes one part of the axis , or gnomon of the dyal , and may be used to shew the hour : this point b , you must imagine to be the center of the earth ( for the vast distance to the sun , maketh the space betwixt the center and superficies of the earth to be insensible ) and from it at all times of the year ( excepting the aequinoctial day ) the sun in its course forms two cones , whose apex is the point b , that next the sun termed conus luminosus or the light cone , the other whereof our author makes use , termed conus umbrosus the dark cone , now in this dark cone , if by any three points equally distant from the apex b , the cone be cut , the section will be a circle parallel to the equinoctial : and thereby , as the author shews many wayes , the position of the axis or gnomon may be found out , and the dyal easily made . now it rests , courteous countryman , that we be very gratefull , and every way forward to encourage mr. d. king , one very industrious in the studies of antiquities and heraldry ; who out of his desire to serve his country , hath caused this piece speak english , hath been very carefull to see the cutts well done , and will ( no doubt ) proceed to cause some of those rare pieces of perspective in french to be translated . then prosper king , untill thy worthy hand , the gallick learning make us understand . jonas moore mathesios professor . books printed for isaac pridmore , and are to be sold at the golden faulcon near the new-exchange . the rogue , or the life of gusman de alpherache the witty spaniard , written in spanish by matthew aleman , servant to his catholick majesty ; the fifth and last edition corrected . a physical discourse , exhibiting the cure of diseases by signatures , whereunto is annexed a philosophical discourse , vindicating the souls prerogative in discerning the truths of christian religion with the eye of reason , by r. bunworth . seif-examination or self-preparation for the worthy receiving of the lord ▪ supper ; delivered in a sermon concerning the sacrament , by daniel cawdrey , sometimes preacher there , with a short chatechism : the third edition . the obstinate lady , a comedy written by sir aston cockaine . sportive elegies written by samuel holland gent. a new discovery of the french disease , and running of the reins , with plain and easie directions for the perfect curing the same , by r. runworths . the vnspotted high court of iustice , erected and discovered in three sermons preached in london and other places , by thomas baker , rector of st. mary the more in oxon. a chain of golden poens , imbellished with wit , mirth , and eloquence , together with two most exelent comedies , viz. the obstinate lady , and trapolin suppos'd a prince , by sir aston cockaine . the ascent to blisse by three steps , viz. philosophy , history , and theology , in a brief discourse of mans felicity , with many rem●●keable examples of divers kings and princes . the heroical loves , or anthcon & fidelta a poem , by thomas bancroft . advice to balams asse , or momus catechised , in answer to a certain scurrulous and abusive scribler by , iohn heydon a●●hor of advice to a daughter , by t. p. gen● . the analysis of all the epistles of the new testament , wherein the chief things of every particular chapter are reduced to heads , for the help of the memory , and many hard places explained , for the help of the understanding , by iohn dale master of arts , and fellow of magdal 〈…〉 in oxford . 1 i figure , to all sorts of people . i come now to the first of those two things that you are to doe for to make one of those dyals , which is the manner how to find the position , or the placing of the axeltree . when you have a mind to find out the right placing of the axeltree of one of those dyals by this general way , mark first which way the light of the sun comes to the place where you will make your dyal , and which way it goes out again . then make fast upon the place , as the figure above doth shew , with cement , plaister , mastick , or the like , a peg or pin , ab , by the great end a , putting the other small or sharp end b as far out of the superficies of that place as you can , in such sort that while the sun doth shine upon that place , the shadow of the end of the pin b may fall always upon this superficies , and for the rest it is no matter how this pin or peg be framed , or placed , or turned , you are only to look to the small end thereof , that must be in such a manner , that you may set or apply upon it one of the feet of the compass . then in a fair sunshiny day , when the light is very clear , and the shadow very clean , whilst it falls upon this superficies in the figure below , mark in it , as the figure shews , in one and the same day , at three several times as far asunder as you can , three several points cdf , each of them at the end of the shadow of this pin ab that answers to the small end of it b. you must nore that there is a certain time and place in which you cannot mark the points of the shadow ; that is when this superficies is flat , and situated after such a manner , that the ground plot thereof being stretch'd at length , answereth and reacheth into the center of the sun . for in that case how short soever the peg or pin b may be , the shadow hereof cannot goe and fall in this superficies , but at the end of an extreme length . therefore when the days are equal with the nights , or very near , you cannot mark in this manner three points of shadow in a flat superficies , which is situated in that manner called parallel to the equator . when you have thus mark'd three points of shadow , you have no more need of the light of the sun , and you may make an end of the rest in any other time and season , as well by night as by day , as i shall say three times together for one and the same manner , in three several ways to be expressed , after i have briefly satisfied the theoriciens that take pleasure to see the reasons of the precepts , or rules of the practice of the arts , before they see the precepts themselves . 1 to the theoriciens . this resolution will serve you . after you have conceived that the sun in his full revolution of a natural day makes a circle parallel to the equator , and the rest of this hypothesis , for dyals . the three beams of the sun or straight lines , bc , bd , bf , make in their point or common end b. some angles or corners equal one to another ; with an other straight one that makes the fourth , which is the axeltree of the dyall . now the position or placing of these three straight lines bc , bd , bf , is given out ; therefore the placing of this fourth which is the axeltree of the dyal is given also . you shall have hereafter in the fourth figure an other resolution of this kind , before you have the way to compose some problemes , or propositions about it . i said to the theoriciens , because if you were not at all versed in any kind of practice , either of geometry or art , you might hardly understand me at first concerning the 2d . & third figures following , because of the short & compendious way whereby i expresse my self unto those that are skilled in geometry : but i can assure you , that when you have understood what is written in order for all sorts of people , if you come again to these second and third figures , you shall know at the very first sight what they mean . for the theoriciens , and for those that are skilled in geometry . the i figure is a plate of some thin , flat , smooth , and solid stuff ▪ as iron tinned , or the like , being round , and having a hole just in the center , greater or lesser , according to the occasion . the ii figure is a straight rod , round , smooth , and solid , as of iron , or the like , of the bigness of the hole in the plate . the iii figure is as it were a whirl made of the plate , and of the rod put thorow the plate , in such sort that it is perpendicular to the said plate , as the squire that turns round about doth represent unto you , and is so fast that it cannot stir or move . in the v figure ab is the peg or pin that hath mark'd unto you the points of the shadow cdf , the rods or sticks bc , bd , bf , are solid and strong , as of wood , or the like , having each of them a slope edge in a direct line all along , going from the point of the peg b to each point of the shadow cdf , and are so turned or ordered , that in applying the whirl unto them , the edge of the plate may goe andtouch the three slope edges of the rods all at once ; and the rods or sticks are made fast in this situation , in such sort that they cannot move nor stir . the rule that crosseth over the three slope edges , bc , bd , bf , toucheth them all three , or else two at the time only , whereby it shews whether those slope edges are all three in one and the same situation , or upon one and the same ground or no , and on which side is their hollowness when there is any . the hand applies the whirl unto it , and keeps it there till the axeltree bo● come to touch the end of the peg or pin b , and that at the same time the edge of the plate edh touch the three slope edges of the rods . and when the whirl is placed or setled after this manner , the rod is the axeltree of the dyal , and placed as it ought to be , and there remains nothing else but to make it fast in this situation or position . the iv figure doth shew , that if you goe to make use of thin and supple strings in this practice or working , in pulling those two mark'd with ie , and ih , to make them fast in direct lines , they would make the two strings mark'd with bc , bf to bend , so that you can doe nothing exactly with them , which is the reason that monsieur desargues hath not thought fit to make use of them for the beams of the sun , but rather of the slope edges of the rods that are both stiff and strong . 2 3 to the theoriciens , and others that are skill'd in geometry . this foregoing figure shews to the eye that all the pieces of the instrument are made so strong and firm , that they cannot bend . ab is the pin , by whose point b , you have had the points of shadow c , d , f. the three sticks or rods bc , bd , bf , have each of them a slope edge in a direct line at length , going from the point of the pin b to the three points of shadow , c , d , f. the slope edges of the two longest sticks or rods , bc , bf , have some portions made in them , equal every one to the third and shortest stick bd. the three sticks ih ▪ id , ie , are every one longer than bd , and all three made even , then they are joyned all by the end to one of the points edh , of the slope edges of the other sticks , bc , bd , bf , and their other ends i , are brought together in one and the same point , i. the rod bi ▪ is straight , round , smooth and strong as of yron or the like , it hath a straight line bi , drawn from one end to an other , and one of the points b , of this line of the said rod toucheth the point of the pin ; and with an other point i of the same it toucheth the point i of the three rods or sticks . this being so , the rod bi comes to be the axeltree of the dyal rightly placed , there remains nothing else but to make it fast in this position or situation . the figure shews in the rods that goe from the point of the pin b , to the points of shadow cde , how one may make fast those rods at one end to the pin , and also all together to one point , by binding them to it ; and how they may be made f●●t at the otherend to one point of the superficies of the dyal , by fastning them to it with mastick , plaster , cement , or thelike . this way is more sure than that with the strings ; but yet it is not the easiest ▪ nor the least troublesome , in my judgement . to the theoriciens , another resolution of the same kind with the former . the position or placing is given of the four points bc df , and the placing of the two straight lines be , bh , that divide in two the angles cbd , and dbf , and of the two ground plots that passe unto those two straight lines be , and bf , and that are perpendicular to the ground plots of those angles cbd and dbf , are given out ; therefore the intersection or intercutting of these two ground plots so perpendicular is given . but this intercutting is the axeltree of the dyal , therefore the position of the axeltree of the dyal is given . any one may frame at his pleasure upon that which is granted concerning this composition , many other resolutions , and divers compositions of problemes , and divers general ways of practice . in the mean time you shall have here three several ways one after another , to see which is the most advantagious for the actual practizing of the art , and to induce you to seek or try if there is any other shorter . 4 for the theoriciens . the composition of the probleme , or proposition , in consequence of the resolution made upon the lowermost figure of the first draught the first figure is the place of the dyal , with the pin and the points of shadow , cdf . make a ground plot of it , ii upon one straight line bd , and with one point b , three angles dbn , dbr , dbh , equal to the three angles of the first figure , that are between the beams of the sun , dbc , dbf , cbf , every one to his own respectively . from the center b , ii figure , and from any space bd , draw a half circle that may meet in the points , dnrh , the straight lines bd , bn , br , bh . make in the third figure a triangle dgv , with three spaces , equal to the three spaces dh , dr , dn , every one to his respectively , as having the condition necessary for that purpose . find the center o of the circle evgd , drawn about this triangle vgd . draw two diametters doe , pob , of this circle , perpendicular one unto another . lengthen one pob sufficiently of one side and on the other . from one of the ends d , from the other eod draw as far as that which is lengthened pob one straight line db , even with the straight line db of the ii figure , for it must reach unto it , viz. in the equinoctial at the point o , and in an other place at an other time , lengthen sufficiently iii figure this straight line bd. make in it the segments or cuttings even with the beams of the sun of the i figure bd , bf , bc. take in the iii figure , in the straight line pob , conveniently a point i other than b. make in the iiii figure three rods or sticks ci , di , fi , each of them sharp at both ends , and equal with the three spaces ci , di , fi , of the third figure . draw a straight line along the axeltree rod , mark in this line of the axeltree conveniently figure iiii , one cut bi equal with the space bi of the iii figure . set figure iiii one of the ends of the stick ci to the point of shadow , c , one of the ends of the stick di to the point of shadow , d , and one end● of the stick fi to the point of shadow f. let the ends of those sticks or rods be so well fastened to the points of shadow cdf , that they cannot stir . bring together the other ends i , of those sticks in one point i. put one of the point ▪ b of the axeltree rod to the point of the pin b , and the other point i with the three ends of the sticks ci , di , fi , set or joyned together . and if you have been very exact in the work , the point i of the pin will go and place it self with the three ends of the sticks set together in the point i , if not , you have not wrought exactly . 6 to the theoriciens . it is no matter whether the figures come right to the compasses , you are only to take notice what this insuing discours ordains you to do . make figure i with three straight lines cqrd , dipe , and cf , a triangle even and like unto the triangle figure iii , of the three points of shadow cdf , upon the straight line cqrd figure i. make a triangle cbd , both like and equal with the triangle figure iii. of the sun-beams cbd , and upon the straight line fpid figure i. make a triangle fdb , like to the triangle figure iii. of the sun-beams fdb , make longer if need be figure i. on the side of d the straight lines cqrd and fpid . by the points b and b draw a straight line brayh perpendicular to cqrd , and a straight line biakl perpendicular to the straight line fpid , find out the end or point a , common to these two straight lines brayh , biakl , and by this end a draw a straight line ae perpendicular to the straight line brayh , and a straight line ag , perpendicular to the straight line biakl , from the point r draw as far as the straight line ae a straight line re even with rb , from the point i draw as far as the straight line ag a straight line ig even with ib. from the point e carry to the straight line braih a straight line eh , perpendicular to the straight line re , from the point g , carrry to the straight line biakl a straight line gl , perpendicular to the straight line ig , from the points b and b carry a straight line bq that may divide in half the angle cbd and a straight line bp that may divide in half the angle dbf . by the points q and h draw a straight line qoh , and by the points p and l draw a straight line pol , find the end or the point o common to the two straight lines qoh and pol , and from the point a for center and space ao draw an half circle that may meet with the straight lines al in k and ah in y. now make in some other place even or flat , as in the second figure in one and the same line bdfc three cuts bc , bd , bf , even with the sun beams , figure iii. bc , bd , bf , each of them to his own from the point b of this second figure for center , and from the interval or space ey or gk , of the first figure , draw an half circle o from the point c figure ii for center , and from the space co , of the first figure draw an other half circle o from the point d of the ii figure for center and space do of the first figure , draw an other half circle o , and from the point f also of the ii figure for center and space fo of the i figure , draw an other half circle o , and if you have done right , all these half circles will meet in the same point o , if not , you have not been exact in working . by the points b and o draw a straight line bo , take in this line a point at discretion , first make three rods even with the spaces ci , di , fi of the second figure , and every one sharp at both ends , make in the length of the axeltree rod figure iii the space bi , even with the space bi of the ii figure . lastly set these rods to the axeltree figure iii as i have said at the end of the fifth table , and the axeltree of the dyal is placed . there are some situations of superficies of dyals , where practising this manner of drawing one or the other of the points lh or o comes so far from the straight line cf , that you should have need of too great a space to come to it . but in what manner soever the superficies of the dyal may be situated , and at all times or seasons of the year , i mean , in any strange or odd kind of example that may be found , you may work or practise these kinds of draughts with as much ease as in the most easy pattern . 5 and by means of these three angles even with those in the air between the beams of the sun , you may chuse at pleasure within the lines that represent those beams , other points cdf and otherwise disposed between them , then those which the shadow of the point of the pin hath given upon the superficies of the dyal , and upon those three points chosen out at pleasure , you may make an other triangle cdf , and practise afterwards this manner of drawing as far as the triangle cbo figure ii than in this triangle ; and in the straight line bc , make bc , bd , bf , even with the beams in the air , bc , bd , bf , of the third figure , contained from the point of the pin b , to the points of shadow cdf in the superficies of the dyal each of them to his own , and after you have taken , as it is said , the point i in the straight line bo , you must make use of the points cdf , last made in the triangle ocb , for to set the rods ci , di , fi , to the axeltree bi then to work on as before . to make other points instead of those of the superficies of the dyal , you need only to make some at the two extremities or furthest ends cf , and make bc , and bc , equal one to the other , and unequal with the middlemost bd but a little bigger , more or lesse according as the angles dbc , dbf , are more or lesse unequal among themselves , and instead of making figure i the triangle cdf of the spaces between the points of shadow cdf of the superficies of the dyal , you shall make it of the spaces between the points that are set in the place of these points of shadow . 6 to the theoriciens . make in one and the same plain , as in the first figure , vith three right lines cgkd , crtf , dief , a triangle , cdf equal and like to the triangle of the three points of shadow , fig. iv. cdf make upon the said three straight lines cgkd , crtf , dief , three other triangles cbd , cbf , dbf , equal and like to the triangles in the air of the beams of the sun , iii. fig. cbd , cbf , dbf every one to his own . by the points b and b i. fig. draw a straight line bqg that may part in two the angle dbf. draw out of the point c at your discretion a straight line aqkty perpendicular to the straight line cgkd , and out of the point f , draw a straight line hpirx perpendicular to the straight line feid , make in the triangle fcb the section or cutting cl , equal with ca , of the triangle cbd , and the section fs , with fh , of the triangle fbd , from the point t center , and space tl , draw a bow lm , from the point k , center and space ka , draw a bow am , that may meet with the bow lm , in m ▪ and draw along the straight line km , from the point r center and space rs , draw a bow sn , from the point i center and space i , h , draw a bow in ( hn ) that may meet with the bow ( sn ) in ( n ) draw along the straight line ( in ) : make in the straight line ( km ) the section or cutting ( ku ) equal with kq. by the point ( u ) bring to the straight line aqkty , a straight line ( uy ) perpendicular to the straight line ( km ) ; make in the straight line ( in ) a section ( iz ) equal with ip ; by the point ( zx ) carry to the straight line ( hpirx ) a straight line ( zx ) perpendicular to the straight line ( in ) finde out the butt end ( y ) common to the two straight lines ( aqkty ) and uy . and also the butt end x common to the straight lines hpirx and zx , draw the straight lines goy , and eox ▪ find the butt end o common to these straight lines gov , eox . make in an other place figure ii. a triangle gqy , of the three straight lines , as , gq , gy , and yu of the first 7 figure ; make in the ii. fig , and in the straight lines gy and gq , the section ( go ) equal to go of the i. fig. and the section gb also equal to go of the i. figure ; draw if you will the straight line ( bo ) of the second figure . make again in another place fig , 3. a triangle ( cbo ) of the three straight lines ( bo ) of the triangle ( gbo ) of the second figure . and of cb and co of the first figure ; and upon bc fig. 3. make the cuts bc , bd , bf , equal to the lines bc , bd , bf , of the first figure , every one to his own respectively . and if you have done rightly , the spaces fo , do , co of the triangle cbo , fig. 3. are equal with the spaces fo , do , co of the first figure , every one to his own respectively . take fig. 3. in the straight line ( bo ) according to your discretion the point ( i ) other then ( b ) make three sticks sharp at both ends , and equal to the three spaces ci , di , si , of the third figure : mark along upon`the rod or axeltree the space ( bi ) equal to the space ( bi ) of the third figure : work as i have said , and as the fourth figure doth shew you , and you shall find the axeltree of the dial placed in his right place . you may after this manner , as in others substitute , or bring in other points cdf in stead of those of shadow of the superficies , or face of the dyal and work by this mean , every where with the like ease . figure 8 , for those that have skill in geometry . the higher figure is the place of the dyal with the face unequal to the pin ab , and to the three points of shadow cdf , all markt , as it is said . get a flat and solid thing , as a slate , a board , paceboard , or the like . draw upon it in the lower figure a straight line bdfc , make in that line three cuts bc , bf , bd , equal with the three spaces bc , bf , bd , of the place of the dyal , each of them to his own respectively , then from the point b of the lower figure for the center , and from the spaces bc , bf , bd , draw some circles dh , fe , cg . by this means you see whether the spaces bc , bd , bf , of the higher figure or dyal , are equal or unequal one unto another , and when these spaces are unequal among themselves , as it happens in this example , you see which is the least , and which is the biggest , as in this example , the space bd , comes to be the shortest of the three . now from the point c of the lower figure for the center , and from the space between the two points of shadow c and f of the higher figure , draw a circle e , that may meet in one point e , the circle of the space bf , viz. the circle fe , for it must meet with it , then draw the straight line fb , that may go and meet in one point h , the circle of the shortest space bd , viz. the circle dh . again from the point c of the lower figure for center , and space between the two points of shadow c and d of the higher figure , draw a circle n , that may go and meet in the point n , the circle of the shortest space bd , viz. the circle dh , for it must meet with it . from the point f in the lower figure for center , and from the space between the two points of shadow fd of the higher figure , draw a circle that may meet in the point r , the circle of the shortest space bd , viz. the circle dh , for it must meet with it ▪ by this means the three spaces or straight lines dh , dr and dn , of the circle dh , which is that of the shore●t space bd , have the conditions that are requisite for the making of a triangle . figure 9 , for those that are skilled in geometry . make in another place , as in the lower figure , a triangle dgv of three straight lines , equal with the three spaces , dh , dr , dn , of the higher figure , every one to its own . find in the lower figure the center o of a circle , the edge whereof may reach to the points vdg , according as the lower figure doth declare . draw a straight line doe , through the diameter or midd'st of this circle . by the point o in the lower figure , draw a straight line poq , perpendicular to this diameter doe . from the point d in the lower figure , for the center and space bd of the higher figure , draw a circle that may meet as in b , the straight line qop , for it must meet with it in one or two points , viz. in the times of the equinoxe in one point only , which is the point o , and all the rest of the year , in two points divided on both sides from the point o. and that you may be exact in working , do as much on the other part , and of e for center . draw in the lower figure the straight line bd , which in the times of the equinoxe is joyned with the straight line od , and all the rest of the year is divided from it , and draw along this straight line bd beyond the point d. make in the straight line bd of the lower figure two sections bc and bf , equal to the two sections bc , & bf of the higher figure , each of them to his own respectively . take in the straight line qop of the lower figure of one side or other of the point b , at your discretion , a point i , besides the point b , and that may stand as far from the point b , as the occasion may give you leave . figure 10 , to those that are skilled in geometry . then according as the lower figure shews you , cut three sticks ci , fi , di , sharp at both ends , and equal to the three spaces ci , fi , di , of the higher figure , each of them to its own space respectively , and upon the rod below , whereunto you mean to make the axeltree of the dyal , make a section bi , equal with the space bi of the higher figure . 8 9 10 11 figure 11 , for those that are skilled in geometry . afterwards as you see in the lower figure set to the points of shadow cdf upon the place of the dyal , the three ends cdf of the sticks ci , di , fi , each of them to his own point ; and the point b of the axeltree rod to the point of the pin ab , then bring into one point alone in the air i , the three other ends i of these three sticks ci , di , fi , and set them to the point i of the axeltree rod . for the ends of these three sticks , and the point i of the axeltree ought to meet all four together in one point in the air i , and then you shall find the axeltree rod placed as it must be in the dyal . so that you need no more but to make it fast afterwards in this position or placing , or else to place an other in an other place that may be a parallel to it . if the matter was only about that which is sufficient to shew geometrically the truth of the proposition , it were sufficient to have either the three sticks only without the spacebi of the axeltree rod , or the spacebi of the axeltree rod with two sticks , without a third . but to make the operation sure and effective , you can not be confident that you have done rightly without a fourth stick that may serve for a proof ; this is that which monsieur de sargues had a mind to impart unto you . figure 8 , the same thing over again , but in other terms . to the workmen of many sorts of arts . vvhen you have markt the three points of shadow cdf in the place where you mean to make one of these dyals , draw with the rule in some even or flat place , as you see in the lower figure , a line bd , fc , and make in that line a prick or point b where you shall think fit , or at your discretion . then go to the place of the dyal in the higher figure , take with the compasses the space from the point b , of the ●pin●●… b to the point of shadow c. and with that space come back to the line bd , fc of the lower figure , set one of the feet of the compasses to the point b , and with the other foot go and mark in that same line bd an an other point or prick c , then with the same space , give a stroak with the compasse cg about the point b. go back again to the place of the dyal above , take with your compasses the space from the point b , of the pin ab to the point of shadow f , and with this space come back to the lower figure ; set again one of the feet of the compasse to the point b , and with the other foot go and mark in that same line bc an other prick or stay f , and draw again about the point b with the same space of the compasse this half circle fe . 8 then look in the lower figure which of the three stroaks cg , fe , dh , is nearest the point b , and which are the furthest off , as in this example you see that the stroak dh is nearer to the point b than any of the two others fe and cg , and if they were either two or three together it were no matter . when you know which of these stroaks of the lower figure cg fe and dh , is the nearest to the point b , and which are the farthest , as here , the stroak dh is the nearest , and the two cg , fe are the farthest off . go to the higher figure to the points of shadow c , and f , which are even with the two stroaks below cg , and fe , which are the furthest from the point b , and open your compasses upon the points of shadow c and f , and remember well the two letters or cotes upon which you have opened your compasses , and with this space come back to the lower figure , and set one of the feet of the compasses to the point c , and with the other foot go and mark a point e , upon the stroak of the compasse fe , for it must reach to it . then draw with the rule by the two points e and b , a line eb , that may go and make a point h upon the stroak dh , which is the nearest to the point b. go back again to the higher figure , and open your compasses upon the points of shadow c and d , and with this space come back to the lower figure ▪ set one foot of the compasse upon the point c , and with the other foot go and mark a point n ▪ upon the stroak dh , which is the nearest to the point b , for it must reach to it . go back again to the higher figure , and open your compasses upon the point of shadow f and d , and with this space come again to the lower figure , and set one foot of the compasse upon the point f , and with the other foot go and make a point r upon the stroak dh , which is the nearest to the point b , for it must reach to it . after that , you have no more to do upon the place of the dyal , till you place the axeltree as it ought to be , and you have in the lower figure upon the stroak da , which is the nearest to the point b , four several points or stayes dnrh to make three point perdus with , as you shall see , in the mean time remember when you open the compasse upon the points of shadow in the place of the dyal , to take great notice upon what letters you have opened your compasse , that you may apply the same space below upon the two stroaks which are equal with the two points of shadow upon which you have opened your compasse , and set one foot upon one of the stroaks , and the other upon the other stroak . and moreover that the points nr , may well come out from betwixt the points d and h ; and that i have caused them to come in so betwixt them , by reason of the smallnesse of the place , and what way soever they come to be disposed , it is but one and the same thing still . fig. 9 , to the workmen of many sorts of arts . set your compasse upon the points d , and h , of the higher figure , and with that space go to some flat or even place in the lower figure , and make two points d , and v , so that the space dv below , may be even with the space dh above . then go to the figure above , and set your compass upon the points d and r , and with this space come back to the figure below , and set one foot of the compasse to the point v , and with the other foot draw a line from the point d , to the point g , so that the space vg below , may be even with the space dr above . go back again to the figure above , and set your compass to the points d & n , and with this space come back to the lower figure , & set one foot of the compass to the point d , and with the other foot draw a line from the point v , to the point g , so that the space dg below , may be even with the space dn above , and may meet in g the other circular line that you have drawn about the point v , for it must meet with it . and so you have made in the lower figure three points vgd that will be perdus or lost . now find a center o , upon which having set one of the feet of the compass , and the other upon d , let this foot in turning the compass about , go and passe by those three points perdus vgd , then draw with the rule by the points , as it were o and d , a line doe , and setting again one foot of the compass to the point o , and turning the other foot to e , make in the line doe , the side oe , even with od. then by the point o , draw a line qop , that may cut the line doe , in two equal parts ; again set your compass to the points b , and d , of the figure above , and with this space go to the figure below , set one foot of the compass to the point d , & with the other foot draw from the point e , a line b , that may meet as it were in the point b , the line qop , and make with this other foot of the compass a point b , in the line qop , for it must meet with it , if you have done exactly . 9 9 when the dayes and the nights are equal , it meets with it in one point alone , viz. o , and at some other times it meets in two points , one of one side of the o , and the other on the other side , as in the point e. then remove your compass out of his place , and with the same space of the points b and d of the figure above , set one foot of the compass to the point e , and with the other foot draw from the point d , with your compass another line b , that may go and meet the line ▪ qop , with the line that you have traced with the compass about the point d , and both of them in one and the same point b , for it must do it if you have been exact : and that serves to mark more exactly this point b , in the line qop , how neer soever it is to the point o. after that , whether the point b of the lower figure meets with the point o or not , draw with the rule by the points b and d , the line bd , and draw this line bd , as you see beyond the point d. that being done , open your compass upon the points b and c , of the higher figure , and carry this space to the line bd of the lower figure , and from b into c. set your compass again upon the points b and f of the higher figure , and bring this space to the line bd of the lower figure , and from b into f. and finally make in the line qop , a point i , at your discretion of one side or other of the point b , and let it be as far distant from the point b , as occasion will give you leave . and so you have in this lower figure from the point i to every one of the points bdfc , all the measures that are necessary for the placing of the axeltree or needle in your dyal , in the manner hereafter following . figure 10 , to the workmen of many sorts of arts . cut three rods or sticks sharp at both ends as you see below , one ci of the length that is betwixt the point c , and the point i of the figure● above ▪ the other fi of the length that is betwixt the point f , and the same point i of the higher figure , the other di of the length that is betwixt the point d , to the same point i of the higher figure , then open your compasse upon the points b and i of the higher figure , and bring down this space upon the axeltree rod , and make in the same as you see , two points b and i with this same space bi of the figure above . 10 11 figure 11 , to the workmen of many sorts of arts . goe to the place of the dyal below which i have expressed again , a purpose to avoid the confusion of lines , and put the end of the rod ci to the point of shadow c , the end f of the rod fi to the point of shadow f , and the end d of the rod di to the point of shadow d , and set the point b of the axeltree rod to the point b of the pin ab . then bring together into one point in the air i , the three other ends of the three rods ci , di , fi , for they must come in there together , and bring the point i , of the axeltree rod , to the same point i in the air together with the three other ends of the rods i , for these four things must come alltogether into one and the same point in the air i , if so be you have been exact in working . and when these three ends of the rods and the point i of the axeltree rod , are all four gathered together into one and the same point in the air i , the axeltree rod will come to be placed directly as it must be in the dyal , and so you need no more but to make it fast in that place , or to fasten an other either near it or farr from it , that may be even with it , or parallel to , or equally distant from it . if the four points i should go and meet in the body of the dyal , you must but take in it's figure the point i nearer , or in the other side of the point b , and make an end of the rest as i have said . 8 figure 8 , i will say the same thing over again , but more at large . to all sorts of people that have neither skill in geometry nor in arts ; but are apt and sit to learn them both ▪ before you undertook to make this dyal , you had nothing about you , nor knew nothing wherewith to further you in it , and going about it , you have made use of the pin ab , as it were at a venture . now you must consider that having placed the pin ab , in this manner , you have given out of your self in the end thereof a point alone unmovable and fixed in the air . then by means , of this fixed point in the air b , and of the sun-beams , you have found out three other unmovable and fixed points of shadow cdf , on the outward face of the place where you have a mind to make your dyal ▪ so you see that by means of this end of pin b , and of the sun-beams , you have established upon the place where you intend to make your dyal four points fixed and divided one from another , viz , one in the air , which is the point or the end b of the pin ab , and three in the superficies of the dyal which are the three points of shadow cdf . whereby you have found also six spaces , that is to say , the lengths of six straight lines unmoveable , fixed , distinct , and divided one from the other . for if you consider well , you shall see that you have found out by this means the spaces , or lengths , or distances that are from the point b of the pin ab , to every one of the three points of shadow cdf , viz. the space from the point of the pin b to the point of shadow c , the space from the same point of the pin b to the point of shadow d , and the space again from the same point of the pin b to the point of shadow f. and for your better instruction , if you will make these three lines visible to the eye , set unto every one of them either a ruler or a string stretch'd out in a direct line from the point of the pin b , to every one of the points of shadow cdf , as the points do shew it unto you ; and so you may see the three lines bc , bd , bf , which otherwise are invisible in the air . and besides these three spaces or lengths , you have also found out the three spaces or lengths that are from every one of the three points of shadow cdf , unto the other , viz. the space from the point of shadow c , to the point of shadow f , the space from the point of shadow c , to the point of shadow d , and the spaces from the point of shadow f to the point of shadow d , as you may see by the points that are there . so you have six spaces or lengths bc , bd , bf , cf , cd , df , which you have already found unmovable , and fixed to the place wherein you intend to make your dyal , which are so great a furtherance unto your work , that there remains nothing else to do , but by the help and means of the said six spaces or lengths , to find also three or four more , that you may have all that is requisite for the placing of the axeltree rod of your dyal as it ought to be . you must know that there are several wayes whereby these six spaces which you have found already , viz , bc , bd , bf , cf , cd , df , are made use of to find out those three or four more , which you must have to inable you to place the axeltree rod of your dyal as it must be . and that of all those several wayes a man may have a liking to one for one reason , and another man to an other for some other reason , and of those several wayes monsieur de sargues hath shewed me three or four at the most , viz. that which he hath set down in the figure of his model or project page , and of the others for which you must know how to make sometimes somekind of alteration , and which i have set down in short , there is one in the sixth figure , and another in the seventh . as for this it is such , that there is no occasion but you may practise it in effectually , and with the like ease every where , without you need either to add or alter any thing , as you shall see presently . draw with the rule , as you see in the figure below in some flat or even place a straight line bd , fc , then go to the figure above , and open your compasse , and set one of the feet to the point b of the pin ab , and the other foot to the point of shadow c , and by that means you shall take with your compasse the space or the lengths that are from the point of the pin b , to the point of shadow c , whereof you will be pleased to remember , to the end that when i shall bid you for brevity sake , take after the same manner with your compasse such a space , you may be able to do with your compasse even as i told you just now of the space bc in the figure above . now with this space bc of the figure above , come back to the figure below , and set at your discretion one of the feet of the compasse upon the straight line that you have drawn there , as for example set it to the point b , then turning the compasse about upon this point b , draw with the other foot a circular line cg , which circle by this means shall have a space bc equal with the space bc of the higher figure , and will meet the line bd for example in the point c. go back again to the figure above , take there after the same manner the space from the end or point of the pin b to the point of shadow d , and with this space come back to the figure below , and set again one of the feet of your compasse to the point b , and holding it still upon this point b draw with the other foot a second circular line dh , that will be equal with the space above bd , and that may meet the line bc , for example in the point d. go back again to the figure above , and take with your compasse the space betwixt the point of the pin b and the point of shadow f , and with this space come back to the figure below set one of the feet of the compasse to the point b , and draw with the other foot a third circular line fe , with the space bf of the figure above , and that may meet the line bd for example in the point f. by this means you have set away and transported the three spaces bc , bd , bf , from the rise or place which they had in the place of the dyal above in a flat and even place below ; and all of them united together in one single line bdfc , in which you may see whether those spaces be equal amongst themselves , as they may be in some occasions which is indifferent , or whether they be unequal , by seeing whether the points cdf , are united together two or three in one single point , or whether they are disunited or divided one from another , and when these three points cdf , are disunited or divided one from the other which happens most commonly , and that these three spaces bc , bd , bf , are unequal amongst themselves , as it falls out in this example , you see which of these spaces are the greatest and which is the least , considering which of these three points cdf is the nearest , and which is the furthest from the point b , that is to say also that by this means you see which of these circular lines cg , ef , dh , is nearest to the point b , and which are furthest as in this example you see that of the three spaces bc , bd , bf , those two bc and bf , are the greatest , and bd is the least ; and of the three half circles cg , fe , dh , you see that dh is the nearest to the point b , and that the circle fb is nearer to it than the circle cg . when you have thus found out which of the three spaces bc , bd , bf is the least , and which of the three circles cg , fe dh is the nearest of the point b. go back again to the higher figure to the points of shadow cdf , and take with the compasse the space betwixt the points of shadow c and f , which are at the ends of the two greatest spaces bc and bf , and with this space cf of the higher figure , go to the lower figure to the same point cf , and set one foot of the compasse upon either of those points c and f , that is the furthest from the point b , viz. c , and holding still the point of the compasse upon this point c , go and mark with the other foot a point for example e , in the circle of the other of these two points c and f , viz. in the circle of the point 〈◊〉 which is the circle fe , for this other foot of the compasse must reach to this circle of the point f , as for example in the point e ; this being done ▪ draw by this point e to the point b a straight line eb , that may go and meet , in one point h the circle of the point d which is the nearest to the point b , and mark in it this point h. then go to the figure above , and take with the compasse the space betwixt the two points of shadow c and d , and with this space cd of the figure above go to the figure below , to the same points c and d set one foot of the compasse to that point of these two c and d , which is the furthest from the point b , viz. c , and keeping this foot of the compasse upon this point c , go and mark with the other a point , for example n in the circle of the other of the two points c and d , viz. in the circle of the point ▪ d with the circle dh , for this other foot of the compasse must reach to the circle of the point d , for example in the point n. go back to the figure above , take with your compasses the space betwixt the two points of shadow f and d , and with this space fd of the figure above , come back to the figure below to the like points f and d set one foot of the compasse upon that point of these two f and d , which is furthest from the point b , viz. f and holding still the point of the compasse upon this point f , go and mark with the other foot a point for example r , in the circle of the other of these two points f and d to wit , in the circle of the point d which is the circle dh , for this other foot of the compasse must reach to the circle of the point d , for example in the point r. this being done , you have no more to do in the place of the dyal , till you go and place the axeltree rod in it as it must be , and in the figure below in the circle of the point d , which is nearest to the point b , you have found by this means four points dn , rh , different and divided one from the other , and when the two points n and r should be found united together , it were no matter . now by the means of these four points you have three spaces amongst others from the point d , to every one of the three points hr and n to wit , the space from d to h , the space from d to r , and the space from d to n , of which spaces you see which is the greatest and which is the least , when they are all three unequal , as in this example , for it may happen that there will be two found equal amongst them . and with the help of these spaces dh , dr , dn , you shall find presently those four that you must have for the placing of the axeltree rod of your dyal , as it ought to be . figure 9 , to all sorts of people ▪ take with the compasse in the figure above , of these three spaces dh , dr dn , that same that is the greatest of all , as in this example here , the space dh and with this space dh of the higher figure , go to some place that is flat , as in the figure below , and set at the same time the two feet of the compasse upon it , as for example , to the two points d and v ▪ and mark these two points as it were d and v , which by this means will be distant one from the other , the length of the space dh of the figure above . go back again to the figure above , take there with the compasse the space from d to r , and with this space go to the figure below , set one of the feet of the compasse to the point v , and from thence draw with the other foot towards the point d a circle g , which by this means shall be made of the space dr of the figure above . go back again to the figure above , take with the compasse the space from d to h , and with this space go to the figure below , set one of the feet of the compasse to the point d , and from thence with the other foot draw towards the point v another circular line that may meet , for example , in g the other circle , which you have drawn about the point v , for this other foot of the compasse must meet the other circle in one or two points , and for example , in the point g for one . 9 figure 9 , to the workmen of many sorts of arts . set your compasse upon the points d and h of the higher figure , and with that space go to some flat or even place in the lower figure , and make two points d and v , so that the space dv below , maybe even with the space dh above . then go to the figure above , and set your compasse upon the points d and r , and with this space come back to the figure below , and set one foot of the compasse to the point v , and with the other foot , draw a line from the point d to the point g , so that the space vg below , may be even with the space dr above . go back again to the figure above , and set your compasse to the points d and n , and with this space come back to the lower figure , and set one foot of the compasse to the point d , and with the other foot draw a line from the point v to the point g , so that the space dg below , may be even with the space dn above , and may meet in g the other circular line that you have drawn about the point v , for it must meet with it . and so you have made in the lower figure three points vgd , that will be perdus or lost . now find a center o , upon which having set one of the feet of the compasse , and the other upon d , let this foot in turning the compasse about go and passe by those three points perdus vgd , then draw with the rule by the points , as it were , o and d , a line doe , and setting again one foot of the compasse to the point o , and turning the other foot to e , make in the line doe , the side oe , even with od. they by the point o , draw a line qop , that may cut the line doe in two equal , parts . again , set your compasse to the points b and d of the figure above , and with this space go to the figure below , set one foot of the compasse to the point d , and with the other foot draw from the point e a line b , that may meet as it were in the point b the line qop , and make with this other foot of the compasse a point b in the line qop , for it must meet with it , if you have done exactly . when the dayes and the nightes are equal , it meets with it in one point alone , viz. o , and at some other times it meets in two points , one of one side of the o , and the other of the other side , as in the point e , then remove your compasse out of his place , and with the same space of the points b and d , of the figure above set one foot of the compasse to the point e , and with the other foot draw from the point d with your compasse another line b , that may go and meet the line qop , with the line that you have traced with the compasse about the point d , and both of them in one and the same point b , for it must do it if you have been exact , and that serves to mark more exactly this point b in the line qop , how near soever it is to the point o. after that , whether the point b of the lower figure meets with the point o or not , draw with the rule by the points b and d the line bd , and draw this line bd , is you see , beyond the point d. that being done , open your compasse upon the points b and c of the higher figure , and carry this space to the line bd of the lower figure , and from b into c. set your compass again upon the points b and f of the higher figure , and bring this space to the line bd of the lower figure , and from b into f. and finally make in the line qop a point i , at your discretion , of one side or other of the point b , and let it be as far distant from the point b , as occasion will give you leave ; and so you have in this lower figure from the point i , to every one of the points bd , fc , all the measures that are necessary for the placing of the axeltree or needle in the dyal , in the manner hereafter following . figure 9 , to all sorts of people . that being done open your compasse at your discretion , and the more that the occasion will permit you to open it , it will be so much the better , and with this opening , set one of the feet of the compasse to the point g of the figure above , then turning about this point of the compass upon this point g , draw with the other foot four circles h , l , m , s , about the point g. viz. two h , and l from the point d , and two m and s from the point v , then remove your compass , and set one of the feet upon the point d , and with the other foot draw from the point g , two circles that may meet in two points , viz. land g , those two circles that you have drawn about the point g , from the point d , if this same foot of the compass could not meet with these two circles hl , that you have drawn about the point g & from the point d , it is because you had not opened your compass enough , before you did set it upon the point g , and in such a case you shall open it more , and set it again upon this point g , and when this same foot meets these two first circles , for example , in h and in l , matk these two points l , and h. afterwards remove the compass and keeping still the same open , set one foot to the point v , and turning the compass upon the point v , draw with the other foot from the point g , two circles that may meet likewise in two points as s , and m , the two circles that you have drawn about the point g , from the point v , and mark those two points s and m , in which those two circles meet with the other two : and therefore note that before you make yout compass to turn upon the point g ▪ you must open it in such a manner , that when you shall set it afterwards upon the points d and v , the other foot may meet with the circles that you have drawn about the point g. 9 then draw by the two points h and l , a long straight line h , l , o , and by the two points s and m , draw an other long line and as straight , m , s , o , and these two lines hl and ms , being sufficiently drawn at length , will meet in one and the same point o. by the two points d and o , draw a straight line do , and draw it at length , as you see from the point o , then set one of the feet of the compass upon the point o , and the other foot upon the point d , and turn that foot of the compass which is upon the point o. the other foot which is upon the point d must go and touch every one of the points g and v , and when this other foot hath gone over the point d , g , v , go and mark at the very same time with it a point , as e in the line doe , and so you shall make the portion oe of the line od , equal with the portion od. then open your compass at discretion more than from the space od , and as much as the occasion will give you leave , the more the better , and the compass being so open at discretion , set one of the feet to the point d of the lower figure , and turning that foot of the compass upon the point d , draw with the other foot from the point o , two circles , as p and q then remove the compass , keeping still the same distance , set one foot upon the point e , and turning it about , draw with the other foot from the point o , two other circles , that may meet in two points with the two circles that you have drawn about the point d , and as for example in the two points q and p , and draw with the rule by these two points , as q and p , a long straight line qp , that must reach to the point o , if you have been very exact in the working ; if it doth not reach to it you have not been very exact , and i advise you to begin it again : if it reaches to it , go back to the figure below , and take with the compass the distance between b and d , then with this space go to the figure below , set one foot of the compass upon the point d , and turning it about , draw with the other foot from the point o , a circle that may meet the line qop , as for example in the point b , for this other foot of the compass must go and meet that straight line poq either in one or in two points , because the space from b to d of the higher figure ought never to be smaller or lesser than the space do of the figure above . it is true that twice in the year , viz. in autumne and in the spring , when the dayes and nights are equal , that space bd of the figure above comes to be equal with the space do , of the figure below , and in those times that other foot of the compasse that tutns about the point d of the figure below , meets the line qop just in the point o. but at all other times the space bd of the figure above is somewhat bigger than the space do of the lower figure ; and then the other foot of the compasse that turns about the point d , meets the line qop , in two points , one of each side of the point o , as for example in b , for one . and that you may be the more exact , remove the compasse from one part of the straight line bo unto the other , and with the same opening of the space bd of the higher figure , set one of the feet upon the point e of the figure below , and turning this foot upon this point e , draw with the other and from the point d an other circle that will meet ( if you have been exact in the working ) the straight line qop , and the circle also that you have drawn about the point d , and both in one point ; as for example in the point b , which will inable you to discern well the point b in the straight line poq ; mark this point b in the line poq , whether you find it united with the point o , and so both of them making but one and the same point , as it falls out , when the days and nights are equal , or whether you find it divided from the point o , as it falls out in other seasons , and as you see in this example ▪ then draw with the rule by these two points b and d a straight line bd , which you shall stretch out sufficiently beyond the point d. when the days and nights are equal , as in autumne and in the spring , and that the point b is found to be united with the point o , the line bd comes likewise to be united with the line od , and both together make but one line ; but at any other time , as the two points b and o are two several points and divided one from the other , so the two lines bd and od , are two several lines ▪ and divided one from the other , this being done , go to the figure above , mea●ure with your compasse the space from b to c , and with this space go to the figure below , set one foot of the compasse upon the line bd , to the point b , and set the other foot in any place of the same line bd where it may light upon ; as for example in the point c , by this means you shall make the portion bc , of the line bd of the lower figure , equal with the portion bc , of the line bd of the higher figure , make after the same manner with the compasse , the portion bf , of the line bd of the lower figure , equal to the portion bf , of the line bd of the figure above . finally , in the same figure below , and in the line qop , mark at your discretion another point , i , of one side or other of the point b , according as you shall find it most convenient for the place of the dyal , and as far from this point b , as occasion will permit , the further the better , and so you have found the four spaces that you wanted for the perfect placing of the axeltree of your dyal . for in so doing , you have found in this figure below the distances that are from every one of the four points bdf c , to one and the same point i , that is to say the space from b to i , the space from d to i , the space from f to i , and the space from c to i , which distances bi , di , fi , and ci , will serve you to place the axeltree of the dyal in the manner following . figure 10 , to all sorts of people . cut ( as you see in the lower figure , ) three sticks sharp it both ends , one ci of the length of the point c , to the point i , otherwise of the space ci of the figure above : the other fi , of the length of the space fi of the higher figure , and take with your compasses the space bi of the figure above , and being so open , see both feet at once upon a straight line , along the axeltree rod of the lower figure , for example , in two points as b and i , and mark these two points b , i , in the axeltree rod . 10 11 figure 11 , to all sorts of people . that being done , go to the place of the dyal , the which , to avoid the confusion or multiplicity of lines , i have set below in the lower figure , set in this lower figure one of the ends of the stick ci , to the point of shadow c , one of the ends f , of the stick fi , to the point of shadow f , and one of the ends d , of the stick di to the point of shadow d , and one of the points b of the axeltree rod , set it to the point b , of the pin ab . and holding thus the three ends cdf , of the three sticks to the points of shadow cdf , every one respectively to his own , and the point b of the axeltree rod , to the point of the pin b , bring together the three other ends i , of the three sticks or rods ci , di , fi , into one point in the air i , for they must meet there ▪ then bring the point i of the axeltree rod , also to the point in the air i , with the three ends i of the sticks , for it must come and meet there exactly , if you have done right , or if the straightnesse of the place hath not hindered you . if the straightnesse of the place of the dyal hinders the three ends i , of the sticks from meeting together in one point in the air i , take the point i in the figure below in the ninth cut , or that above in the tenth figure in an other place , than in that where you had taken it and according to the occasion , then bring the sticks to it as before ( for you may take it anywhere , or in any place of the line poq , of one side or other of the point b ; ) but the further you can take it from the point b , will be better , and take it in so many-places , that having set the sticks of the points cdf , to this point i , and mark'ed the space bi upon the axeltree rod , the four points i , may at last meet together in one point in the air i. and when the point b of the axeltree rod , is at the point b , of the pin ab , and when the three ends i , of the sticks , and the point i of the axeltree rod are met , as you see in the lower figure , all four together in one and the same point in the air i. the axeltree rod will come then to be placed , just as it ought to be in the dyal . that if you do not care to be sure that your dyal must be as just , as it is possible for art to do , in such a case , you may spare one of the four lengths ci , di , fi , bi , and content your self with three only , as being sufficient for the theorie : but the fourth will serve you for a proof , to see whether or no you have been very exact in working , and will justifie the three others . figure 12 , to all sorts of people . the figure above shews you how that which you have done with three sticks , may be done either with many compasses , with the help of some body , or else with other kinds of branches tyed or fastened one with another . the same figure above , as also the figures below , shew how every one of those branches may be of two several pieces , which go in by couples into one hoop or ring , and slide along one by another , and are made fast with a screw to the measure where you will have them to stand upon , and these pieces may be made of tinn'd yron , or of yron , if you are afraid that their points will grow dull by often using them . or otherwise they shew you that insteed of one stick , you may have two , both sharp at one end , which you shall fasten and bind together at the other end , of what length or measure you please . the same figures do shew you also , that two divers branches , viz. ci , and fi , may be fastened together in the place where you will have them to stand together , with a presse and a screw to fasten them with . the higher figure shews you besides , that you may ●●●●en or bind with strings or threds , the axeltree rod with the point b , of the pin ab , and the two branches ci , fi , with the axeltree rod , to make them stand fast of themselves in their place . when you have found thus the placing of the axeltree rod , it is in your choice , either to seal it and fasten it in that place , or to place another insteed of it , that may go the same way , and that may be every way equally distant from it ; but that you may be the more exact , it will be as good to seal or fasten that in the place , where the practice of the draught hath caused it to meet , than to place another , unlesse there was some occasion or necessity for it . 12 figure 3 , to those that have understood what hath been said before . having understood what i have said before , concerning those many wayes of finding the position of the axeltree of the dyal , you may compose others besides , making use partly of that of one figure , and partly of that of an other . for example , here is one way composed of two of those that are afore . ovt of the third or fifth figure , you shall take in the sun-beams or sticks bc , bd , bf , three spaces equal each unto the other ; and out of the 5 and 6 figures , you shall make a triangle of three lines equal to the three spaces he , de , dh of the third figure , and you shall find the center o of the circle , circumscribed about this triangle . you shall find also within the ground plot of the points hde , the points like to a & o of the 6 figure or cut , which in this case come to be united together in one and the same point o. that is to say , having found one of these two points a & o , you have found also the other , because they are united or gathered together into one . so you have in the second figure of the third cut , the spaces do , and di , for two sides of a triangle with straight angles or corners odi , whose side di , holds up the straight angle , and the sides do , and di , do contain or comprehend it . make this triangle odi , with three sticks , or with any other thing that may be strong and small as you will , so that you may at your need lengthen the side io , from the right angle o. 3 set the point d , of this triangle dio , to the point of shadow d , and holding this point of triangle to this point of shadow d , make the side io , of this triangle ( drawn at length if need be ) come and touch the point b , of the pin ab for if you have been very exact in working , it must touch it . take a stick hi , of the length of di , set one of the ends to the point h , and bring the other end to the point i , of the triangle odi , without the side io , leave the end b of the pin ab , for it must be so , if you have wrought exactly as you ought . you may have also an other stick ei , of the length di , and set one end to the point e , and bring the other likewise to the point i , of the triangle dio , without the side oi , leave the end b of the pin ab . that being done the rod bi , comes to be the axeltree of the dyal , and placed as it ought to be , and so of all the other wayes that you find besides . you may , if you will make use of a triangle rectangular eoi. and of the stick hi , content your self only with the three equal lengths ei , di , hi , to find out the point thereby , that you may draw from thence a line to the point b , without making use of any thing else to know if you have done exactly or no , you can not be sure whether you have done well or ill . but when you have together with that , either a fourth length bi ▪ or the straight angle doi , that will serve you to try whether or no you have been exact in your operations , for as concerning an effectual execution , unlesse you have from time to time such a kind of proof , to shew whether you have wrought exactly as you ought , you cannot assure your self that your work is as well as it can be done . one thing i must tell you , that in some certain occasions according to the times and the placing , or according as the superficies of your dyal is , the shadow of the pin comes to be of such a length , and the extremity or end thereof so weakned , and so diminished in strength , and so confuse in the superficies of the dyal , that it is very hard to find out figure 13 , to all sorts of people . i come now to the next and second thing that you are to do , which is to trace out the lines of the hours . in this example i suppose that the axeltree rod doth not meet the superficies of the dyal , about the place that you work in ; and therefore i represent it suspended in the air , with two or three supporters as you see , i suppose also that the superficies of the dyal is not smooth , but rough and uneven as i have said . when you have placed the rod bi , which is the axeltree of the dyal , as you see both in the higher and lower figure , you have made an end then of the first of those two things that you were to do , for the making of your dyal : now there remains but the second to be done , which is the finding and the tracing out of the lines of the hours in the dyal ; and for that purpose . consider in your higher figure , that the superficies and the axeltree of your dyal are two divers things , and differing one from an other , and there is no such communication from the one to the other , as that with them alone you may find out directly the place of the lines of the hours , without making use of a third thing that may be a means betwixt those two . the meanest and the least thing that you can have to be a means betwixt the superficies and the axeltree of the dyal , is a ruler . 13 to the end that this middle rule may serve you alike in all occasions , it must have all the conditions that you see represented in the figure below . first it must be as long as the place will give you leave , and it must crosse over if need be the whole superficies of the dyal , and reach over on both sides if it be possible . secondly it must be in the air , and suspended between the superficies and the axeltree of the dyal . thirdly it must be placed as far from the axeltree rod , as possible may be . fourthly it must be placed like a crosse , in regard of the same axeltree rod . figure 14 , to all sorts of people . to place this middle rule well , and as it ought to be betwixt the superficies and the axeltree of the dyal . chuse along the axeltree rod bi , of the higher figure , some fit or convenient place , as in the point o , and make a round and fixed stay in that place , by winding , or tying some strong thing about this axeltree rod , as the figure doth shew . tye a string to the axeltree rod bi , by the means of a ring that may be so big , that you may turn the string with it about the axeltree rod easily , as the lower figure shews you ; then with the corner of a squire ed , in the lower figure , thrust on the ring where the string is , and put it close to this stay o , and holding the string fast between the stay o , and the squire ed , set the back of one of the sides oe of this squires length , to the axeltree rod bi , and by this means , the other side do , of this squire , will shoot out into the air like a wing from the axeltree rod bi , then stretch out the string in a straight line from the stay o of the axeltree , along the back of the other side od of the squire . and holding still in this manner the ring close to the stay of the axeltree , by means of the squire , and the back of one of the sides joyned at length to the axeltree rod , and the other side of the squire like a wing , and the string stretcht out in a straight line along this wing , turn both the squire and the string altogether still in this same manner about the axeltree rod , as the lower figure doth shew . 14 when you have found out those two places that are farthest one from another , in which this string turning in this manner with the squire along the side like a wing , may go and meet the superficies of the dyal , as here the places g and h. make with mastick , or plaster , or cement , or such like stuff , a little knob flat at the top in each of these places , as for example , one in g , and another as in h , which two knobs may shoot out of the superficies of the dyal , in such sort , that you may lay a ruler on the top of them , going from one of the knobs to the other , as you see here in the lower figure . 15 figure 15 , to all sorts of people . vvhen you have thus made those two knobbs g , h , in the lower figure , take the squire again and the string , and set them again close to the stay o , of the axeltree rod , as you know they were . and make them go about again as before , both together about the axeltree bi , and while you are turning about , the string will fall right over against the two knobs , shorten or lengthen it so , that it may go and touch a point , at the top of each one of the two knobs , one after another ▪ viz. a point as p , at the top of the knob g , and a point as q , at the top of the knob h , and mark these two points q and p , upon these two knobs . when you have mark'd two points in this manner , set a ruler in the lower figure upon these two knobs , and place it so , that it may passe from one to the other , by those two points q and p , and make the ruler fast in this place with cement or plaster , or the like , in such wise that it may not stir any way . and this rule so placed , is the third and middle piece between the superficies , and the axeltree of the dyal , by means whereof you shall cause , as i shall say hereafter , this superficies and this axeltree to have what communication soever you please one with another . after you have placed this middle rule in this manner , between the superficies and the axeltree of the dyal . consider that in france now they reckon 34 hours , for one day and a night , and that these 24 hours , are divided in twice twelve hours , and that every one of these 12 hours is subdivided in twice 6 hours . so that in the 24 hours of one day and one night , as they are now reckoned in france , there are two hours that are each of them of 12 , that is to say , one hour of 12 in the middest of the night , and another hour of twelve in the middest of the day , these two hours of 12 , are called midnight and midday , then there are two other hours , each of them of 6. viz. an hour of 6 in the evening , and another of 6 in the morning . where you must note , that both the two hours of 12 , and the two hours of 6 , come alwayes to meet together in one and the same line , though it may be lengthened if need be , viz. the two of twelve in one line , and the two of 6 in an other . and you shall know , that it is an infallible thing , that within the compasse of the superficies of the dyal where you work in , if you have placed the axeltree pretty near it , there must needs be either one of the hours of 12 , or one of the hours of 6 , and sometimes they meet there both at once , that is to say , one of the hours of 12 , and one of the hours of 6. there be many situations of superficies of dyals , in which , within the compasse , where one may trace in the hours , there is only the line of the hours of 12 , and there is found not any one of the hours of 6 , and others in which there is found only the line of the hours of 6 , and not any one of the hours of 12. but there is no dyal in which , within the compasse where it is traced in , but the hours of 12 , and any one of the hours of 6 are found in it , i mean that one may find in it , either one or other of the hours of 12 , and of the hours of 6 , by setting the axeltree near enough to the superficies of the dyal . and now since you are sure , that there is without doubt in your dyal , either one of the hours of 12 , or one of 6. you shall begin to seek in it , first the place of that sort of hours of twelve or of six , that may be in it . and when you have found the point , either of one of the hours of six , or of one of twelve ; you shall find afterwards the points of the other hours , that meet with it in the dyal it is in your choice to begin to seek the point , of which of the two sorts of hours of 6 , or of 12 , you will , and i will show you two wayes of seeking them out , both one after another , that when they come to be both in the dyal , you may find them out both there if you will , for they serve for a proof one unto another , if you have been exact in your operation . that you may finish your dyal as you ought , seek in the middle rule the point which is there to be found , either of the hours of twelve , or of the hours of six , for it is set there on purpose to serve for that chiefly . for example , seek out first in it the point of one of the hours of six , as i am going to shew , then i will shew you the way to find out the point of one of the hours of 12 , and afterwards i will shew you the way to find out the points of all the other hours of the day . 16 figure 16 , to all sorts of people . to seek out in the middle rule , whether the point of one of the hours of 6 be there . take the string that is made fast to the axeltree , set it very close again to the point or stay o , as it was when you made it turn about the axeltree , then stretch it out in a straight line from the point o of the stay of the axeltree , to the middle rule pq , and holding still this string so stretcht out , make it turn about the point o , carrying it from one to the other of the points q and p ▪ along the rule qp , and making it longer or shorter if need be , and set a carpenters level or triangle over it , to see , while it turns thus , stretch't out in a straight line about this point o , keeping along the middle rule , whether there is any place , wherein it comes to be found levell , as for example , you see in the higher figure ( over the leaf ) and when you find it to be level , you must make it fast there . and that your level may be more fast , you may set it by the middle upon the axeltree rod close to the stay , or you may set close to the stay a ruler notch't at one end , as you see the ruler n , is notcht , then guide it with the string , and it will serve to fasten the level upon it . or to say it again otherwise , you must , as you know , cause to go about the axeltree rod the squire and the string stretcht out , as i have said , in a straight line , and made longer if need be , this string will go and passe all along the ruler pq . and if it happens that the string , so guided or carryed along the middle rule pq , come to be found level , as the higher figure shews , mark in this middle rule the point 6 , where this string toucheth or reacheth unto when it is so level , and remember that this point 6 , is the point of one of the hours of 6 , either of the evening , or of the morning . this is that which concerns one point of one of the hours of 6 , that if the string in turning thus , comes to passe from one end of the middle ●●…e to the other , without falling to be level it is a sign , that not one of the hours of 6 , comes to be found in this dyal , to frame it from the point that you have taken for a rest or stay . now that you may seek out the point of the hours of 12 , in the lower figure . fasten the center of a hanging plummet , with a string s , to the middle of the body of the axeltree rod , above it or under , as the lower figure shews , it matters not , and as the occasion will permit or require , and set this plummet so , that it may come and fall as near the middle rule as you can . then tye to the axeltree rod , as far as you can from the superficies of the dyal , another string i with a loose knot , and when the plummet of the first string s , comes to be at rest , make it so fast that it may not stir , then stretch out this second string i in a straight line , in such sort , that comming from the axeltree rod , it may go and touch the string of the hanging plummet s , without breaking ( or stirring ) the string nor the lead , and so holding this second string i , stretcht out close to the string with the plummet on s , see whether this second string stretcht out in this manner can , being made shorter or longer , if need be , go and meet the middle rule , in a point , or not . and when this second string so stretch'd out comes to meet the middle rule , in one point , as 12 , mark in the middle rule this point 12 , in which this string so stretcht out doth meet with it , and note that that point 12 , is the point of one of the hours of 12. when you have found out and markt in the middle rule the point of one or other of the hours , either of 6 , as you see in the figure above , or of 12 , as you see in the figure below , if you have them both , they shall serve for a proof one to another , if you have but one , you may make use of that alone . let us suppose first , that it is the point of one of the hours of 6 , as the point , 6 ▪ 〈◊〉 you shall go on in finding out the points of the other hours , which may be found in your dyal , in this manner following . figure 17 , to all sorts of people . mark at your discretion in the rule pq . two several points mn , and consider the point in the middle of the body of the axeltree , close by the stay o , that is the point about which you have turned the string with the corner of the squire . you see there three several points unmoveable , and fixed , viz. the point m , and the point n , in the middle rule , and the point o , in the middle of the body of the axeltree rod , close to the stay . and so having those three points fixed , m , n , o , you have by this means the three several distances , viz. the measures of the distances that are from one of these three points , to the two others , viz. the space or distance from the point m , to the point n , the distance from the point m to the point o and the space from the point n , to the point o. remember two things , one is , that the point o , is in the middle of the body , that is to say of the bignesse , and not in the out side of the axeltree rod . the other is , that these two points mn , that you have mark'd at discretion in the middle rule , are not for all that certainly the points of hour , and that they are to serve you to find out the points of hour , and perhaps they may chance to be some of them ; and may be not , and perhaps you must blot them out after you have found out the points of hour . this being done so , take with your compasse upon the middle rule , the distance from the point m , to the point n , and with this space go to some place that is flat or smooth , and set both the feet of your compasse therein at once , as in the figure below in the points m and n , and by these two points , draw a straight line mn , as long at either end as the rule pq . then go back to the dyal above , take with the compasse the distance which is from the posnt m , to the middle of the bigness of the axeltree close by the stay o , or else otherwise , take the distance which is from the point m , to the axeltree towards the stay o , and adde unto it half of the bigness of the axeltree ; and with this space mo , come back to the figure below , set one of the feet of the compasse to the point m , and turning this foot about upon this point m , trace with the other foot a line crooked like a bow o , go back to the figure above , take again with your compass the distance which is from the point n , to the middle of the bigness of the axeltree close by the stay o , and with this space come back to the lower figure , set one of the feet of the compass to the point n , and turning this foot upon this point n , trace with the other foot another crooked line that may meet with the other in one point , as o , for it must meet with it . then open yout compass at discretion , rather more than lesse , and set one of the feet of the compass so open at discretion to the point o , and turning this foot of the compass upon this point o , trace with the other foot a round rgsh . go back to the dyal in the figure above , take with your compass upon the rule qp , the distance which is from one of the points m or n , to the point of 6 hours , and with this space , for example of m6 , come back to the lower figure , set one of the points or feet of the compass upon this point m , go and mark with the other foot in the line m , a point as 6 , of the same side upon the rule . and so you have in the line mn , one and the same thing as you have in the dyal in the middle rule , viz. the three points mn and 6 , of the same distance , in each of these two straight lines . this being done , draw in the figure below by the two points o and 6 , a straight line o , 6 , which may divide the round rgsh , in two halfs rgs , and rhs. open the compass at your discretion , and as much as the space will give you leave , and keeping your compass so open at discretion , set one of the feet to the point s , and turning this foot upon this point s , trace with the other foot , two crooked lines l and d , then with the same space , remove your compass out of his place , and set one of the feet to the point r , and turning this foot about upon this point r , trace with the other foot two other crooked lines , that may meet in two points l and d , the two crooked lines that you have drawn about the point s , and mark those two points l and d , and draw by those two points a straight line ld , which may passe by the point o , if you have been exact in your operations . so you have divided this round , into four quarters of a round , with the two straight lines sor , lod , and if the straight line lod , drawn in length comes to meet the line mn , in a point as 12 , it shews that there is also the point of the hours of 12 , in your dyal , viz. in the middle rule between the superficies and the axeltree ; now divide with your compass every one of these quarters of the round , into six parts equal , as you see in the points that are upon the brim of the round rgsh , and by the center or middle point of this round o , and by every one of the points of these divisions of the edge or brim of the round , draw some lines or beams , as you see some drawn already , that may go and meet the straight line mn , as in the points , 5 , 4 , 3 , 2 , 1 , 11. and these points are the points of the other hours , that are to be found in your dyal . 18 17 figure 18 , to all sorts of people . now take with the compass in the figure above , the space from 6 to 5 , and with this space go to the dyal in the figure below , set one of the feet of the compass to the point 6 , and keeping this foot of the compass upon this point 6 , go and mark with the other foot in the middle rule another point 5 , and by this means you shall transport with your compass the space 6 , 5 , ●●rom the line of the figure above , which represents your table , or the flat place in the dyal of the figure below , upon the middle rule mn , & so accordingly take with your compass every one of the other spaces , 5 , 4 , 4 , 3 , 3 , 2 , 2 , 1 , 1● , 12 , 11 , from the higher figure , and bring them in this manner to the dyal upon the middle rule , in the lower figure , and so you have done in this middle rule in the dyal of the lower figure , all and the same spaces as those are , that are upon the table of the lower figure : and those points of the middle rule of the lower figure , are as many points of hours that will be in the dyal , among which you know that the point 6 , is the point of one of the hours of six , either of the evening , or of the morning , whereby you shall come to know which are the other hours , whereof you have the points so mark'd in the rule of the dyal . as for to let you know , whether this point of 6 hours , is either of those in the morning , or of those of the evening , i will not trouble this paper with it , because that is plain enough of it self : and you see well enough , whether the shadow of the axeltree rod will fall upon this point , either in the morning about the beginning of the day , or else in the evening about the shutting up of the day . by this means you may see well enough , whether the hours of your dyal are of those of the forenoon , or of the afternoon , for to mark them accordingly , without speaking any further about it . if you have found upon the middle rule mn , the point of one of the two hours of 12 , and not the point of one of the hours of 6 , you are but to do with that point of hour of twelve , the same thing that i said you should do with the point of one of the hours of 6. when you have so transported the points of the hours from the table of the figure above , to the dyal in the lower figure upon the middle rule ▪ that which remains to do , is to transport those points of hours from the middle rule , into the superficies of the dyal in the manner following , and to trace in it afterwards the line of the hours , as i will shew you . you see , there are two strings tyed to the axeltree rod , in the figure below , set the rings of these two strings , as far as you can one from another , and as from r in s , then take one of these two strings , as that which comes from the point r , carry it stretcht out in a direct line from the axeltree rod , to one of the points of hours that are mark'd in the middle rule , for example in the point of hour 12 , and cause this string comming so from the axeltree , to passe to this point of hour 12 , of the middle rule , and to go altogether in a straight line as far as the superficies of the dyal ; and mark in the superficies of the dyal the point in which this string so carryed , meets it ; by this means you shall transport this point or hour 12 , from the middle rule into the superficies of the dyal to the point xii . and after the same manner , you shall transport one after another the points of hour 11 , 12. 1 , 2 , 3 , 4 , 5 , 6 , from the middle rule , into the superficies of the dyal to the points xi , xii ▪ i , ii , iii , iv , v , vi . figure 19 , to all sorts of people . how to trace the lines of the hours upon the superficies of the dyal . of the two strings , fig. above , that are made fast to the axeltree rod , stretch out one in a straight line , from the axeltree rod , as from the point r to a point of hour of the middle rule , as to the point of hour i ▪ and holding this string so stretcht , take the other or second string comming from the point s , and stretching it likewise in a straight line , make it crosse over the first string bi ▪ and let it touch it without breaking his straight line , and let it go in a straight line from thence , to the superficies of the dyal as to the point d , and mark the point d , in the superficies of the dyal , wherein this second string so carried , comes to touch it ; then make this second string to go and touch again the first in another place , and with this second string go and touch in the same manner , another point e , in the superficies of the dyal e , and so remove this second string along the first string , as many times as you shall have need to mark any several points , as d , e , l , in the superficies of the dyal , to trace the line of that hour there , then draw a line as fine and delicate as you can by all those points del , in the superficies of the dyal , that line shall be the line of that hour i. and after this manner , you shall trace in the superficies of the dyal , the lines of all the other hours that are in the middle rule , and your dyal will be finished . the lower figure shews you to the eye , how that after you have transported as above said , all the points of hour from the middle rule into the superficies of the dyal , you may take away this middle rule , and the two knobs that hold it up , and make an end of tracing the rest of the lines in the superficies of the dyal , as i have said , with the strings comming from r and i , and by means of the points of hours xi , xii , i , ii , iii , iv , v , vi . figure 20 , to all sorts of people . after that you have placed the axeltree of the dyal as it must be , if you desire to find the points of the hours in the superficies , with some extraordinary instruments , that which is the plainest of all , viz. a round flat plate , and stiff , as of tinn'd yron , or the like , and divided into 24 parts equal one unto another , and set up in the manner of a rotunda or whirl , by the squire , or with right angles about the axeltree of the dyal , as the figure below doth shew , is the most common and the shortest way of all . the figure h , shews you this round alone , and how it is open of one side , that the center thereof may be placed with the center of the axeltree . the 1. figure shews the neck that may be applyed unto this round about the center , to the end that one may with this neck , set the round to the axeltree of the dyal by the squire , or with angles straight between themselves , as you see in the second figure . when you have thus set this round to the axeltree of the dyal , the lower figure shews you how you must place the string of the plummet , hanging upon the axeltree by a point of one of the divisions of the edge of this round , that it may give you the points of the hours in the superficies of the dyal . the strings which comming from the axeltree passe afterwards to the points of the division of this round in it's 24 parts , shew you , how you must afterwards bring the strings from the axeltre , by the points of the division of this round in 24 parts , equal to the superficies of the dyal , that you may have the points of the hours in this superficies . the string ls , xii , that passeth to the string with the plummet rs , gives the point of the hours of 12. the string lvi , that passeth to one of the points of this division in 24 , and is found to be level , gives the point of the hours of 6. the other strings shew you , that the way of tracing the points of the other hours is the same as above . 20 figure 21 , to all sorts of people . vvhen you have brought , as i have said , by means of this rotunda and the strings , all the points of the hours , into the superficies of the dyal ; you may take away the rotunda if you will , and make an end of tracing the lines of the hours as before with the strings , and by means of the points of hours , which you have brought into the superficies of the dyal , as you see in the figure below , the line delkpqsyizg . and for this purpose by means of the said strings , carry a string in a straight line from the axeltree to the point of hour , for example i , and holding it there stretcht in a straight line , carry of one side or other according to the occasion , an other string comming also from the axeltree , as from i , or from b , that may go in a straight line as far as the superficies of the dyal , and let it go and touch , and crosse over the string ir , several times in several places , and at every time go with this second string to touch and mark a point in the superficies of the dyal , until you have enough , as you see the points d , e , l , k , p , q , y , i , z , g , and carry by these points a line sweetned , that shall be a line of hour , do the like for the lines of the other hours , and you have done . when you have mark'd in the superficies of the dyal , a point of every one of the hours that are to be found in it : if you desire to trace the lines of the hours every one at once , without making use of the strings , as in the figure above , you may do it when it is dark , as by night , with the light of a candle , in that manner as it is exprest in the lower figure . set a light behind the axeltree rod of the dyal , and turn the same lightabout this axeltree , untill the shadow of this axeltree come to one of the points of hour i , and trace in the superficies of the dyal a line delkpqsyizg , all along this shadow of the axeltree , that line shall be a line of hour , do the like for every one of the other points of hour , and you have finish'd your dyal . 22 instruments to work with all , 21 figure 22 , several instruments to work withall in these occasions hereafter specified . i did not intend to burden my memory with any thing in this matter , but with monsieur de sargues universal rules for the placing of the axeltree , and for the tracing in a dyal the hours after the french way , without medling with the rest , which is more curious than useful . but to follow the advice of many considerable persons whom i do honour , i have set down also the way to mark that which is commonly called the signs : the hours after the italian or babylonian way : the hours after the manner of the ancients : the elevations of the sun above the horizon ; and the rising of the same . and for as much as none can do any of these things universally , without using these instruments more or lesse ; this table following shews to the eye all the pieces that are used in those several occasions . these instruments are first a circle , a half circle , or the quarter of a circle , which is all one which is made to turn about its diameter set fast in it 's due and convenient place , or down right , as in the fourth figure , or level , as in the second or third figure , or else inclining or hanging downward , as in the first figure . the way to make this circle to move in all kinds of positions , is to set two rings in it's diameter , through which one may put in a stick straight , round , and smooth , about which this circle may turn round like a weather cock about his needle or spear , as in the second figure , and there must be within those hoop rings , a screw to fasten this circle in that place , or which way soever you will have it to stand . the sticks or rods are represented by the 7th . figure , with a fork at the end of every one , bored in the cheek , to put a pin through , as you see , that one may be set plum or down right , and the other level , being made fast at one end to the axeltree rod as in the 5th . figure . and for this purpose also the axeltree rod is bored in o. the 8th . figure shews the axeltree rod by it self bored with o , and the pin q , put through the hole , to shew more plainly that which the 5th figure represents , viz. all the pieces set or joyned together being mark'd with qo . you see that the hoop rings are near the edge or brim of the circle , a purpose to leave the center o , and a space about it free , having commonly a piece taken off , that this circle may turn freely about the forked end , that is to say about his center , without any let or hinderance at all . 23 for the signes . figure 23 , how to mark the signs . get , figure 2 below , a half circle both thin and stiff ctsrd , draw there a beam ozs , perpendicular to the diameter cpoqd , take on both sides of this beam os , 23 degrees and half , for example 23 degrees and half from s towards t , and as much from s towards r , draw the straight line r , t , make upon diameter t , r , a half , circle tzr , divide the edge or brim of that half circle in six equal parts , as in the points that you see there ; draw by those points as far as the half circle ctsru , some straight lines that may be perpendicular to the straight line r , t , bring from the center o , by the points that those perpendicular have made upon the edge of the half circle ctsrd , some straight lines , as you see that the strings shew you , and with those lines drawn out sufficiently , you shall mark the signs in the dyal , as i shall say . you see that the half circle is cut thorow , or made hollow from the point p , to the point q , round about the center o , according to the circumference pzq , which is notch'd also in the points that you see in it , which are betwixt every degree of the half circle ; and the center o , and these notches , are there a purpose to fasten a string upon them , insteed of bringing it from the center o. the two figures s32tgez , s45rbcz , both on the right and on the left side of the half circle ctsrd , shew as you may judge by their letters or coats , each of them one half of the figure tsrz , of the half circle ctsrd , which i have made thus bigger than each half of this figure , that one may set in the letters ge , cb , and some figures , 2345 , about the edges of the two half circles , and also the signs , as you see , which i could not do in the middle figure without confusion . the lines comming from the points t23s , s45r , towards the lowest past of the figure or plate , comming near one an other , go and seek the center of the half circle t23s45r , every one of the three saces between those straight lines is to hold two signes , mark them there in the same order that you see them , close by these straight lines , one of one side , and an other on th'other side . and by this meanes the straight line of the half circle t23s45r , which from the center of the half circle passeth to the points , is that of the signes of aries , and of libra ; that which passeth to the point 3 , is that of taurus , and of virgo ; that which passeth to the point 2 , is that of gemini , or the twinnes , and of leo ; that which passeth to the point t , is that of cancer ; that which passeth to the point r , is that of capricornus ; that which passeth to the point 5 , is that of sagittarius and aquarius ; that which passeth to the point 4 , is that of the scorpion , and of the fishes . the two figures s32cgez , s45rbcz , shew you also that with one quarter of circe mark'd on both sides with 6 signes in one part , and 6 others in the other part , you may do the same thing as well as with the half circle , by turning this quarter of circle , as you see in the said figures , once of one side , and once of th'other . i shall for all that speak to you alwayes as if you had the half circle in your hand . therefore when you will mark the lines of the signes in the superficies of the dyall , the first figure shewes you how you must set up your half circle with the axeltree , for to turn it about the same , without going up or down along the axeltree . the first figure above shewes how you must make your half circle , viz. about the axeltree , set then the half circle to the axeltree of the dyall , as you see in the figure that is under the first . tye a string with a loose knot , just in the center of the circle . turn the half circle about the axeltree , cause at the same time the string comming from the center , to passe by one of the lines of the signes , holding it longer or shorter as need shall require , go and touch with the string many several points in the superficies of the dyall one after an other . draw a line sweetned , by all those points , and it is the line of the signes that are markt along the straight line of the half circle which the string doth cover , in turning with it about the axeltree . do the like for every line of the signes , mark the signes in the dyall by the lines so drawn according to the situation , in regard of the country and of the place of the dyall , and as the figure shews , you have marked the signes in the dyall . and if the strings could not come from the center , fasten them with a knot to the beames , comming from the center , in the notches or clefts of the circumference pzq . set a button or an other mark in the axeltree , in the place where the center of the half circle ctsrd hath been , and the shadow of the button will go and mark the signe that the sun is in . figure 24. to mark the houres after the italian or babylonian way . the first figure shews how you must set on your half circle , and how to make it turn about the axeltree . moreover , the line no ( shews you what kind of line comming from the center o of this half circle , you must make use of , in making the half circle to turn about . when you have drawn the lines of the houres after the french way at length , in the superficies of the dyall , as the figure below doth shew . set on , as the same figure shewes you also , the half circle o t rto the exeltree of the dyall with a string on in it's center . let this half circle hang down right or plum , and when this half circle is just down and at rest , draw the string on , in a straight line comming from the center o , and closing with the half circle , in such sort as it may go , and touch it all at length , turn this string as a beam of the half circle about the center o , till it be very levell , as the figure shewes by the setting on of the carpenters levell a. when the string on is stretcht out very levell close to the half circle , mark exactly upon the edge of this half circle , the point wherein the string on toucheth it , as doth the letter e. then let the half circle turn about the axeltree . make in the mean time the string on , to passe by the point t , which you have markt upon the edge of the half circle , and making it shorter or longer according as need shall require , go and touch with this string many several points one after an other in the superficies of the dyall in divers places , 1 , 2 , 3 , 4 , 5 , 6. draw an obscure line by those points , as you see the line bowed or crooked , 23 , 24 , 1 , 2 , 3 , 4 , 5 , 6 , and which reacheth beyond the axeltree towards h. this line crosseth over the superficies of the dyall , out of the equinoctial line pq , and meets by the way all the lines of the houres after the french way , as you see it doth in 23 , 24 , 1 , 2 , 3 , 4 , 5 , 6. there remaines to trace the lines of these houres after the italian or babylonian way in the superficies of the dyall and when you know how to trace one , you shall be able also to trace the other . therefore to trace a line of those kinds of houres , it is no matter which you begin to trace first ; count upon the equinoctiall line pq fixs paces of houres equall , after the french way , one after an other , as from xii . to vi . afterwards follow the lines of the houres , after the french way , that passe by the points xii . & vi the two extremities of these six spaces every one to the aforefaid line of which you have found the place , by turning the string with the half circle about the axeltree by the point t , as you see , as farre as the points 24. and 6. take conveniently in these two lines of hours after the french way , in each of them one of the points wherein it meets with the equinoctial line pq , or else the said line found with the string 1 , 2 , 3 , 4 , 5 , 6 , that is to say in one , the point that the line placed with the string makes in it , and in the other ; the point that the equinoctial line makes in it , for example , in the line , xii , 24 , take in it the point 24. wherein it meets with the line found with the string : and in the other , vi , take in it the point vi . wherein it meets with the equinoctiall line . set either a string or a rule by these two points so taken 24. and vi , as you see the line , 24. 6. then with the string comming from the center of the half circle q go razing or laying even the string 24. vi by making it longer or shorter as need requires ; as you see in o , g , mark many several points in the superficies of the dyall one after an other , as for example 24 , g , vi , more or lesse , according as the superficies of the dyall is more or lesse uneven . draw an obscure line by the points 24 , g , vi , and it will be a line of houres after the italian or babylonian way , and so of all the rest . the string h , oh shews that you may if need be , do the like both of one and of the other side of the center o , to go and place of one part or other according to the occasion , the line , as 2 , 3 , 4 , 5. and if you have a straight line , as might be o , q , which may turn about the center o , and be perpendicular to the axeltree bi , and you hold the half circle with this straight line , set one at a convenient or reasonable distance from the other : and let it be alwayes exactly of the distance of six hours after the french way : first of all , this string describes the equinoctial line in the superficies of the dyall , secondly when one of the two , either the half circle or the straight line o q , is found in one of the points of the hours of the equator , th'other is likewise found in it , in an other point of hour , then drawing with a string coming from the center o , a straight line that may go from the point , as t , to the end of the straight line o q which you shall go drawing with this string made shorter or longer , as need requires , and mark some points of line of hour , after the italian or babylonian way in the superficies of the dyall . and for this purpose there is nothing so easie as to have a circle of equator , that may be fitted to the half circle , and where you may have alwayes a space ready made for it's hour . 25 to mark the houres after the manner of the iewes . 24 for the houres after the italian way . figure 25 , to mark the hours after the manner of the ancients or the jewes . you must know first that it would be very troublesome to draw in the superficies of the dyall , the lines of this kind of hours in such a manner as that they might be alwayes just and right in theory , all the year long . and therefore it is sufficient to draw them just by demonstration in three points onely , viz , in their points of both ends , and of the middle , which are the points of those circles that appear the greatest above the horizon being parallel to the equator , and of the equator it self . the rest goes as it may , and therefore it may be said , that the lines of such hours traced in this manner are false in the rest of their length , yet curiositie makes them passe for current . wherefore to mark this kind of lines of hours . the higher figure 4 shewes which way you must make this half circle to turn about , viz. about a straight axeltree line placed levell in the center of the axel-tree of the dyall . and to be short , set up and make very fast a rod in a straight line passing to the center o , and let it be first within the joynt of the axeltree rod , secondly let it be level , as the figures do shew of a plummet p. and of a level a , this being done , tye some strings with a loose knot to this rod so levelled nl as you see nr , and lt. take the string from about the center o , stretch it out in a direct or straight line from the center o to one of the points of hour , after the french way , of the equinoctial line of the dyall , for example to the point of 1 hour , as you see the string oi . this string being thus strecht out , take the other strings of one or th'other end nl , and crosse over this string oi with them , and so go and mark many points in the superficies of the dyall , as tir . draw an obscure line by those points as tir , it is a line of hours after the manner of the ancients or the jews , do the like with the other hours and half hours of the equinoctial line . if you leave a rod in the dyall , as nol , the shadow thereof will go and shew these hours continually at length , if you will not leave it in , the button or center o of the axeltree of the dyall will shew them . 26 for the hight of the sun . figure 26. how to mark the elevation of the sun above the horizon . the higher fig. 3. shews which way you must turn the half circle , viz. about a straight axel-line hanging down right . set up your half circle , so that it may turn like a weather-cock about a rod hanging down right , or plum , above or below the axeltree of the dyal , it matters not which . vvhilest you turn it thus as it is above said , cause in the mean time the string comming from the center o to passe by one of the degrees of the edge of the circle ; and make the string shorter or longer as need shall require , mark with it many several points in the superficies of the dyal , according as you see them rankt one by an other , in four places . draw a small or obscure line through all these points , and this will be one of the lines of the elevation of the sun . count the degrees in the edge of the circle , beginning at the first of the beam which is level , and ending at the 90. beam which is down right or plum . mark in the line of the dyal , the number of the degrees of the border of the circle , where the string passes that hath mark't the points of that line , and so of all the others , and the shadow of the button of the axeltree which is in the center of the circle , will shew the elevation of the sum above the horizon . figure 27 : how to mark the sun rising , or east rising of the sun . the figure 2 above shews how you must place the half circle , viz. parallel unto the horizon , i would not put a levell to it to avoid confusion . it shews also that one of the diameters of the circle must be set within the center of the dyal , that is to say , thar it must go directly from the south to the north , and accordingly the diameter which is perpendicular to it , will go from east to west . when your circle is set fast in this position , let a plummet op in the lower figure hang from the center o. this being done , from each point of degree of the edge of the circle , as from x and from z. mark with a string xt or zr . many points in the superficies of the dyal . draw a small or obscure line through these points , as ty or sr. it is a line of the suns eastrising mark in it the number of degrees of the point of the circle from whence the string comes , according as you will count them , to begin either from the east , or from the south . and so of all the other degrees accordingly . and the shadow of the button o will shew which way the sight of the sun comes upon the dyal . i will take here occasion to tell you , that if for some reason or other , you could observe , in one and the same day , but two shadows of the sun in stead of three , as we have said in the placing of the axeltree in the dyal , the declining of the sun in that day , will serve you for a third shadow , or else two other shadows observed in an other day . i mean you may find equally the placing of the axeltree by one or other of those ways above mentioned , and with 3 shadows ; and with 2 shadows , and the declining of the sun in that day , and with 4 shadows , two of one day , and two of an other , which are three wayes that come all to one . 27 for the eastrising of the sun . 28 figure 28. i do not specifie in this volum these kinds of flat dyals , wherein you may work without knobs or middle rule : and where you may draw the equinoctial line ; trace out and divide the circle equator : in a word , where you may do all : yea and in the very superficies of the dyal , you may easily come to know them you self , by putting this universall way into practice . here is onely a way how to trace out all the twelue lines of the hours equal , after the french way , in the flat dyals where the axeltree meets the superficies athwart in the space that you work in , so that you shall have no need of a greater place . and what i have already said , and what i am now going to say again , will serve to find out the way to do the like in all kinds of dyals universally . when you have drawn upon your dyal the equinoctial line m 12 m , drawn conveniently and divided the circle equator q 12 q bring to the equinoctial line , the beam of the 12 hours q , 12. draw of both sides of the equator , and from the equinoctial line a straight line mq parallel to the beam of 12. hours o 12. bring the beams of the other hours to the first , which they shall find of the equinoctial in rt : and of mq in c , d●g , q , bring in the dyal the line of the twelve hours b. 12. draw by the point m , of the equinoctial line , and from the center of the dyal b a straight line ml parallel to the line of twelve hours b 12 ; make upon this line ml and upon the point m a triangle lmn like to the triangle in the aire ob 12. and let the angles of these triangles in the points l and b be equal one unto an other , carry the spaces mq , mg , md , mc , from the straight line mq into the straight line mn , viz. from m into n , into u , into i , into o , bring by the points n , u , i , o , some straight lines nl , ub , if , oh , parallel to the side , nl of the triangle lmn : carry from the center of the dyal b by the points r , t , h , f , o , l , some straight lines , bl , bh , bf , bh , bt , br ; these are such lines of hours as you may continue beyond the center b , and mark them according to their orders . the end . notes, typically marginal, from the original text notes for div a35744e-3300 i.e. that are made without any aim , or heed . the semicircle on a sector in two books. containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. by j. t. taylor, john, 1666 or 7-1687. 1667 approx. 163 kb of xml-encoded text transcribed from 77 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-10 (eebo-tcp phase 1). a64223 wing t533b estc r221720 99832990 99832990 37465 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a64223) transcribed from: (early english books online ; image set 37465) images scanned from microfilm: (early english books, 1641-1700 ; 2155:11) the semicircle on a sector in two books. containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. by j. t. taylor, john, 1666 or 7-1687. [8], 144 p. printed for william tompson, bookseller at harborough in leicestershire, london : 1667. j.t. = john taylor. in two books; book ii has caption title (the first word of which is in greek characters); register and pagination are continuous. pages stained with some loss of print. reproduction of the original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one copy (usually the first edition) of every monographic english-language title published between 1473 and 1700 available in eebo. eebo-tcp aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the text encoding initiative (http://www.tei-c.org). the eebo-tcp project was divided into two phases. the 25,363 texts created during phase 1 of the project have been released into the public domain as of 1 january 2015. anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source. users should be aware of the process of creating the tcp texts, and therefore of any assumptions that can be made about the data. text selection was based on the new cambridge bibliography of english literature (ncbel). if an author (or for an anonymous work, the title) appears in ncbel, then their works are eligible for inclusion. selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. in general, first editions of a works in english were prioritized, although there are a number of works in other languages, notably latin and welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so. image sets were sent to external keying companies for transcription and basic encoding. quality assurance was then carried out by editorial teams in oxford and michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet qa standards were returned to the keyers to be redone. after proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. any remaining illegibles were encoded as <gap>s. understanding these processes should make clear that, while the overall quality of tcp data is very good, some errors will remain and some readable characters will be marked as illegible. users should bear in mind that in all likelihood such instances will never have been looked at by a tcp editor. the texts were encoded and linked to page images in accordance with level 4 of the tei in libraries guidelines. copies of the texts have been issued variously as sgml (tcp schema; ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng mathematics -early works to 1800. navigation -early works to 1800. dialing -early works to 1800. 2000-00 tcp assigned for keying and markup 2001-00 aptara keyed and coded from proquest page images 2001-08 sean norton sampled and proofread 2005-03 olivia bottum text and markup reviewed and edited 2005-04 pfs batch review (qc) and xml conversion the semicircle on a sector : in two books . containing the description of a general and portable instrument ; whereby most problems ( reducible to instrumental practice ) in astronomy , trigonometry , arithmetick , geometry , geography , topography , navigation , dyalling , &c. are speedily and exactly resolved . by j. t. london , printed for william tompson , bookseller at harborough in leicestershire . 1667. to the reader . all that is intended in this treatise , is to acquaint thee with an instrument , that is both portable and general , of no great price , easie carriage , yet of a speedy and accurate dispatch in the most difficult problems in astronomy , &c. the lines for the most part have been formerly published by mr. gunter , the famous mr. foster , mr. white , &c. the reduction of the 28. cases of spherical triangles unto ii. problems , i first learned from the reverend mr. palmers catholick planisphere . many of the proportions in the treatise of dyalling are taken from ( though first compared with the globe ) my worthy friend ( to whom i am indebted in all the obligations of civility , and without whose encouragement this had never adventured the publick test ) mr. john collins . the applying mr. fosters line of versed sines unto the sector was first published by mr. john brown , mathematical instrument-maker , at the sphere and sun-dial in the minories , london , anno 1660. who bath very much assisted me , by making , adding unto , and giving me freely the perusal of many instruments , according to any directions for improvement , that was proposed to him . after this account , what hath been my part in this work , i hazard to thy censure ; and when i see others publish a more convenient , speedy , accurate , and general instrument , i assure them to have as low thoughts of this , as themselves . but here is so large a catalogue of errata's as would stagger my confidence at thy pardoning , had they not been irrevocably committed before i received the least notice of them . the printer writing me word ( after i had corrected so much as came to my sight ) that he could alter no mistakes until the whole book was printed : by which means he enforced me to do pennance in his sheets , for his own crimes : did not one gross mistake of his become my purgation , viz. in lib. 2. ( throughout chap. 3. ) where instead of the note of equality ( marked thus = ) he hath inserted the algebraick note of subtraction , or minoration ( marked thus ) nor hath the engraver come behinde the composer , who so miserably mangleth fig. 13. that ( at first sight ) it would endanger branding of a mans brains to spell the meaning thereof , either in it self , or in reference to the book . all that i can help thee herein , is this ; whereas the book mentions that figure for an east dyal , if you account it ( as now cut ) a west dyal , and alter the names of the hours , by putting figures for the afternoon , in the place of those there for the morning ; you will then have a true west dyal of that figure . the correction of punctations would be an endless task ; for i finde some to be resolved , ever since valentine , to recreate themselves at spurn-point . what other material mistakes are in the book ( which ought to be corrected before reading thereof ) you will finde mentioned in the errata . farewel . march 29. 1667. j. t. errata . page 4. line 12. signs , r. sines . p. 8. l. 5. seconds , r. secants . p. 12. l. 9. all , r. allone . p. 13. l. 9 , and 10 . sec . r. min. p. 14. l. 17. sec. r. min. p. 22. l. 14. any , r. what . p. 23. l. 3. exact , r. erect . p. 24. l. ult . adde lib. 2. p. 25. l. 2. signs , r. sines . l. 5 . sign . r. sine . p. 35. l. 3 . a mark , r. an ark . p. 37. l. 22. 20. r. 22. and 42. r. 20. p. 44. l. 5. at , r. at. p. 49. l. 1 . divided by , r. dividing . p. 62. l. 15. dele , a. in lib. 1. chap. 9. the pages are false numbred . but in chap. 9. p. 62. l. 11 . next , r. exact . l. 15 . gauger , r. gauge . l.24.the , r. what . p. 73.l . 5 . whereas , r. where i. p. 80. l. 13 . wherein , r. whereof . l.22.pont , r. point . p. 91 . l.1 . the co-tangent , r. half the co-tangent . l. 19 . l. r. p. p. 92. l. 21 . pz . r. ps . p. 93. l. 12 . nsp , r. nsz . 6.angle , r. ark . throughout page 96. fig. 3. r. fig. 7. and fig. 4. r. fig. 8. p. 98. l. 13. serve , r. scrue . p. 99 . l.12 . places , r. plates . l. 13. proportion , r. perforation . l.9.serve , r. scrue . l.4.serve , r. scrue . 21.serve , r : scrue . p. 111. l. 20. lay in , r. laying . l.14.of , ●● and. p. 128. l. 3. fig. 12. r. fig. 13. l. 22 . ed. r. ec . in fig. 12. c. r. a. at the end of the line g. to the right honorable , the lord sherard , baron of letrim . my lord , since the trifling treatise of an almanack hath usurped a custom to pinnion some honourable name to the patronage of the authors follies ; had we not certain evidence from the uncertainty of their predictions , that their brains ( like their great oracles the planets ) are often wandring ; it might be deemed a crime , beyond the benefit of the clergy , to prefix before any book , a dedication to a noble person : or when i read the unreasonableness of others in those addresses , imploring their patrons to be their dii tutelares , and prostrate their reputes to the unmannerly mangling of every censurist , under the notion of protecting ( that is adopting ) the authors ignorance , or negligence ; it s enough to tempt the whole world to turn democritians , and hazard their spleens in laughing at such mens madness . my present design is only to give your lordship my observance of your commands about the description and improvement of the sector ; and wherein i have erred ( through mistake , or defect ) i despair not , but from your honor i shall meet with a pardon of course to be granted unto your lordships most humble servant j. t. the semicircle on a sector . lib . i. chap. i. a description of the instrument , with the several lines inscribed thereon . the instrument consists of three rulers , or pieces ; two whereof are joyned together by a river that may open and shut to any angle , in fashion of the sector ; or to use a courser comparison , after the manner of compasses . the third piece is loose , or separable from them , to be put into the tenons at the end of the inward ledge of the joyned pieces , and thereby constituting an aequilateral triangle . on these rulers ( after this manner put together ) we take notice ( for distinction sake ) of the sides , ledges , ends , and pieces . the sides are thus differenced , one we call the quadrantal , the other the proportional , or sector side . the ledges are distinguished by naming one the inward , the other the outward ledge . the ends are known in terming one the head ( viz. where the two pieces are riveted together ) the other the end . the pieces are discovered by styling one the fixed piece , viz. that which hath the rivet upon it ; the second the movable piece , which turns upon that rivet , and the last the loose piece , to be put into the tenons , as before expressed . the quadrantal side of the joyned pieces is easily discerned , having the names of the moneths stamped on the movable piece , and par. scale on the fixed piece . the quadrantal side of the loose piece is known by the degrees on the inward and outward limb . these directions are sufficient to instruct you how to put the instrument together . imagining the instrument thus put together , the lines upon the quadrantal side are these . first on the fixed piece , next the outward limb , is a line of 12 equal parts ; and each of those parts divided into 30 degrees , marked from the end towards the head with ♈ ♉ ♊ , &c. representing the 12 signs of the zodiack ; the use of this line , with the help of those under it , was intended to find the hour of the night by the moon . the next line to this is a line of twice 12 , or 24 equal parts , each division whereof cuts every 15th degree of the former line : and therefore if the figures were set to every 15th degree on this side the former line , this second line would be useless , and the former perform its office more distinctly . this line was intended an assistant for finding the hour by the moon ; but is very ready to find the hour by any of the fixed stars . the third line is a line of 29● equal parts , serving for the dayes of the moons age ; in order to find the hour of the night by the moon . but the operation is so tedious , and far from exactness , that i have no kindness for it ; and should place some other lines in the room of this and the former , did i not resolve to impose upon no mans phantasie . the fourth line is a line of altitudes for a particular latitude , noted at the end , par. scale , &c. this helpeth to find the hour and azimuth of the sun , or any fixed star very exactly . the fifth line is a line of natural sines ; at the beginning whereof there is a pin , or else an hole to put a pin into , whereon to hang the thread and plummet for taking of altitudes . to this line of sines may be joyned a line of tangents to 45 degr . the use of the sines alone , is to work proportions in signs . the use of the sines with the tangent line may be for any proportion in trigonometry ; but that i leave to liberty . the sixth line , and last on this side the fixed piece is a line of versed sines , numbred from the centre at the head to 180 at the end . on the quadrantal side of the movable piece , the first line next the inward edge , is a line of versed sines answering to that on the fixed piece . the use of these versed sines is various at pleasure . the second line from the inward edge , is a line of hours and azimuths serving to find the hour by the sun or stars , or the azimuth of the sun , or any fixed star from the south . the third and fourth lines are lines of moneths , marked with the respective names , and each moneth divided into so many parts as it contains dayes . the fifth line is a line of signs marked ♈ taurus ; gemini ; , &c. each sign being divided into 30 degr . and proceeding from ♈ , or aries ( which answers to the tenth of march ) in the same order as the moneths . the use of this line , with help of the moneths , is to find the suns place in the zodiack . the sixth line is a line of the suns right ascension , commonly noted by hours from 00 to 24. but better if divided by degrees or sines from 00 to 360 , and both wayes proceeding backward and forward as the signs of the zodiack , or dayes of the moneth . lastly the outward edge or limb of the movable and loose piece both , is graduated unto 180 degrees , or two quadrants ; whose centre is the pin , or pin-hole before mentioned , at the beginning of the sines on this side the fixed piece . the perpendicular is at 0 / 60 upon the loose piece ; from whence reckoning along the outward edge of the loose piece , till it intersects the produced line of sines at the end of the fixed piece , you have 90 degrees . or counting from 0 / 60 on the loose piece , and continuing it along the degrees of the outward limb of the movable piece , until they intersect the produced line of sines on the fixed piece at the head , you have again 90 degrees , which compleat the semicircle . the use of this line is for taking of altitudes , counting upon the former 90. degr . when you hold the head of the fixed piece toward the sun : and numbring upon this latter , when you hold ( which is best because the degrees are largest ) the end of the fixed piece toward the sun. there are other wayes of numbring these degrees for finding of the azimuth , &c. which shall be mentioned in their proper places . on the quadrantal side of the loose piece , the inward edge or limb is graduated unto 60 degrees ( or twice 30 , which you please ) whose centre is a pin at the head . the use of this is to find the altitude of the sun , or any star , without thread and plummet ; or to perform some uses of the cross-staff . this is for large rules or instruments , and therefore not illustrated here . in the empty spaces upon the quadrantal side may be engraven the names of some fixed stars , with their right ascensions and declinations . on the proportional side the lines issuing from the centre are the same upon the fixed and movable piece , but happily transplaced ( thanks to the first contriver ) after this manner : the line that lies next the inward edge on the fixed piece hath his fellow or correspondent line toward the outward edge on the movable piece ; by which means these lines all meeting at the centre , stand all at the same angle , and give you the freedom from a great deal of trouble , in working proportions by sines and tangents , or laying down any sine or tangent to any radius given , &c. the lines issuing from the centre toward the outward edge of the movable piece , whose fellow is next the inward edge of the fixed piece , is a line of natural sines on the outward side , marked at the end s , and on the inward side a line of lines or equal parts , noted at the end l ; the middle line serving for both of them . the lines issuing from the centre next the inward edge of the movable piece , whose fellows are toward the outward edge of the fixed piece , are lines of natural tangents , which on the outward , side of the line is divided to 45 the radius ; and on the inward side of the lines ( the middle line serving for both ) at a quarter of the former radius from the centre is another radius noted 45 at the beginning , and continued to the tangent 75. these lines are noted at the end t. the use of these you will find chap. 3,4,5 . betwixt the lines of sines and tangents , both upon the fixed and movable pieces , is placed a line of seconds , continued unto 60 , and marked at the end . se. next to the outward edge on the fixed and movable piece ( which is best discerned when those pieces are opened to the full length ) is a line of meridians divided to 85 ; whose use is for navigation , in describing maps or charts , &c. in the vacant spaces you may have a line of chords , sines , and tangents , to any radius the space will bear ; and what other any one thinks best of , as a line of latitudes and hours , &c. on the proportional side of the loose piece are lines for measuring all manner of solids , as timber , stone , &c. likewise for gaging of vessels either in wine or ale measure . on the outward ledge of the movable and fixed piece , both ( which in use must be stretched out to the full length ) is a line of artificial numbers , sines , tangents , and versed sines . the first marked n , the second s , the third t , the fourth vs . on the inward ledge of the movable piece is a line of 12 inches divided into halfs , quarters , half-quarters . next to that is a prick'd line , whose use is for computing of weight and carriages . lastly a line of foot measure , or a foot divided into ten parts ; and each of those subdivided into ten or twenty more . on the inward ledge of the loose piece you may have a line of circumference , diameter , square equal , and square inscribed . there will still be requisite sights , a thread and plummet . and if any go to the price of a sliding index to find the shadow from the plains perpendicular , in order to taking a plains declination , and have a staff and a ballsocket , the instrument is compleated with its furniture . proceed we now to the uses . onely note by the way that mr. brown hath ( for conveniency of carrying a pair of compasses , pen , ink , and pencil ) contrived the fixed piece and movable both to be hollow , and then the pieces that cover those hollows do , one supply the place of the loose piece for taking altitudes ; the other ( being a sliding rule ) for measuring solids , and gaging vessels without compasses . chap. ii. some uses of the quadrantal side of the instrument . probl. 1. to find the altitude of the sun or any star. hang the thread and plummet upon the pin at the beginning of the line of sines on the fixed piece , and ( having two sights in two holes parallel to that line ) raise the end of the fixed piece , toward the sun until the rayes pass through the sights ( but when the sun is in a cloud , or you take the altitude of any star , look along the outward ledge of the fixed piece , until it be even with the middle of the sun or star ) then on the limb the thread cuts the degree of altitude , if you reckon from 0 / 60 on the loose piece toward the head of the movable piece . probl. 2. the day of the moneth given , to find the suns place , declination , ascensional difference , or time of rising and setting , with his right ascension . the thread laid to the day of the moneth gives the suns place in the line of signs , reckoning according to the order of the moneths ( viz. forward from march the 10th . to june , then backward to december , and forward again to march 10. ) in the limb you have the suns declination , reckoning from 60 / 0 on the movable piece towards the head for north , toward the end for south declination . again , on the line of right ascensions , the thread shews the suns right ascension , in degrees , or hours , ( according to the making of your line ) counting from aries toward the head , and so back again according to the course of the signs unto 24 hours , or 360 degrees . lastly on the line of hours you have the time of sun rising and setting , which turned into degrees ( for the time from six ) gives the ascensional difference . ex. gr . in lat . 52. deg . 30 min. for which latitude i shall make all the examples . the 22 day of march i lay the thread to the day in the moneths , and find it cut in the signs 12 deg . 20 min. for the suns place , on the limb 4 deg . 43 min. for the suns declination north. in the line of right ascensions it gives 46 min. of time , or 11 deg . 30 min. of the circle . lastly , on the line of hours it shews 28 min. before six for the suns rising ; or which is all , 7 deg . for his ascensional difference . probl. 3. the declination of the sun or any star given to find their amplitude . take the declination from the scale of altitudes , with this distance setting one point of your compasses at 90 on the line of azimuth , apply the other point to the same line it gives the amplitude , counting from 90 ex. gr . at 10 deg . declination , the amplitude is 16 deg . 30 min. at 20 deg . declination , the amplitude is 34 deg . probl. 4. the right ascension of the sun , with his ascensional difference , given to find the oblique ascension . in northern declination , the difference betwixt the right ascension and ascensional difference , is the oblique ascension . in southern declination take the summ of them for the oblique ascension , ex. gr . at 11 deg . 30 sec. right ascension , and 6 deg . 30 sec. ascensional difference . in northern declination the oblique ascension will be 5 deg . in southern 18 deg . probl. 5. the suns altitude and declination , or the day of the moneth given to find the hour . take the suns altitude from the scale of altitudes , and laying the thread to the declination in the limb ( or which is all one , to the day in the moneths ) move one point of the compasses along the line of hours ( on that side the thread next the end ) until the other point just touch the thread ; then the former point shews the hour ; but whether it be before or after noon , is left to your judgment to determine . ex. gr . the 22 day of march , or 4 deg . 43 min. north declination , and 20 deg . altitude , the hour is either 47 minutes past 7 in the morning , or 13 minutes past 4 afternoon . probl. 6. the declination of the sun , or day of the moneth , and hour given to find the altitude . lay the thread to the day or declination , and take the least distance from the hour to the thread , this applyed to the line of altitudes , gives the altitude required . ex. gr . the 5 day of april or 10 deg . declination north , at 7 in the morning , or 5 afternoon , the altitude will be 17 deg . 10 sec. and better . probl. 7. the declination and hour of the night , given to find the suns depression under the horizon . lay the thread to the declination on the limb ; but counted the contrary way , viz. from 60 / 0 on the movable piece toward the head for southern ; and toward the end for northern declination . this done take the nearest distance from the hour to the thread , and applying it to the line of altitudes , you have the degrees of the suns depression . ex. gr . at 5 deg . northern declination , & 8 hours afternoon , the depression is 13 deg . 30 min. probl. 8. the declination given to find the beginning and end of twilight , or day-break . lay the thread to the declination counted the contrary way , as in the last problem , and take from your scale of altitudes 18 deg . for twilight , and 13 deg . for day-break , or clear light ; with this run one point of the compasses along the line of houres ( on that side next the end ) until the other will just touch the thread , and then the former point gives the respective times required . ex. gr . at 7 deg . north declination , day breaks 8 minutes before 4 : but twilight is 3 houres 12 minutes in the morning , or 8 hours 52 minutes afternoon . probl. 9. the declination and altitude of the sun or any star , given to find their azimuth in northern declination . lay the thread to the altitude numbred on the limb of the moveable piece from 60 / 0 toward the end ( and when occasion requires , continue your numbring forward upon the loose piece ) and take the declination from your line of altitude ; with this distance run one point of your compasses along the line of azimuths ( on that side the thread next the head ) until the other just touch the thread , then the former point gives the azimuth from south . ex. gr . at 10 deg . declination north , and 30 deg . altitude , the azimuth from south is 64 , deg . 40 min. probl. 10. the suns altitude given to find his azimuth in the aequator . lay the thread to the altitude in the limb , counted from 60 / 0 on the loose piece toward the end , and on the line of azimuths it cuts the azimuth from south . ex. gr . at 25 deg . altitude the azimuth is 53 deg . at 30 deg . altitude the azimuth is 41 deg . 30 min. fere . probl. 11. the declination and altitude of the sun , or any star given to find the azimuth in southern declination . lay the thread to the altitude numbred on the limb from 60 / 0 on the moveable piece toward the end , and take the declination from the scale of altitudes ; then carry one point of your compasses on the line of azimuths ( on that side the thread next the end ) until the other just touch the thread , which done , the former point gives the azimuth from south . ex. gr . at 15 deg . altitude and 6 deg . south declination the azimuth is 58 deg . 30 min. probl. 12. the declination given to find the suns altitude at east or west in north declination , and by consequent his depression in south declination . take the declination given from the scale of altitudes , and setting one point of your compasses in 90 on the line of azimuths , lay the thread to the other point ( on that side 90 next the head ) on the limb it cuts the altitude , counting from 60 / 0 on the moveable piece . ex. gr . at 10 deg . declination the altitude is 12 deg . 40 min. probl. 13. the declination and azimuth given to find the altitude of the sun or any star. take the declination from the scale of altitudes ; set one point of your compasses in the azimuth given , then in north declinanation turn the other point toward the head , in south toward the end ; and thereto laying the thread , on the limb you have the altitude , numbring from 60 / 0 on the moveable piece toward the end . ex. gr . at 7 deg . north declination , and 48 deg . azimuth from south , the altitude is 35 deg . but at 7 deg . declination south , and 50 deg . azimuth the altitude is onely 18 deg . 30 min. probl. 14. the altitude , declination , and right ascension of any star with the right ascension of the sun given , to find the hour of the night . take the stars altitude from the scale of altitudes , and laying the thread to his declination in the limb , find his hour from the last meridian he was upon , as you did for the sun by probl. 5. if the star be past the south , this is an afternoon hour ; if not come to the south , a morning hour ; which keep . then setting one point of your compasses in the suns right ascension ( numbred upon the line twice 12 or 24 next the outward ledge on the fixed piece ) extend the other point to the right ascension of the star numbred upon the same line , observing which way you turned the point of your compasses , viz. toward the head or end . with this distance set one point of your compasses in the stars hour before found counted on the same line , and turning the other point the same way , as you did for the right ascensions , it gives the true hour of the night . ex. gr . the 22 of march i find the altitude of the lions heart 45 deg . his declination 13 d. 40 min. then by probl. 5. i find his hour from the last meridian 10 houres , 5 min. the right ascension of the sun is 46 m. of time , or 11 d. 30 m. of the circle , the right ascension of the lions heart , is 9 hour 51 m. fere , of time , or 147 deg . 43 m. of a circle ; then by a line of twice 12 , you may find the true hour of the night , 7 hour 13 min. probl. 15. the right ascension and declination of any star , with the right ascension of the sun and time of night given , to find the altitude of that star with his azimuth from south , and by consequent to find the star , although before you knew it not . this is no more than unravelling the last problem . 1 therefore upon the line of twice 12 or 24 , set one point of your compasses in the right ascension of the star , extending the other to the right ascension of the sun upon the same line , that distance laid the same way upon the same line , from the hour of the night , gives the stars hour from the last meridian he was upon . this found by probl. 5. find his altitude as you did for the sun. lastly , having now his declination and altitude by probl. 8. or 10. according to his declination , you will soon get his azimuth from south . this needs not an example . by help of this problem the instrument might be so contrived , as to be one of the best tutors for knowing of the stars . probl. 16. the altitude and azimuth of any star given to find his declination . lay the thread to the altitude counted on the limb from 60 / 0 on the moveable piece toward the end , setting one point of your compasses in the azimuth , take the nearest distance to the thread ; this applyed to the scale of altitudes gives the declination . if the azimuth given be on that side the thread toward the end , the declination is south ; when on that side toward the head , its north. probl. 17. the altitude and declination of any star , with the right ascension of the sun , and hour of night given to find the stars right ascension . by probl. 5. or 14. find the stars hour from the meridian . then on the line twice 12 , or 24 , set one point of your compasses in the stars hour ( thus found ) and extend the other to the hour of the night . upon the same line with this distance set one point of your compasses in the right ascension of the sun , and turning the other point the same way , as you did for the hour , it gives the stars right ascension . probl. 18. the meridian altitude given to find the time of sunrise and sunset . take the meridian altitude from your particular scale , and setting one point of your compasses upon the point 12 on the line of hours ( that is the pin at the end ) lay the thread to the other point , and on the line of hours the thread gives the time required . probl. 19. to find any latitude your particular scale is made for . take the distance from 90 , on the line of azimuth unto the pin at the end of that line , or the point 12 : this applyed to the particular scale , gives the complement of that latitude the instrument was made for . probl. 20. to find the angles of the substile , stile , inclination of meridians , and six and twelve , for exact declining plains , in that latitude your scale of altitudes is made for . sect. 1. to find the distance of the substile from 12 , or the plains perpendicular . lay the thread to the complement of declination counted on the line of azimuths , and on the limb it gives the substile counting from 60 / 0 on the moveable piece . sect. 2. to find the angle of the stile 's height . on the line of azimuths take the distance from the plains declination to 90. this applyed to the scale of altitudes gives the angle of the stile . sect. 3. the angle of the substile given to find the inclination of meridians . take the angle of the substile from the scale of altitudes , and applying it from 90 on the azimuth line toward the end ; the figures shew the complement of inclination of meridians . sect. 4. to find the angle betwixt 6 and 12. take the declination from the scale of altitudes , and setting one point of your compasses in 90 on the line of azimuths , lay the thread to the other point and on the limb it gives the complement of the angle sought , numbring from 60 / 0 on the moveable piece toward the end . this last rule is not exact , nor is it here worth the labour to rectifie it by another sine added ; sith you have an exact proportion for the problem in the treatise of dialling chap. 2 . sect. 5. paragr . 4. chap. iii. some uses of the line of natural signs on the quadrantal side of the fixed piece . probl. 1. how to adde one sign to another on the line of natural sines . to adde one sine to another , is to augment the line of one sine by the line of the other sine to be added to it . ex. gr . to adde the sine 15 to the sine 20 , i take the distance from the beginning of the line of sines unto 15 , and setting one point of the compasses in 20 , upon the same line , turn the other toward 90 , which i finde touch in 37. so that in this case ( for we regard not the arithmetical , but proportional aggregate ) 15 added to 20 , upon the line of natural sines , is the sine 37 upon that line , and from the beginning of the line to 37 is the distance i am to take for the summe of 20 and 15 sines . probl. 2. how to substract one sine from another upon the line of natural sines . the substracting of one sine from another , is no more than taking the distance from the lesser to the greater on the line of sines , and that distance applyed to the line from the beginning , gives the residue or remainer . ex. gr . to substract 20 from 37 i take the distance from 20 to 37 that applyed to the line from the beginning gives 15 for the sine remaining . probl. 3. to work proportions in sines alone . here are four cases that include all proportions in sines alone . case 1. when the first term is radius , or the sine 90. lay the thread to the second term counted on the degrees upon the movaeble piece from the head toward the end , then numbring the third on the line of sines , take the nearest distance from thence to the thread , and that applyed to the scale from the beginning gives the fourth term . ex. gr . as the radius 90 is to the sine 20 , so is the sine 30 to the sine 10. case 2. when the radius is the third term . take the sine of the second term in your compasses , and enter it in the first term upon the line of sines , and laying the thread to the nearest distance , on the limb the thread gives the fourth term . ex. gr . as the sine 30 is to the sine 20 , so is the radius to the sine 43. 30. min. case 3. when the radius is the second term . provided the third term be not greater than the first , transpose the terms . the method of transposition in this case is , as the first term is to the third , so is the second to the fourth , and then the work will be the same as in the second case . ex. gr . as the sine 30 is to the radius or sine 90 , so is the sine 20 to what sine ; which transposed is as the sine 30 is to the sine 20 , so is the radius to a fourth sine , which will be found 43 , 30 min. as before . case 4. when the radius is none of the three terms given . in this case when both the middle terms are less than the first , enter the sine of the second term in the first , and laying the thread to the nearest distance , take the nearest extent from the third to the thread : this distance applyed to the scale from the beginning gives the fourth . ex. gr . as the sine 20 to the sine 10 , so is the sine 30 to the sine 15. when only the second term is greater than the first , transpose the terms and work as before . but when both the middle tearms be greater than the first , this proportion will not be performed by this line without a paralel entrance or double radius ; which inconveniency shall be remedied in its proper place , when we shew how to work proportions by the lines of natural sines on the proportional or sector side . these four cases comprizing the method of working all proportions by natural sines alone , i shall propose some examples for the exercise of young practitioners , and therewith conclude this chapter . probl. 4. to finde the suns amplitude in any latitude . as the cosine of the latitude is to the sine of the suns declination , so is the radius to the sine of amplitude . probl. 5. to finde the hour in any latitude in northern declination . proport . 1. as the radius to the sine of the suns declination , so is the sine of the latitude to the sine of the suns altitude at six . by probl. 2. substract this altitude at six from the present altitude , and take the difference . then proport . 2. as the cosine of the latitude is to that difference , so is the radius to a fourth sine . again proport . 3. as the cosine of the declination to that fourth sine , so is the radius to the sine of the hour from six . probl. 6. to finde the hour in any latitude when the sun is in the equinoctial . as the cosine of the latitude is to the sine of altitude , so is the radius to the sine of the hour from six . probl. 7. to finde the hour in any latitude in southern declination . proport . 1. as the radius to the sine of the suns declination , so is the sine of the latitude to the sine of the suns depression at six ; adde the sine of depression to the present altitude by probl. 1. then proport . 2. as the cosine of the latitude is to that summe , so is the radius to a fourth sine . again , proport . 3. as the cosine of declination is to the fourth sine , so is the radius to the sine of the hour from six . probl. 8. to finde the suns azimuth in any latitude in northern declination . proport . 1. as the sine of the latitude to the sine of declination , so is the radius to the sine of altitude at east , or west . by probl. 2. substract this from the present altitude , then , proport . 2. as the cosine of the latitude is to that residue , so is the radius to a fourth sine . again , proport . 3. as the cosine of the altitude is to that fourth sine , so is the radius to the sine of the azimuth from east or west . probl. 9. to finde the azimuth for any latitude when the sun is in the equator . proport . 1. as the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth sine . proport . 2. as the cosine of altitude to that fourth sine , so is the radius to the sine of the azimuth from east , or west . probl. 10. to finde the azimuth for any latitude in southern declination . proport . 1. as the cosine of the latitude to the sine of altitude , so is the sine of the latitude to a fourth . having by probl. 4. found the suns amplitude , adde it to this fourth sine by probl. 1. and say as the cosine of the altitude is to the sum , so is the radius to the sine of the azimuth from east or west . the terms mentioned in the 5th . 7th . 8th . 10th . problems are appropriated unto us that live on the north side the equator . in case they be applyed to such latitudes as lie on the south side the equator . then what is now called northern declination , name southern , and what is here styled southern declination , term northern , and all the proportion with the operation is the same . these proportions to finde the hour and azimuth , may be all readily wrought by the lines of artificial sines , only the addition and substraction must alwayes be wrought upon the line of natural sines . chap. iv. some uses of the lines on the proportional side of the instrument , viz. the lines of natural sines , tangents , and secants . probl. 1. to lay down any sine , tangent , or secant to a radius given . see fig. 1. if you be to lay down a sine , enter the radius given in 90 , and 90 upon the lines of sines , keeping the sector at that gage , set one point of your compasses in the sine required upon one line , and extend the other point to the same sine upon the other line : this distance is the length of the sine required to the given radius . ex. gr . suppose a. b. the radius given , and i require the sine 40. proportional to that radius . enter a. b. in 90 , and 90 keeping the sector at that gage , i take the distance , twixt 40 on one side , to 40 on the other , that is , c. d. the sine required . the work is the same , to lay down a tangent to any radius given , provided you enter the given radius in 45 , and 45 , on the line of tangents . only observe if the tangent required be less than 45. you must enter the radius in 45. and 45 next the end of the rule . but when the tangent required exceeds 45. enter the radius given in 45 , and 45 'twixt the center and end , and keeping the sector at that gage , take out the tangent required . this is so plain , there needs no example . to lay down a secant to any radius given , is no more than to enter the radius in the two pins at the beginning of the line of secants ; and keeping the sector at that gage , take the distance from the number of the secant required on one side , to the same number on the other side , and that is the secant sought at the radius given . the use of this problem will be sufficiently seen in delineating dyals , and projecting the sphere . probl. 2. to lay down any angle required by the lines of sines , tangents , and secants . see fig. 2. there are two wayes of protracting an angle by the line of sines , first if you use the sines in manner of chords . then having drawn the line a b at any distance of your compass , set one point in b , and draw a mark to intersect the line b a , as e f. enter this distance b f in 30 , and 30 upon the lines of sines , and keeping the sector at that gage , take out the sine of half the angle required , and setting one point where f intersects b a , turn the other toward e , and make the mark e , with a ruler draw b e and the angle e b f is the angle required , which here is 40. d. a second method by the lines of sines is thus , enter b a radius in the lines of sines , and keeping the sector at that gage , take out the sine of your angle required with that distance , setting one point of your compasses in a , sweep the ark d , a line drawn from b by the connexity of the ark d , makes the angle a b c 40 d. as before . to protract an angle by the lines of tangents is easily done , draw b a the radius upon a , erect a perpendicular , a c , enter b a in 45 , and 45 on the lines of tangents , and taking out the tangent required ( as here 40 ) set it from a to c. lastly , draw b c , and the angle c b a is 40 d. as before . in case you would protract an angle by the lines of secants . draw b a , and upon a erect the perpendicular a c , enter a b in the beginning of the lines of secants , and take out the secant of the angle , with that distance , setting one point of your compasses in b , with the other cross the perpendicular a c , as in c. this done , lay a ruler to b , and the point of intersection , and draw the line b c. so have you again the angle c b a. 40. d. by another projection . these varieties are here inserted only to satisfie a friend , and recreate the young practitioner in trying the truth of his projection . probl. 3. to work proportions in sines alone , by the lines of natural sines on the proportional side of the instrument . the general rule is this . account the first term upon the lines of sines from the center , and enter the second term in the first so accounted , keeping the sector at that gage , account the third term on both lines from the center , and taking the distance from the third term on one line to the third term on the other line , measure it upon the line of sines from the beginning , and you have the fourth term . ex. gr . as the radius is to the sine 30 , so is the sine 40 to the sine 18. 45. there is but one exception in this rule , and that is when the second term is greater than the first ; yet the third lesser than the first , and in this case transpose the terms , by chap 3. probl. 3. case 3. but when the second term is not twice the length of the first , it may be wrought by the general rule without any transposition of terms . ex. gr . as the sine 30 is to the sine 50 , so is the sine 20 to the sine 31. 30. min. and by consequent , when the third term is greater than the first , provided it be not upon the line , double the length thereof , it may be wrought by transposing the terms , although the second was twice the length of the first . ex. gr . as the sine 20 is to the sine 60 , so is the sine 42 , to what sine ? which transposed is , as the sine 20 is to the sine 42 , so is the sine 60 to the sine 35. 30. this case will remove the inconveniency mentioned , chap. 3. probl. 3. case 4. of a double radius . i intended there to have adjoyned the method of working proportions by natural tangents alone , and by natural sines , and tangents , conjunctly : but considering the multiplicity of proportions when the tangents exceed 45. i suppose it too troublesome for beginners , and a needless variety for those that are already mathematicians . sith , both may be eased by the artificial sines and tangents on the outward ledge , where i intend to treat of those cases at large , and shall in this place only annex some proportions in sines alone , for the exercise of young beginners . probl. 4. by the lines of natural sines to lay down any tangent , or secant required to a radius given . in some cases , especially for dyalling , your instrument may be defective of a tangent , or secant for your purpose , ex. gr . when the tangent exceeds 76 , or the secant is more than 60. in these extremities use the following remedies . first , for a tangent . as the cosine of the ark is to the radius given , so is the sine of the ark to the length of the tangent required . secondly , for a secant . as the cosine of the ark is to the radius given , so is the sine 90 to the length of the secant required . probl. 5. the distance from the next equinoctial point given to finde the suns declination . as the radius to the sine of the suns greatest declination , so is the sine of his distance from the next equinoctial point to the sine of his present declination . probl. 6. the declination given to finde the suns equinoctial distance . as the sine of the greatest declination is to the sine of the present declination , so is the radius to the sine of his equinoctial distance . probl. 7. the altitude , declination , and distance of the sun from the meridian given to finde his azimuth . as the cosine of the altitude , to the cosine of the hour from the meridian , so is the cosine of declination to the sine of the azimuth . chap. v. some uses of the lines of the lines , on the proportional side of the instrument . probl. 1. to divide a line given into any number of equal parts . see fig. 3. suppose a b a line given to be divided into nine equal parts . enter a b in 9 , and 9 on the lines of lines , keeping the sector at that gage , take the distance from 8 , on one side , to 8 on the other , and apply it from a upon the line a b , which reacheth to c ; then is c b a ninth part of the line a b. by this means you may divide any line ( that is not more than the instrument in length ) into as many parts as you please , viz. 10 , 20 , 30 , 40 , 50 , 100 , 500 , &c. parts according to your reckoning the divisions upon the lines , ex. gr . the line is actually divided into 200 parts , viz. first into 10 , marked with figures , and each of those into twenty parts more . again , if the line represents a 1000 , then every figured division is 100 , the second or shorter division is 10 , and the third or shortest division is 5. in case the whole line was 2000 , then every figured division is 200 , every smaller or second division is 20 , every third or smallest division is 10 , &c. suppose i have any line given , which is the base of a triangle , whose content is 2000 poles , and i demand so much of the base as may answer 1750 poles . enter the whole line in 10 , and 10 at the end of the lines of lines , and keep the sector at that gage . now the whole line representing 2000 poles , every figured division is 200 ; therefore 1700 is eight and an half of the figured divisions , and 50 is five of the smallest divisions more ( for in this case every smallest division is 10 , as was before expressed ) wherefore setting one point of the compasses in 15 of the smallest divisions beyond 8 on the rule . i extend the other point to the same division upon the line on the other side , and that distance is 1750 poles in the base of the triangle proposed . how ready this is to set out a just quantity in any plat of ground , i shall shew in a scheam , chap. 12. probl. 2. to work proportions in lines , or numbers , or the rule of three direct by the lines of lines . enter the second term in the first , and keeping the sector at that gage , take the distance 'twixt the third on one line , to the third on the other line , that distance is the fourth in lines , or measured upon the line from the centre , gives the fourth in numbers , ex. gr . as 7 is to 3 , so is 21 to 9. probl. 3. to work the rule of three inverse , or the back rule of three by the lines of lines . in these proportions there are alwayes three terms given to finde a fourth , and of the three given terms two are of one denomination ( which for distinction sake i call the double denomination ; ) and the third term is of a different denomination from those two , which i therefore call the single denomination , of which the fourth term sought must also be . now to bring these into a direct proportion , the rule is this . when the fourth term sought is to be greater than the single denomination ( which you may know by sight of the terms given ) say , as the lesser double denomination is to the greater double denomination , so is the single denomination to the fourth term sought . the work is by probl. 2. if 60 men do a work in 5 dayes , how long will 30 men be about it ? as 30 is to 60 , so is 5 to 10. the number of dayes for 30 men in the work . again , when the fourth term is to be less than the single denomination , say , as the greater double denomination is to the lesser double denomination , so is the single denomination to the fourth term sought . if 30 men do a work in 5 dayes , how long shall 60 be doing of it ? as 60 is to 30 , so is 5 to 2½ . the time for 60 men in the work . probl. 4. the length of any perpendicular , with the length of the shadow thereof given , to finde the suns altitude . at the length of the shadow upon the lines of lines , is to the tangent 45 , so is the length of the perpendicular numbred upon the lines of lines , to the tangent of the suns altitude . probl. 5. to finde the altitude of any tree , steeple , &c. at one station . at any distance from the object ( provided the ground be level ) with your instrument , look to the top of the object along the outward ledge of the fixed piece , and take the angle of its altitude . this done , measure by feet or yards , the distance from your standing to the bottom of the object . then say , as the cosine of the altitude is to the measured distance numbred upon the lines of lines , so is the sine of the altitude to a fourth number of feet or yards ( according to the measure you meeted the distance ) to this fourth , adde the height of your eye from the ground , and that sum gives the number of feet or yards in the altitude . chap. vi. how to work proportions in numbers , sines , or tangents , by the artificial lines thereof on the outward ledge . the general rule for all of these , is to extend the compasses from the first term to the second ( and observing whether that extent was upward or downward ) with the same distance , set one point in the third term , and turning the other point the same way , as at first , it gives the fourth . but in tangents when any of the terms exceeds 45 , there may be excursions , which in their due place i shall remove . probl. 1. numeration by the line of numbers . the whole line is actually divided into 100 proportional parts , and accordingly distinguished by figures , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , and then , 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100. so that for any number under 100 , the figures readily direct you , ex. gr . to finde 79 on the line of numbers , count 9 of the small divisions beyond 70 , and there is the point for that number . now as the whole line is actually divided into 100 parts , so is every one of those parts subdivided ( so far as conveniency will permit ) actually into ten parts more , by which means you have the whole line actually divided into 1000 parts . for reckoning the figures impressed , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , to be 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , and the other figures which are stamped 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , 90 , 100 , to be 100 , 200 , 300 , 400 , 500 , 600 , 700 , 800 , 900 , 1000. you may enter any number under 1000 upon the line , according to the former directions . and any numbers whose product surmount not 1000 , may be wrought upon this line ; but where the product exceeds 1000 , this line will do nothing accurately : wherefore i shall willingly omit many problems mentioned by some writers to be wrought by this line , as squaring , and cubing of numbers , &c. sith they have only nicety , and nothing of exactness in them . probl. 2. to multiply two numbers , given by the line of numbers . the proportion is this . as 1 on the line is to the multiplicator , so is the multiplicand to the product . ex. gr . as 1 is to 4 , so is 7 , to what ? extend the compasses from the first term , viz. i unto the second term , viz. 4. with that distance , setting one point in 7 the third term , turn the other point of the compasses toward the same end of the rule , as at first , and you have the fourth , viz. 28. there is only one difficulty remaining in this problem , and that is to determin the number of places , or figures in the product , which may be resolved by this general rule . the product alwayes contains as many figures as are in the multiplicand , and multiplicator both , unless the two first figures of the product be greater than the two first figures in the multiplicator , and then the product must have one figure less than are in the multiplicator , and multiplicand both . ex. gr . 47 multiplied by 25 , is 2175 , consisting of four figures ; but 16 multiplied by 16 , is 240 , consisting of no more than three places , for the reason before mentioned . i here ( for distinction sake ) call the multiplicator the lesser of the two numbers , although it may be either of them at pleasure . probl. 3. to work division by the line of numbers . as the divisor is to 1 , so is the dividend to the quotient . suppose 800 to be divided by 20 , the quotient is 40. for , as 20 is to 1 , so is 800 to 40. to know how many figures you shall have in the quotient , take this rule . note the difference of the numbers of places or figures in the dividend and divisor . then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend , the quotient shall have one figure more than the number of difference : but where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend , the quotient will have only that number of figures noted by the difference . ex. gr . 245 divided by 15. will have two figures in the quotient ; but 16 divided 〈…〉 ●●ve only one figure in the quotient . probl. 4. to finde a mean proportional 'twixt two numbers given by the line of numbers . divide the space betwixt them upon the line of numbers into two equal parts , and the middle point is the mean proportional . ex. gr . betwixt 4 and 16 , the mean proportional is 8. if you were to finde two mean proportionals , divide the space 'twixt the given numbers into three parts . if four mean proportionals divide it into five parts , and the several points 'twixt the two given numbers , will show the respective mean proportionals . probl. 5. to work proportions in sines alone , by the artificial line of sines . extend the compasses from the first term to the second , with that distance set one point in the third term , and the other point gives the fourth . only observe that if the second term be less than the first , the fourth must be less 〈…〉 or if the second term exceed 〈…〉 fourth will be greater than the third . this may direct you in all proportions of sines and tangents singly or conjunctly , to which end of the rule to turn the point of your compasses , for finding the fourth term . ex. gr . as the sine 60 is to the sine 40 , so is the sine 20 to the sine 14. 40. again , as the sine 10 is to the sine 20 , so is the sine 30 to the sine 80. probl. 6. to work proportions in tangents alone by the artificial line of tangents . for this purpose the artificial line of tangents must be imagined twice the length of the rules , and therefore for the greater conveniency , it is doubly numbred , viz. first from 1 to 45 , which is the radius , or equal to the sine 90 : in which account every division hath ( as to its length on the rule ) a proportional decrease . secondly , it s numbred back again from 45 to 89 , in which account every division hath ( as to its length on the line ) a proportional encrease . so that the tangent 60 you must imagine the whole length of the rule ; and so much more as the distance from 45 unto 30 or 60 is . this well observed , all proportions in tangents are wrought after the same manner of extending the compasses from the first term to the second , and that distance set in the third , gives the fourth , as was for sines and numbers . but for the remedying of excursions , sith the line is no more than half the length , we must imagine it . i shall lay down these cases . case 1. when the fourth term is a tangent exceeding 45 , or the radius . ex. gr . as the tangent 10 is to the tangent 30 , so is the tangent 20 , to what ? extending the compasses from 10 on the line of tangents to 30 , with that distance i set one point in 20 , and finde the other point reach beyond 45 , which tells me the fourth term exceeds 45 , or the radius ; wherefore with the former extent , i set one point in 45 ; and turning the other toward the beginning of the line , i mark where it toucheth , and from thence taking the distance to the third term , i have the excess of the fourth term above 45 in my compass : wherefore with this last distance setting one point in 45 , i turn the other upon the line , and it reacheth to 50 , the tangent sought . case 2. when the first term is a tangent exceeding 45 , or the radius . ex. gr . as the tangent 50 is to the tangent 20 , so is the tangent 30 , to what ? because the second term is less than the first , i know the fourth must be less than the third . all the difficulty is to get the true extent from the tangent 50 to 20. to do this , take the distance from 45 to 50 , and setting one point in 20 , the second term , turn the other toward the beginning of the line , marking where it toucheth , extend the compasses from the point where it toucheth to 45 , and you will have the same distance in your compasses as from 50 to 20 , if the line had been continued at length unto 89 tangents , with this distance , set one point in 30 the third term , and turn the other toward the beginning ( because you know the fourth must be less ) and it gives 10 the tangent sought . case . 3. when the third term is a tangent exceeding 45 , or the radius . as the tangent 40 is to the tangent 12 , 40 min. so is the tangent 65 , to what ? extend the compasses from 40 to 12 d. 40 min. with distance , setting one point in 65. turn the other toward 45 , and you will finde it reach beyond it , which assures you the fourth term will be less than 45. therefore lay the extent from 45 toward the beginning , and mark where it toucheth , take the distance from that point to 65 , and laying that distance from 45 toward the beginning it gives 30 , the tangent sought . these cases are sufficient to remove all difficulties . for when the second term exceeds the radius , you may transpose them , saying , as the first term is to the third ; so is the second to the fourth , and then it s wrought by the third case . i suppose it needless to adde any thing about working proportions by sines and tangents conjunctly , sith , enough hath been already said of both of them apart , in these two last problems ; and the work is the same when they are intermixed . only some proportions i shall adjoyn , and leave to the practice of the young beginner , with the directions in the former cases . probl. 7. to finde the suns ascensional difference in any latitude . as the co-tangent of the latitude is to the tangent of the suns declination , so is the radius to the sine of the ascensional difference . probl. 8. to finde at what hour the sun will be east , or west in any latitude . as the tangent of the latitude is to the tangent of the suns declination , so is the radius to the cosine of the hour from noon . probl. 9. the latitude , declination of the sun , and his azimuth from south , given to finde the suns altitude at that azimuth . as the radius to the cosine of the azimuth from south , so is the co-tangent of the latitude , to the tangent of the suns altitude in the equator at the azimuth given . again , as the sine of the latitude is to the sine of the suns declination , so is the cosine of the suns altitude in the equator ( at the same azimuth from east or west ) to a fourth ark . when the azimuth is under 90 , and the latitude and declination is under the same pole , adde this fourth ark to the altitude in the equator . in azimuths exceeding 90 , when the latitude and declination is under the same pole , take the equator altitude out of the fourth ark . lastly , when the latitude and declination respect different poles , take the fourth ark out of the equator altitude , and you have the altitude sought . probl. 10. the azimuth , altitude , and declination of the sun , given to finde the hour . as the cosine of declination is to the sine of the suns azimuth , so is the cosine of the altitude to the fine of the hour from the meridian . proportions may be varied eight several wayes in this manner following . 1. as the first term is to the second , so is the third to the fourth . 2. as the second term is to the first , so is the fourth to the third . 3. as the third term is to the first , so is the fourth to the second . 4. as the fourth term is to the second , so is the third to the first . 5. as the second term is to the fourth , so is the first to the third . 6. as the first term is to the third , so is the second to the fourth . 7. as the third term is to the fourth , so is the first to the second . 8. as the fourth term is to the third , so is the second to the first . by thesse any one may vary the former proportions , and make the problems three times the number here inserted . ex. gr . to finde the ascensional difference in problem 10 , of this chapter , which runs thus . as the co-tangent of the latitude is to the tangent of the suns declination , so is the radius to the sine of ascensional difference . then by the third variety you may make another problem , viz. as the radius is to the co-tangent of the latitude , so is the sine of the suns ascensional difference to the sine of his declination . again , by the fourth variety you may make a third problem , thus , as the sine of the suns ascensional difference is to the tangent of the suns declination , so is the radius to the co-tangent of the latitude . by this artifice many have stuffed their books with bundles of problems . chap. vii . some uses of the lines of circumference , diameter , square equal , and square inscribed . all these are lines of equal parts , bearing such proportion to each other , as the things signified by their names . their use is this , any one of them given in inches or feet , &c. to finde how much any of the other three are in the same measure . suppose i have the circumference of a circle , tree , or cylinder given in inches , i take the same number of parts ( as the circle is inches ) from the line of circumference , and applying that distance to the respective lines , i have immediately the square equal , square inscribed , and diameter , in inches , and the like , if any of those were given to finde the circumference . this needs no example . the conveniency of this line any one may experiment in standing timber ; for taking the girth , or circumference with a line , finde the diameter ; from that diameter abate twice the thickness of the bark , and you have the true diameter , when it s barked , and by chap. 9. probl. 5. you will guess very near at the quantity of timber in any standing tree . chap. viii . to measure any kinde of superficies , as board , glass , pavement , walnscot , hangings , walling , slating , or tyling , by the line of numbers on the outward ledge . the way of accounting any number upon , or working proportions by , the line of numbers , is sufficiently shewn already , chap. 6. which i shall not here repeat , only propose the proportions for these problems , and refer you to those directions . probl. 1. the breadth of a board given in inches , to finde how many inches in length make a foot at that breadth , say , as the breadth in inches is to 12 , so is 12 to the length in inches for a foot at that breadth . ex. gr . at 8 inches breadth you must have 18 inches in length for a foot . probl. 2. the breadth and length of a board given to finde the content . as 12 is to the length in feet and inches , so is the breadth in inches to the content in feet . ex. gr . at 15 inches breadth , and 20 foot length , you have 25 foot of board . probl. 3. a speedy way to measure any quantity of board . the two former problems are sufficient to measure small parcels of board . when you have occasion to measure greater quantities , as 100 foot , or more , lay all the boards of one length together , and when the length of the boards exceeds 12 foot , use this proportion . as the length in feet and inches is to 12 , so is 100 to the breadth in inches for an 100 foot . ex. gr . at 30 foot in length 40 inches in breadth , make an 100 foot of board ( reckoning five score to the hundred . ) this found with a rule or line , measure 40 inches at both ends in breadth , and you have 100 foot . when one end is broader than another , you may take the breadth of the over-plus of 100 foot at both ends , and taking half that sum for the true breadth of the over-plus by probl. 2. finde the content thereof . when your boards are under 12 foot in length , say , as the length in feet and inches is to 12 , so is 50 unto the breadth in inches for 50 foot of board , and then you need only double that breadth to measure 100 foot as before . in like manner you may measure two , three , four , five , a hundred , &c. foot of board speedily , as your occasion requires . probl. 4. to measure wainscot , hangings , plaister , &c. these are usually computed by the yard , and then the proportion is . as nine to the length in feet , and inches , so is the breadth or depth in feet and inches to the content in yards , ex. gr . at 18 foot in length , and two foot in breadth , you have four yards . probl. 5. to measure masons , or slaters work , as walling , tyling , &c. the common account of these is by the rood , which is eighteen foot square , that is 324 square foot in one rood , and then the proportion is . as 324 to the length in feet , so is the breadth in feet to the content in roods . ex. gr . at 30 foot in length , and 15 foot in breadth , you have 1 rood 3 / 10 and better , or one rood 126 / 124 parts of a rood . chap. ix . the mensuration of solids , as timber , stone , &c. by the lines on the proportional side of the loose piece . these two lines meeting upon one line in the midst betwixt them ( for distinction sake ) i call one the right , the other the left line , which are known by the hand they stand toward when you hold up the piece in the right way to read the figures . the right line hath two figured partitions . the first partition is from 3 , at the beginning to the letters sq. every figured division representing an inch , and each subdivision quarters of an inch . the next partition is from the letters sq. unto 12 , at the end , every figured division signifying a foot , and each sub-division the inches in a foot . the letters t r , and t d , are for the circumference and diameter in the next measuring of cylinders . the letters r and d. for the measuring of timber , according to the vulgar allowance , when the fourth part of the girt is taken , &c. the letters a and w are the gauger points for ale and wine measures . lastly , the figures 12 'twixt d and t d. are for an use , expressed probl. 2. the left line also hath two figured partitions , proceeding first from 1 at the beginning to one foot , or 12 inches , each whereof is sub-divided into quarters . from thence again to 100 , each whereof to 10 foot , is sub-divided into inches , &c. and every foot is figured . but from 10 foot to 100 , only every tenth foot is figured ; the sub-divisions representing feet . the method of working proportions by these lines ( only observing the sides ) is the same as by the line of numbers , viz. extending from the first to the second , &c. probl. 1. to reduce timber of unequal breadth and depth to a true square . as the breadth on the left is to the breadth on the right , so is the depth on the left , to the square on the right line . at 7 inches breadth , and 18 inches depth , you have 11 inches ¼ and better for the true square . probl. 2. the square of a piece of timber given in inches , or feet , and inches to finde how much in length makes a foot. as the square in feet and inches on the right is to one foot on the left , so is the point sq. on the right to the number of feet and inches on the left for a foot square of timber . at 18 inches square , 5 inches ¼ and almost half a quarter in length makes a foot . when your timber ( if it be proper to call such pieces by that name ) is under 3 inches square , account the figured divisions on the right line from the letters sq. to the end , for inches , and each sub-division twelve parts of an inch . so that every three of them makes a quarter of an inch . then the proportion is , as the inches and quarters square on the right is to 100 on the left , so is the point 12 'twixt d , and t d , on the right , to the number of feet in length on the left to make a foot of timber . as 2 inches ½ square you must have 23 foot 6 inches , and somewhat better for the length of a foot of timber . probl. 3. the square and length of a plece of timber given to finde the content . as the point sq. on the right is to the length in feet and inches on the left , so is the square in feet and inches on the right to the content in feet on the left . at 30 foot in length , and 15 inches square , you have 46 foot ½ of timber . at 20 foot in length , and 11 inches square you have 16 foot , and almost ¼ of timber . when you have a great piece of timber exceeding 100 foot ( which you may easily see by the excursion upon the rule ) then take the true square , and half the length , sinde the content thereof by the former proportion , and doubling that content , you have the whole content . probl. 4. the circumference , or girth , of a round piece of timber , being given , together with the length , to finde the content . as the point r. on the right , is to the length in feet and inches on the left , so is the circumference in feet and inches on the right , to the content in feet on the left . at 20 foot in length , and 7 foot in girth , you have 60 foot of timber for the content . this is after the common allowance for the waste in squaring ; and although some are pleased to quarrel with the allowancer , as wronging the seller , and giving the quantity less , than in truth it is ; yet i presume when they buy it themselves , they scarcely judge those chips worth the hewing , and have as low thoughts of the over-plus , as others have of that their admonition . if it be a cylinder that you would take exact content of , then say , as the point t r , on the right , is to the length on the left , so is the girt on the right to the exact content on the left . at 15 foot in length , and 7 foot in girt , you have 59 foot of solid measure . the diameter of any cylinder given , you may by the same proportion finde the content , placing the point d , instead of r , in the proportion for the usual allowance , and the point t d , for the exact compute . probl. 5. to measure tapered timber . take the square or girt at both ends , and note the sum and difference of them . then for round timber , as the point r. on the right , is to the length on the left , so is half the sum of the girt at both ends on the right , to a number of feet on the left . keep this number , and say again , as the point r. on the right , is to the third part of the former length on the left , so is half the difference of the girts on the right , to a number of feet on the left ; which number added to the former , gives the true content . the same way you may use for square timber , only setting the feet and inches square instead of the girt , and the point sq. instead of the point r. at 30 foot in length , 7 foot at one end , and 5 at the other in girt , half the sum of the girts is 6 foot , or 72 inches ; the first number of feet found 67 , half the difference of the girts is 1 foot , or 12 inches , the third part of the length 10 foot ; then the second number found will be 7 foot , one quarter and half a quarter . the sum of both ( or true content ) 74 foot , one quarter , and half a quarter . for standing timber , take the girt about a yard from the bottom , and at 5 foot from the bottom , by chap. 7 , set down these two diameters without the bark ; and likewise the difference 'twixt them . again , by chap. 6. probl. 4. finde the altitude of the tree , so far as it bears timber ( or as we commonly phrase it , to the collar ) this done , you may very near proportion the girt at the collar and content of the tree , before it falls . in case any make choice of the hollow contrivance mentioned , chap. 1. they need no compasses in the mensuration of any solid ; provided the lines for solid measure , and gauging vessels , be doubly impressed ( only in a reverted order , one pair of lines proceeding from the head toward the end , and the other pair from the end toward the head ) upon the sliding cover , and its adjacent ledges . this done , the method of performing any of the problems mentioned in this chapter , is easie . for whereas you are before directed to extend the compasses from the first term to the second ; and with that distance setting one point in the third term , the other point gave the fourth , or term sought . so here , observing the lines as before , slide the cover until the first term stand directly against the second ; then looking for the third on its proper line , it stands exactly against the fourth term , or term sought on the other line . only note , that when the second term is greater than the first , it s performed by that pair of lines proceeding from the head toward the end : but when the first term is greater than the second , it is resolved by that pair of lines which is numbred from the end toward the head . chap. x. the gauge vessels , either for wine , or ale , measure . probl. 1 . the diameter at head , and diameter at boung , given in inches , and tenth parts of an inch , to finde the mean diameter in like measure . take the difference in inches , and tenth parts of an inch , between the two diameters . then say by the line of numbers , as 1 is to 7 , so is the difference to a fourth number of inches , and tenth parts of an inch . this added to the diameter at head , gives the mean diameter . ex. gr . at 27 inches the boung , and 19 inches two tenths at the head , the difference is 7 inches , 8 tenths . the fourth number found by the proportion will be 5 inches , 4 tenths , and one half , which added to the diameter at the head , gives 24 inches , 6 tenths , and one half tenth of an inch for the mean diameter . probl. 2. the length of the vessel , and the mean diameter given in inches , and tenth parts of an inch , to finde the content in gallons , either in wine or ale measure . note first , that the point a. on the right is the gauge point for ale measure , and the point w. on the right , is the gauge point for wine measure . then say , as the gauge point on the right to the length in inches , and tenth parts of an inch on the left , so is the mean diameter in inches and tenth parts of an inch on the right , to the content in gallons on the left . chap. xi . some uses of the lines on the inward ledge of the moveable piece . the line of inches and foot measure do by inspection ( only ) reduce either of the measures into the other . ex. gr . three on the line of inches stands directly against 25 cents of foot measure . or 75 cents of foot measure is directly against 9 on the inches . another use of these lines ( welcome perhaps to them that delight in instrumental computations ) is to know the price of carriage for any quantity , &c. by inspection . for this purpose , the line of inches represents the price of a pound , every inch being a penny , and every quarter a farthing . the prickt line is the price of an hundred pound at five score and twelve to the hundred , every division signifying a shilling . the line of foot measure is the price of an hundred pound at five score to the hundred , every division standing for a shilling . ex. gr . at 3 pence the pound on the inches , is 28 shillings on the prickt line , and 25 shillings on the foot measure , for the price of an hundred , the like for the converse . otherwise , the price of a pound being given , the rate of an hundred is readily computed without the rule . for considering the number of farthings is in the price of a pound , twice that number of shillings , and once that number of pence , is the price of an hundred , reckoning five score to the hundred ; of twice that number of shillings , and once that number of groats , is the price of an hundred , at five score and twelve to the hundred . ex. gr . at three half pence the pound , the number of farthings is six . therefore , twice six shillings , and once six pence ( that is 125. 6d . ) is the price of an hundred , at five score to the hundred . again , twice six shillings , and once six groats ( that is 14s . ) is the price of an hundred , at five score and twelve to the hundred . but of this enough , if not too much already . chap. xii . to divide a plot of ground into any proposed quantities . see fig. 4 suppose a b c d e f g h i k l , a plot of ground , containing 54 acr . 2 roods , 28 poles , from the point a , i am required to shut off 18 acres , next the side b c. draw a d. and measure the figure a b c d , which is 14 acr : 2 r. 3 p. that is 3 acr . 1. r. 37 p. or 557 poles too little . draw again a f , and measure a d f , which is 1309. poles . then by chap. 5. probl. 1. entring d f , the base of the triangle in 1309 on the lines of lines , and taking out 557 , set it from d to e , and draw a e. so have you the figure a b c d e 18 acres . again , from the point g , i would set off 20 acres next the side a e , draw a g , and measure a e f g 15 acr . or . 2 pol. whereof want 4 acr . or . 38 pol. that 678 poles , then draw g k , and measure g a k 1113 poles . lastly , by chap. 5. probl. i. enter a k , the base of the triangle in 1113 upon the lines of lines , and taking out 678 , set it from a , and draw g l. so have you the figure a e f g l , twenty acres . how ready the instrument would be for surveying with the help of a staff , ballsocket , and needle , is obvious to any one that considers its graduated into 180 degrees . chap. xiii . so much of geography as concerns finding the distance of any two places upon the terrestrial globe . here are three cases , and each of those contains the same number of propositions . case 1. when the two places differ in latitude only . prop. 1. when one place lies under the equator , having no latitude . the latitude of the other place turned into miles , ( reckoning 60 miles , the usual compute , for a degree ) is the distance sought . prop. 2. when both places have the same pole elevated , viz. north , or south . take the difference of their latitudes , and reckoning 60 miles for a degree ( as before ) you have their distance . prop. 3. when the two places have different poles elevated , viz. one north , the other south . adde two latitudes together , and that sum turned into miles is the distance . case 2. when the two places differ in longitude only . prop. 1. when neither of them have any latitude , but lie both under the equator . their difference of longitude turned into miles ( as before ) is their distance . prop. 2. when the two places have the same pole elevated . the proportion is thus . as the radius is to the number 60 , so is the cosine of the common latitude to the number of miles for one degree of longitude . multiply this number found by the difference of longitude , and that product is the distance in miles . prop. 3. when the two places have different poles elevated . as the radius is to the cosine of the common latitude , so is the sine of half the difference of longitude to the sine of half the distance . wherefore this sine of half the distance doubled and turned into miles , is the true distance . case 3. when the two places differ in longitude and latitude both . prop. 1. when one of the places lies under the equator , having no latitude . as the radius is to the cosine of the difference of longitude , so is the cosine of the latitude to the cosine of the distance . prop. 2. when both the places have the same pole elevated . as the radius is to the cosine of the difference in longitude , so is the co-tangent of the lesser altitude , to the tangent of a fourth ark . subtract this fourth ark out of the complement of the lesser latitude , and keep the remain . then , as the cosine of the fourth ark is to the cosine of the remain , so is the sine of the lesser latitude to the cosine of the distance . prop. 3. when the two places have different poles elevated . as the radius is to the cosine of the difference in longitude , so is the co-tangent of either latitude to the tangent of a fourth ark . subtract the fourth out of the latitude not taken into the former proportion , and note the difference . then , as the cosine of the fourth ark is to the cosine of this difference , so is the sine of the latitude first taken , to the cosine of the distance . chap. xiv . some uses of the instrument in navigation , or plain salling . here it will be necessary to premise the explication of some terms , and adjoyn two previous proportions . 1. the compass being a circle , divided into 32 equal parts , called rumbs ; one point or rumb is 11 d. 15 min. of a circle from the meridian : two points or rumbs is 22 d. 30 min. &c. of the rest . 2. the angle which the needle , or point of the compass under the needle , makes with the meridian , or north and south line is called the course or rumb ; but the angle which it makes with the east , and west line , or any parallel , is named the complement of the course or rumb . 3. the departure is the longitude of that port from which you set sail . 4. the distance run , is the number of miles , or leagues ( turned into degrees ) that you have sailed . 5. when you are in north latitude , and sail north-ward , adde the difference of latitude to the latitude you sailed from ; and when you are in north latitude , and sail southward , subtract the difference of latitude from the latitude you sailed , and you have the latitude you are in . the same rule is to be observed in south latitude . 6. to finde how many miles answer to one degree of longitude in any latitude . as the radius is to the number 60 , so is the cosine of the latitude to the number of miles for one degree . 7. to finde how many miles answer to one degree of latitude on any rumb . as the cosine of the rumb from the meridian , is to the number 60 , so is the radius to the number of miles . the most material questions in navigation are these four . first , to finde the course . secondly , the distance run . thirdly , the difference of latitude . fourthly , the difference in longitude ; and any two of these being given , the other two are readily found by the square and index . these two additional rulers were omitted in the first chapter of this treatise ; sith they are only for navigation , and large instruments of two , or three foot in length , which made me judge their description most proper for this place : because , such as intend the instrument for a pocket companion , will have no use of them . the square is a flat rule , having a piece , or plate fastened to the head , that it may slide square , or perpendicular to the outward ledge of the fixed piece . it hath the same line next either edge on the upper side , which is a line of equal parts , an hundred , wherein is equal to the radius of the degrees on the outward limb of the moveable piece . the index is a thin brass rule on one side , having the same scale as the square . on the other side is a double line of tangents , that next the left edge , being to a smaller ; that next the right edge , to a larger radius . for the use of these rulers , you must have a line of equal parts adjoyning to the line of sines on the fixed piece , divided into 10 parts , stamped with figures , each of those divided into 10 parts more ; so that the whole line is divided into 100 parts , representing degrees . lastly , let each of those degrees be sub-divided into as many parts , as the largeness of your scale will permit , for computing the minutes of a degree . the index is to move upon the pin on the fixed piece ( where you hang the thread for taking altitudes ) and that side of the index ( in any of the four former questions ) must be upward , which hath the scale of equal parts . the square is to be slided along the outward ledge of the fixed piece . then the general rules are these . the difference of latitude is accounted on the line of equal parts , adjoyning to the sines on the fixed piece . the difference of longitude is numbred on the square . the distance run is reckoned upon the index . the course is computed upon the degrees on the limb from the head toward the end of the moveable piece . but when any would work these problems in proportions , let them note , the distance run , difference of longitude , and difference of latitude , are all accounted on the line of numbers ; the rumb or course is either a sine or tangent . this premised . i shall first shew how to resolve any problem by the square and index ; and next adjoyn the proportions for the use of such as have only pocket instruments . probl. 1. the course and distance run given , to finde the difference of latitude , and difference of longitude . apply the index to the course reckoned on the limb from the head , and slide the square along the outward ledge of the fixed piece , until the fiducial edge intersect the distance run on the fiducial edge of the index . then at the point of intersection you have the difference of longitude upon the square , and on the line of equal parts on the fixed piece , the square shows the difference of latitude . the proportion is thus , as the radius is to the distance run , so is the cosine of the course to the difference of latitude . again , as the radius is to the distance run , so is the sine of the course to the difference in longitude . probl. 2. the course and difference of latitude given to find the distance run , and difference in longitude . slide the square to the difference of latitude on the line of equal parts upon the fixed piece , and set the index to the course on the limb . then at the point of intersection of the square and index , on the square is the difference of longitude , on the index the distance run . the proportion is thus . as the cosine of the course is to the difference of latitude , so is the radius to the distance run . again , as the radius is to the sine of the course , so is the distance run to the difference of longitude . probl. 3. the course and difference in longitude given , to finde the distance run , and difference of latitude . apply the index to the course on the limb , and the difference of longitude on the square to the fiducial edge of the index . then at the point of intersection you have distance run on the index , and upon the line of equal parts on the fixed piece , the square shows the difference of latitude . the proportion is thus . as the sine of the course is to the difference of longitude , so is the radius to the distance run . again , as the radius is to the distance run , so is the cosine of the course to the difference of latitude . probl. 4. the distance run , and difference of latitude given to finde the course , and difference in longitude . slide the square to the difference of latitude on the line of equal parts on the fixed piece , and move the index until the distance run numbred thereon , intersect the fiducial edge of the square ; then at the point of intersection you have the difference of longitude on the square , and the fiducial edge of the index on the limb shows the course . as the distance run is to the difference of latitude , so is the radius to the cosine of the course . again , as the radius is to the distance run , so is the sine of the course to the difference of longitude . probl. 5. the distance run , and difference of longitude given to finde the course , and difference of latitude . apply the distance run numbred on the fiducial edge of the index , to the difference of longitude , reckoned on the fiducial edge of the square . then on the line of equal parts upon the fixed piece , the square shows the difference of latitude , and the index shows the course on the limb . the proportion is thus . as the distance run is to the difference of longitude , so is the radius to the sine of the course . again , as the radius is to the distance run , so is the cosine of the course , to the difference of latitude . probl. 6. the difference of latitude , and difference of longitude given , to finde the course , and distance run . apply the square to the difference of latitude on the scale of equal parts upon the fixed piece , and move the index until its fiducial edge intersect the difference of longitude , reckoned on the square . then at the point of intersection you have the distance run upon the index , and the fiducial edge of the index upon the limb shows the course . the proportion is thus . as the difference of latitude is to the difference of longitude , so is the radius to the tangent of the course . again , as the sine of the course is to the radius , so is the difference of longitude to the distance run . probl. 7. sailing by the ark of a great circle . for this purpose the tangent lines on the index will be a ready help , using the lesser for small , and the greater tangent line , for great latitudes . the way is thus , account the pont 60 / 0 , on the outward limb of the moveable piece to be the point , or port of your departure ; thereto lay the fiducial edge of the index , and reckoning the latitude of the port you departed from upon the index , strike a pin directly touching it , into the table your instrument lies upon . this pin shall represent the port of your departure . therefore hanging a thread , or hair , on the center , whereon the index moves ; and winding it about this pin . count the difference of longitude 'twixt the port of your departure , and the port you sail toward , from 60 / 0 on the moveable piece toward 0 / 60 , on the loose piece ; and thereto laying the same fiducial edge of the index , reckon the latitude of this last port upon the index , directly touching of it , strike down another pin upon the table , and draw the thread strait about this pin fastening it thereto . this done , the thread betwixt the two pins represents the ark of your great circle ; and laying the fiducial edge of the index to any degree of difference of longitude accounted from 60 / 0 on the moveable piece , the thread shows upon the index what latitude you are in , and how much you have raised , or depressed the pole since your departure . on the contrary , laying the latitude you are in ( numbred upon the index ) to the thread , the index shows the difference of longitude upon the limb ; counting from 60 / 0 on the moveable piece . so , that were it possible to sail exactly by the ark of a great circle , it would be no difficulty to determine the longitude in any latitude you make . but i intend not a treatise of navigation ; wherefore let it suffice , that i have already shown how the most material problems therein , may easily , speedily , and ( if the instrument be large ) exactly , be performed by the instrument without the trouble of calculation , or projection . chap. xv. the projection and solution of the sixteen cases in right angled spherical triangles by five cases . see fig. 5. the fundamental circle n b z c , is alwayes supposed ready drawn , and crossed into quadrants , and the diameters produced beyond the circle . case 1. given both the sides z d , and d r , to project the triangle . by a line of chords , prick off z d , upon the limb , and draw the diameter d a e. again , by a line of tangents , set half co-tangent d r , upon a d , from a to r , then have you three points , viz. n r z , to draw that ark , and make up the triangle . the center of which ark always lies on ac , ( produced beyond c , if need requires ) and is found by the intersection of the two arks made from r , and z. case 2. given one side z d , and the hypothenuse z r , to project the triangle . prick off z d , and draw d a e , by case 1. again , set half the co-tangent z r , on the line a z , from a to f , and the tangent z r , set from f to p , with the extent f p , upon the center p , draw the ark v f i , and where it intersects the diameter d a e , set r ; then have you three points n r z , to draw that ark , as in the former case . case 3. given the hypothenuse z r , and the angle d z r , to project the triangle . prick half the co-tangent d z r , from a to s , and the secant d z r , from s to t , upon the center t , with the extent t s , draw the ark n s z. again , by case 2 , draw the ark h f i , where these two arks intersect each other , set r. lastly , lay a ruler to a r , and draw the diameter drae , and your triangle is made . case 4. given one side z d , and its adjacent angle d z r , to project the triangle . prick off z d , and draw d a e , by case 1. again , by case 3 , draw the ark n s z , where this ark intersects the diameter dae , set r , and your triangle is made . or , given the side d r , and its opposite angle d z r , you may project the triangle . draw the ark n s z , as before . again , take half the co-tangent d r , and with that extent upon a , the center , cross the ark n s z , setting r , at the point of intersection . lastly , lay a ruler to a r , and draw the diameter d r a e , which makes up your triangle . the triangle projected in any of the four former cases , to measure any of the sides , or angles , do thus . first , the 〈◊〉 z d , is found by applying it to a line of chords . secondly , r a , applyed to a line of tangents , is half the co-tangent d r. thirdly , a s , applyed to a line of tangents is the co-tangent d z r. fourthly , set half the tangent d z r , from a to l , then is l the pole point , and laying a ruler to l r , it cuts the limb at v , and the ark z v , upon the line of chords , gives the hypothenuse z r. fifthly , prick z d from c to k , a ruler laid from r to l , cuts the limb at g , then g k , upon a line of chords , is the quantity of z r d. case . 5. the two oblique angles , d z r , and z r d given , to present the triangle . see fig. 6. this is no more than turning the former triangle . thus , draw the ark n s z , by case 1 , and set half the tangent of that ark from a to l. again , set half the co-tangent d r z , from a to f , and the secant of d r z , from f to l , upon the center a , with the extent a p. draw the ark p g , and with the extent f p , from l , cross the ark p g in g. lastly , upon the center g , with the extent g l. draw the ark r d f l , and your triangle is made . the triangle projected you may measure off the sides and hypothenuse . thus , first , the hypothenuse z r , is measured by a line of chords . secondly , a ruler laid to l d , cuts the limb at h , and z h , upon a line of chords , is the measure of the ark z d. thirdly , draw a g , and set half the tangent d r z , from a to v , apply a ruler to v d , it cuts the limb at e , then r e , upon a line of chords , measure the ark r d. note . the radius to all the chords , tangents , and secants , used in the projection , and measuring , any ark or angle , is the semidiameter of the fundamental circle . chap. xvi . the projection and solution of the 12 cases in oblique angled spherical triangles in six cases . see fig. 7. the fundamental circle n h z m , is alwayes supposed ready drawn , and crossed into quadrants , and the diameters produced beyond the circle . case 1. the three sides , z p , p z , and z s , given , to project the triangle . by a line of chords prick off z p , and draw the diameter p c t , crossing it at right angles in the center with ae c e , set half the co-tangent p s , from c to g , and he secant p s from c to r , upon the center r , with the extent r g , draw the the ark fgl . again , set half the co-tangent z s , from c to d , and the tangent z s , from d to o , with the extent o d , upon the center o , draw the ark b d p , mark where these two arks intersect each other as at s. then have you three points t s p , to draw that ark , and the three points n s p , to draw that ark , which make up your triangle . case 2. given two sides z s , and z p , with the comprehended angle p z s , to project the triangle . prick off z p , and draw pct , and aece , and the ark b d p , by case 1. again , set the tangent of half the excess of the angle p z s above 90 , from c to w , and co-secant of that excess from w to k , upon the center k , with the extent k w ; draw the ark n w z , which cuts the ark b d p in s. then have you the three points t s p , to draw that ark which makes up the triangle . case 3. two angles s z p , and z p s , with the comprehended side z p , given , to project the triangle . prick off z p , and draw the lines p c t , and ae c e , by case 1 , and the angle nwz . by case 2. lastly , set half the co-tangent zps from c to x , and the secant z p s , from x to v , upon the center v , with the extent v x , draw the ark t x s p , and the triangle is made . case 4. two sides , zp , and ps , with the angle opposite to one of them szp , given , to project the triangle . prick off zp , and draw pct , and aece , by case 1. and the angle szp by case 2. lastly , by case 1. draw the ark fgl , and mark where it intersects nwz , as at s , then have you the three points tsp , to draw that ark , and make up the triangle . case 5. two angles szp , and zps , with the side opposite to one of them zs , given , to project the triangle . draw the ark bdp , by case 1. and the ark nwz , by case 2. at the intersection of these two arks , set s , with the tangent of the angle zps , upon the center c. sweep the ark vδi . again , with the secant of the ark zps upon the center s , cross the ark vδi , as at the points v and i. then in case the hypothenuse is less than a quadrant ( as here ) the point v , is the center , and with the extent vs , draw the ark tsp , which makes up the triangle . but in case the hypothenuse is equal to a quadrant , δ , is the center ; if more than a quadrant , i , is the center ; in which cases the extent from δ , or i , to s , is the semidiameter of the ark tsp . case 6. three angles zps , and pzs , and zsp , given , to project the triangle . see fig. 7. and 8. the angles of any spherical triangle may be converted into their opposite sides , by taking the complement of the greatest angle to a semicircle for the hypothenuse , or greatest side . wherefore by case 1. make the side zp , in fig. 4. equal to the angle zsp , in fig. 3 , and the side zs , in fig. 4. equal to the angle zps , in fig. 3. and the side ps , in fig. 4. equal to the complement of the angle pzs , to a semicircle in fig. 3. then is your triangle projected where the angle zps in fig. 4. is the side zs , fig. 3. again , the angle zsp , fig. 4. is the side zp , in fig. 3. lastly , the complement of the angle pzs to a semicircle in fig. 4. is the measure of the hypothenuse , or side p s , in fig. 3. the triangle being in any of the former cases projected , the quantity of any side or angle may be measured by the following rules . first , the side z p , is found by applying it to a line of chords . secondly , cx , applyed to a line of tangents , is half the co-tangent of the angle zps . thirdly , cw applyed to a line of tangents , is half the co-tangent of the excess of the angle szp , above 90. fourthly , set half the tangent of the angle zps , from c , to π , a ruler laid to πs , cuts the limb at f ; then pf , applyed to a line of chords , gives the side ps . fifthly , take the complement of the angle pzs , to a semicircle , and set half the tangent of that complement from c , to λ , a ruler laid to λs , cuts the limb at b , and zb , applyed to a line of chords , gives the side zs . sixthly , a ruler laid to sλ , cuts the limb at l. again , a ruler laid to sπ , cuts the limb at φ , and l φ , applyed to a line of chords , gives the angle zsp . the end of the first book . an appendix to the first book . the sights which are necessary for taking any altitude . angle , or distance ( without the help of thread or plummet ) are only three , viz. one turning sight , and two other sights , contrived with chops , so that they may slide by the inward or outward graduated limbs . the turning sight hath only two places , either the center at the head , or the center at the beginning of the line of sines on the fixed piece ; to either of which ( as occasion requires ) it s fastened with a sorne . the center at the head serving for the graduations next the inward limb of the loose piece . and the center at the beginning of the line of sines serving for all the graduations next the outward limb of the moveable and loose piece both . yet because it is requisite to have pins to keep the loose piece close in its place . you may have two sights more to supply their place ( which sometimes you may make use of ) and so the number of sights may be five , viz. two sliding sights , one turning sight , and two pin sights , to put into the holes at the end of the fixed and moveable piece , to hold the tenons of the loose piece close joynted . every one of these sights hath a fiducial ( or perpendicular ) line , drawn down the middle of them , from the top to the bottom , where this line toucheth the graduations on the limb , is the point of observation . the places of these sights have an oval proportion , about the middle of them , only leaving a small bar of brass , to conduct the fiducial line down the oval cavity , and support a little brass knot ( with a sight hole in it ) in the middle of that bar , which is ever the point to be looked at . there are two wayes of observing an altitude with help of these sights . the one when we turn our face toward the object . this is called a forward observation in which you must alwayes set the turning sight next your eye . this way of observation will not exactly give an altitude above 45 degrees . the other way of observing an altitude is peculiar to the sun in a bright day , when we turn our back toward the sun. this is termed a backward observation ; wherein you must have one of the sliding sights next your eye , and the turning sight toward the horizon . this serves to take the suns altitude without thread , or plummet , when it is near the zenith . probl. 1. to finde the suns altitude by a forward observation . serve the turning sight to the center of those graduations you please to make use of ( whether on the inward or outward limb ) and place the two sliding sights upon the respective limb to that center ; this done , look by the knot of the turning sight ( moving the instrument upward or downward ) until you see the knot of one of the sliding sights directly against the sun , then move the other sliding sight , until the knot of the turning sight , and the knot of this other sliding sight be against the horizon ; then the degrees intercepted 'twixt the fiducial lines of the sliding sights on the limb , shew the altitude required . probl. 2. to finde the distance of any two stars , &c. by a forward observation . serve the turning sight to either center , and apply the two sliding sights to the respective limb ( holding the instrument with the proportional side downward ) and applying the turning sight to your eye , so move the two sliding sights either nearer together , or further asunder , that you may by the knot of the turning sight see both objects even with the knots of their respective sliding sights , then will the degrees intercepted 'twixt the fiducial lines of the object sights on the limb show the true distance . by this means you may take any angle for surveying , &c. probl. 3. to finde the suns altitude by a backward observation . serve the turning sight to the center at the beginning of the line of sines , and apply one of the sliding sights to the outward limb of the loose piece , and the other to the outward limb of the moveable piece ; and turning your back toward the sun , set the sliding sight upon the moveable piece next your eye ; and slide it upward or downward toward the end , or head , until you see the shadow of the little bar , or edge , of the sight on the loose piece fall directly on the little bar on the turning sight ; and at the same time the bar of the sight next your eye , and the bar of the turning sight to be in a direct line with the horizon . then will the degrees on the limb intercepted 'twixt the fiducial lines of the sliding sights ( if you took the shadow of the bar ) or 'twixt the fiducial line of the sliding sight next your eye , and the edge of the other sliding sight ( when you took the shadow of the edge ) be the true altitude required . σκιογραφια , or , the art of dyalling for any plain superficies . lib . ii. chap. i. the distinction of plains , with rules for knowing of them . all plain superficies are either horizontal , or such as make angles with the horizon . horizontal plains are those , that lie upon an exact level , or flat . plains , that make angles with the horizon are of three sorts . 1. such as make right angles with the horizon , generally known by the name of erect , or upright plains . 2. such as make acute angles with the horizon , or have their upper edge leaning toward you , usually termed inclining plains . 3. such as make obtuse angles with the horizon , or have their upper edge falling from you , commonly called reclining plains . all these three sorts are either direct , viz. east , west , north , south . or else declining from south , toward east , or west . from north , toward east , or west . all plain superficies whatsoever are comprized under one of these terms . but before we treat of the affections , or delineation of dials for them ; it will be requisire to acquaint you with the nature of any plain , which may be found by the following problems . probl. 1. to finde the reclination of any plain . apply the outward ledge of the moveable piece to the plain with the head upward , and reckoning what number of degrees the thread cuts on the limb ( beginning your account at 30. on the loose piece , and continuing it toward 60 / 0 on the moveable piece ) you have the angle of reclination . if the thread falls directly on 60 / 0 upon the moveable piece , it s an horizontal ; if on 30. on the loose piece , it s an erect plain . probl. 2. to finde the inclination of any plain . apply the outward ledge of the fixed piece to the plain , with the head upward , and what number of degrees the thread cuts upon the limb of the loose piece , is the complement of the plains inclination . probl. 3. to draw an horizontal line upon any plain . apply the proportional side of the instrument to the plain , and move the ends of the fixed piece upward , or downward , until the thread falls directly on 60 / 0. upon the loose piece ; then drawing a line by the outward ledge of the fixed piece , its horizontal , or paralel to the horizon . probl. 4. to draw a perpendicular line upon any plain . when the sun shineth hold up a thread with a plummet against the plain , and make two points at any distance in the shadow of the thread upon the plain , lay a ruler to these points , and the line you draw is a perpendicular . probl. 5. to finde the declination of any plain . apply the outward ledge of the fixed piece to the horizontal line of your plain , holding your instrument paralel to the horizon . this done , lift up the thread and plummer , until the shadow of the thread fall directly upon the pin hole on the fixed piece ( where you hang the thred to take altitudes ) then observe how many degrees the shadow of the thread cuts in the limb , either from the right hand , or from the left hand 0 / 60. upon the loose piece ; and immediately taking the altitude of the sun. by lib. 1. cap. 2. probl. 9 , 10 , 11. finde the suns azimuth from south . and , when you make this observation in the morning , these cases determine the declination of the plain . case 1. when the shadow of the thread upon the limb falls on the right hand 0 / 60 on the loose piece , take the difference of the shadow , and azimuth ( by subtracting the lesse out of the greater ) and the residue or remain is the plains declination . from south toward east , when the azimuth is greater than the shadow . from south toward west , when the shadow is greater than the azimuth , when the shadow and azimuth are equal , it s a direct south plain . when the difference is just 90. its a direct east , when above 90. subtract the difference from 180. and the remain is the declination from north toward east . case 2. when the shadow falls on the left hand 0 / 60. adde the azimuth and shadow together , that sum is the plains declination ; from south toward east , when under 90 ; if it be just 90 , it s a direct east . if above 90. subtract it from 180. the remain is the declination from north toward east . when the sum is above 180. subtract 180 from it , and the remain is the declination from north toward west . case 3. when the shadow falls upon 0 / 60. the azimuth is the plains declination . when under 90 , its south-east , when equal to 90. direct east , when above 90. subtract it from 180. the remain is the declination from north toward east . if you make the observation afternoon , the following cases will resolve you . case 4. when the shadow falls on the left hand 0 / 60. the difference 'twixt the shadow and azimuth is the declination ; when the shadow is more than the azimuth it declines south-east , when less , south-west . when the shadow and azimuth are equal , it s a direct south plain , when their difference is equal to 90. it s a direct west ; when the difference exceeds 90. subtract it from 180. the remain is the declination north-west . case 5. when the shadow falls on the right hand 0 / 60. take the sum of the shadow and azimuth , and that is the declination from south toward west , when under 90. when just 90. its a direct west plain ; when more than 90. subtract it from 180. the remain is the declination north-west ; when the sum is above 180. subtract 180 from it , and the remain is the declination from north toward east . case 6. when the shadow falls upon 0 / 60. the azimuth is the quantity of declination . from south toward west , when under 90. when equal to 90. its direct west ; when more than 90. subtract it from 180. the remain is the declination from north toward west . probl. 6. to draw a meridional line upon a horizontal plain . draw first a circle upon the plain , and holding up a thread and plummet ( when the sun shines ) so that the shadow of the thread may pass through the center of the circle , make a point in the circumference where the shadow intersects it . at the same time finding the suns azimuth from south , by a line of chords , set it upon the limb of the circle from the intersection of the shadow toward the south , and it gives the true south point . wherefore laying a ruler to this last point and the center , the line you draw is a true meridian . chap. ii. the affections of all sorts of plains . sect. 1. the affections of an horizontal plain . 1. the style , or cock of every horizontal dial , is an angle equal to that latitude for which the dial is made . 2. the place of the style is directly upon the meridional , or twelve a clock line , and the angular point must stand in the center of the hour lines . 3. the rule for drawing the hour lines before six in the morning , is to draw the respective hour lines afternoon beyond the center ; or for the hour lines after six at evening , draw the respective hour lines in the morning beyond the center . 4. to place an horizontal dial upon the plain ; first draw a meridian line upon the plain by cap. 1. probl. 6. and lay in the line of 12. exactly thereon , with the angular point of the style toward the south , fasten the dial upon the plain . sect. 2. the affections of erect , direct south and north plains . 1. the style in both these is an angle equal to the complement of the latitude for which the dial is made to stand upon the meridian line , or perpendicular of your plain , with the angular point in the center of the hour lines , and that point in south alwayes upward , in north alwayes downward . 2. to prick off the dial from your paper draught upon the plain , lay the hour line of 6. and 6. upon the plains horizontal ; and applying a ruler to the center , and each hour line , transmit the hour lines from your paper draught to the plain . sect. 3. the affections of erect , direct east and west plains . 1. in both these the style may be a pin or plate , equal in length or heighth to the radius of the tangents , by which you draw the dial. 2. the style in both of them is to stand directly upon the hour of six , and perpendicular to the plain . sect. 4. the affections of erect declining plains . 1. these are of two sorts , either such as admit of centers to the hour lines and style , or such as cannot with conveniency ( because of the lowness of the style , and nearness of the hour lines to each other ) be drawn with a center to those lines . of this latter sort are all such plains , whose style is an angle less than 15 degrees . for where the angle of the style is more than 15 degrees ; those dials may be drawn with a center to the hour lines . 2. in all erect declining plains with centers the meridian is the plains perpendicular . in those that admit not of centers in their delineation , the meridian is parallel to the plains perpendicular . 3. in all declining plains the substile , or line whereon the style is to stand , must be placed on that side the meridian , which is contrary to the coast of declination ; and also in such decliners as admit of centers , the angle 'twixt 12. and 6. is to be set to the contrary coast to that of declination . 4. in all decliners , without centers , the inclination of meridians is to be set from the substyle toward the coast of declination . 5. the proportions in all erect decliners , for finding the height of the style , the distance of the substyle from the meridian , the angle of twelve and six , with the inclination of meridians ( all which may be wrought either by the canon , or exactly enough for this purpose by the instrument ) are as followeth . to finde the styles height above the substile . as the radius is to the cosine of the latitude , so is the cosine of declination , to the sine of the styles height . to finde the substyles distance from the meridian . as the radius is to the sine of declination , so is the co-tangent of the latitude to the tangent of the substyle from the meridian . to finde the angle of twelve and six . as the co-tangent of the latitude is to the radius , so is the sine of declination , to the co-tangent of six from twelve . to finde the inclination of meridians . as the sine of the latitude is to the radius , so is the tangent of declination to the tangent of inclination of meridians . 6. all north decliners with centers have the angular point of the style downward , and all south have it upward . 7. all north decliners without centers have the narrowest end of the style downward , all south have it upward . 8. in all decliners without centers take so much of the style as you think convenient , but make points at its beginning and end upon the substyle of your paper draught , and transmit those points to the substyle of your plain , for direction in placing your style thereon . 9. in all north decliners the meridian , or inclination of meridians is the hour line of twelve at mid-night : in south decliners , at noon , or mid-day . this may tell you the true names of the hour lines . 10. in transmitting these dials from your paper draught to your plain , lay the horizontal of your paper draught , upon the horizontal line of the plain , and prick off the hours and substyle . sect. 5. the affections of direct reclining plains inclining plains . for south recliners , and north incliners . 1. the difference 'twixt your co-latitude , and the reclination inclination is the elevation , or height of the style . 2. when the reclination inclination exceeds your co-latitude , the contrary pole is elevated so much as the excess . ex. gr . a north recliner , or south incliner 50. d. in lat . 52. 30. min. the excess of the reclination inclination , to your co-latitude is 12. d. 30. min. and so much the north is elevated on the recliner , and the south pole on the incliner . 3. when the reclination inclination is equal to the co-latitude , it s a polar plain . for south incliners , and north recliners . 1. the sum of your co-latitude , and the reclination inclination is the styles elevation . 2. when the reclination inclination is equal to your latitude , it s an equinoctial plain , and the dial is no more than a circle divided into 24. equal parts , having a wyer of any convenient length placed in the center perpendicular to the plain for the style . 3. when the reclination inclination is greater than your latitude , take the summe of the reclination inclination of your co-latitude from 180. and the residue , or remain is the styles height . but in this case the style must be set upon the plain , as if the contrary pole was elevated , viz. these north recliners must have the center of the style upward , and the south incliners have it downward . note . in all south re-in-cliners north re-in-cliners , for their delineation the styles height is to be called the co-latitude , and then you may draw them as erect direct plains , for south , or north ( as the former rules shall give them ) in that latitude , which is the complement of the styles height . for direct east and west recliners incliners . 1. in all east and west recliners incliners , you may refer them to a new latitude , and new declination , and then describe them as erect declining plains . 2. their new latitude is the complement of that latitude where the plain stands , and their new declination is the complement of their reclination inclination : but to know which way you are to account this new declination , remember all east and west recliners are north-east , and north-west decliners . all east and west incliners are south-east , and south-west decliners . 3. their new latitude and declination known , you may by sect 4. par . 5. finde the substyle from the meridian , height of the style , angle of twelve and six , and inclination of meridians , using in those proportions the new latitude and new declination instead of the old . 4. in all east and west recliners incliners with centers , the meridian lies in the horizontal line of the plain ; in such as have not centers , its paralel to the horizontal line . sect. 6. the affections of declining reclining plains inclining plains . the readiest way for these , is to refer them to a new latitude , and new declination , by the subsequent proportions . 1. to finde the new latitude . as the radius is to the cosine of the plains declination , so is the co-tangent of the reclination inclination to the tangent of a fourth ark . in south recliners north incliners get the difference 'twixt this fourth ark , and the latitude of your place , and the complement of that difference is the new latitude sought . if the fourth ark be less than your old latitude , the contrary pole is elevated ; if equal to your old latitude , it s a polar plain . in south incliners north recliners the difference 'twixt the fourth ark , and the complement of your old latitude is the new latitude . if the fourth ark be equal to your old co-latitude , they are equinoctial plains . 2. to finde the new declination . as the radius is to the cosine of the reclination inclination , so is the sine of the old declination to the sine of the new . 3. to finde the angle of the meridian with the horizontal line of the plain . as the radius is to the tangent of the old declination , so is the sine of reclination inclination , to the co-tangent of the angle of the meridian with the horizontal line of the plain . this gives the angle for its scituation . observe , in north incliners less than a polar , the meridian lyes . above that end of the horizontall line contrary to the coast of declination .     below   south recliners more than a polar , the meridian lyes . below that end of the horizontall next the coast of declination .     above   north recliners less than an equinoctial , the meridian lyes above that end of the horizontalnext the coast of declination .     below in north recliners this meridian is 12. at midnight .   equal to an equinoctial the meridian descends below the horizontal at that end contrary to the coast of declination , and the substyle lies in the hour line of six .     south incliners more than an equinoctial , the meridian lyes . below that end of the horizontal contrary to the declination .     above in south incliners this meridian is only useful for drawing the dial , and placing the substyle , for the hour lines must be drawn through the center to the lower side . after you have by the former proportions and rules found the new latitude , new declination , the angle and scituation of the meridian , your first business in delineating of the dial will be ( both for such as have centers , and such as admit not of centers ) to set off the meridian in its proper coast and quantity . this done , by sect. 4. paragr . 5. of this chapter , finde the substyles distance from the meridian , the height of the style , angle of twelve and six ( and for dials without centers , the inclination of meridians ) in all those proportions , using your new latitude and new declination , instead of the old , and setting them off from the meridian , according to the directions in paragr . 3. and 4. you may draw the dials by the following rules , for erect declining plains . in placing of them , lay the horizontal line of your paper draught upon the horizontal line of your plain , and prick off the substyle and hour lines . only observe . that such south-east , or south-west recliners , as have the contrary pole elevated must be described as north-east , and north-west decliners , and such north-east , and north-west incliners as have the contrary pole elevated , must be described as south-east , and south-west decliners , which will direct you which way to set off the substyle , and hour line of six from the meridian in those oblique plains , which admit of centers , or the substyle from the meridian , and the inclination of meridians from the substyle in such as admit not of centers . 4. declining polar plains must have a peculiar calculation for the substyle and inclination of meridians , which is thus . to finde the angle of the substile with the horizontal line of the plain . as the radius is to the sine of the polar plains reclination , so is the tangent of declination to the co-tangent of the substyles distance from the horizontal line of the plain . to finde the inclination of meridians . as the radius is to the sine of the latitude , so is the tangent of declination to the tangent of inclination of meridians . 5. the reclining declining polar hath the substyle lying below that end of the horizontal line that is contrary to the coast of declination . the inclining declining polar hath the substyle lying above that end of the horizontal line , contrary to the coast of declination . chap. iii. the delineation of dials for any plain superficies . here it will be necessary to premise the explication of some few terms and symbols , which for brevity sake we shall hereafter make use of . ex. gr . rad. denotes the radius , or sine 90. or tangent 45. tang. is the tangent of any arch or number affixed to it . cos. is the cosine of any arch or number of degrees , or what it wants of 90. ex. gr . cos . 19. is what 19. wants of 90. that is 71. co-tang is the co-tangent of any ark or number affixed to it , or what it wants of 90. ex. gr . co-tangent 30. is what 30. wants of 90. that is 60. = this is a note of equality in lines , numbers , or degrees . ex. gr . ab = cd . that is , the line ab . is equal unto , or of the same length as the line cd . again , abc = efg . that is the angle abc . is of the same quantity , or number of degrees , as the angle efg . once more ab = cd = fg = tang . 15. that is ab . and cd . and fg. are all of the same length , and that length is the tangent of 15. d. ♒ this is a note of two lines being paralel unto , or equidistant from each other . ex. gr . fi. ♒ rs. that is the line fl. is paralel unto , or equi-distant from the line rs. sect. 1. to delineate an horizontal dial. see fig. 9. first draw the square bcde . of what quantity the plain will permit . then make af = ag = hd = he = sine of the latitude , and ah = radius . enter hd . in tang . 45. and keeping the sector at that gage , set off hi = hk = tang . 15. and ho = hl = tang . 30. again , enter fd. in tang . 45. and set off fq = gn = tang . 15. and fp = gm = tang . 30. this done , draw aq . ap. ad. ao . ai. for the hour lines of 5. 4. 3. 2. 1. afternoon . again , draw ak . al. ae . am. an. for the hour lines of 11. 10. 9. 8. 7. before noon . the line fag . is for six and six . in the same manner you may prick the quarters of an hour , reckoning three tangents , and 45. minutes , for a quarter . how to draw the hour lines before , and after six , was mentioned , chap. 2. sect. 1. sect. 2. to describe an erect , direct south dial. see fig. 10. draw abcd. a rect-angle parallellogram . then make ae = eb = cf = fd = cos . of your latitude . and ef = ac = bd = sine of your latitude . enter cf. in tang . 45. and lay down fk = fl = tang . 15. and fi = fm = tang . 30. again , enter ac . in tang . 45. and lay down ag = bo = 15. and ah = bn = tang . 30. with a ruler draw the lines eg . eh . ec . ef. ek . for the hour lines of 7. 8. 9. 10. 11. in the morning . and eo . en . ed. em . el. for 5. 4. 3. 2. 1. afternoon , the line aeb . is for six and six . the line ef. for twelve . the description of a direct north-dial differs nothing from this , only the hour lines from sun rise to six in the morning , and from six in the evening , until sun set , must be placed thereon , by drawing the respective morning and evening hours beyond the center as in the horizontal . see fig. 11. sect. 3. to describe an erect , direct east dial. see fig. 12. having drawn abcd. a rectangle paralellogram , fix upon any point in the lines ab . and cd . for the line of six , provided the distance from that point to a. being entred radius in the line of tangents , the distance from thence to b. may not exceed , nor much come short of the tangent 75. this point being found , enter the distance from thence to a. ( which we shall call 6. a. ) radius in the line of tangents , and keeping the sector at that gage , lay down upon the lines ab . and cd . 6. 11. = tang . 75. 6. 10 = tang . 60. and 6. 9 = 6. a. = tang . 45. and 6. 8. = 6. = 4. = tang . 30. lastly , 6. 7 = 6. 5 = tang . 15. draw lines from these points on ab . to the respective points on cd . and you have the hours . to place it on the plain , draw the angle dce . = co-latitude , and laying ed. on the horizontal line of the plain , prick off the hours . the same rules serve for delineation of a west dial , only as this hath morning , that must be marked with afternoon hours . sect. 4. to describe an erect declining dial , having a center . see fig. 14. draw the square bcde , and make ac = ak of quantity what you please . again , draw ag. 12. ♒ ce. and khf. ♒ ag. 12. by a line of chords , set off the angles of the substyle , style , and hour of six from twelve ; having first found these angles by chap. 2. sect. 4. paragr . 5. this done , make a mark in a 6. where it intersects kf as at h. then enter ak . in the secant of the plains declination , and keeping the sector at that gage , take out the secant of the latitude , which place from a to g. upon the the line a. 12. and again , from h. ( which is the intersection of the paralel fk . with the line of six ) unto f. this done , lay a ruler to the points f. and g. and draw a line until it intersects ce. as fg. 3. lastly , enter gf . in tang . 45. and set off gl = gn = tang . 15. and gm = go = tang . 30. again , enter hf. in tang . 45. and set off hr = tang . 15. and hp = tang . 30. a ruler laid to these points , and the center , you may draw the hours lines from six in the morning unto three afternoon . for the other hour lines , do thus , produce the line ec . and likewise ha. beyond the center , until they intersect each other as at s. then setting off st = hr . and s. 4. = hp . you have the hour points after three in the afternoon , until six , although none are proper beyond the hour line of four ; only by drawing them on the other side the center , they help you to the hour lines before six in the morning . sect. 5. to describe an erect declining plain without a center . see fig. 12. the delineation of these dials is the most difficult of any , and therefore i shall be the larger in their description . 1. by chap. 2. s. 4. paragr . 5. finde the angles of the style , substyle , and inclination of meridians . 2. having found the inclination of meridians , make the following table for the distance of every hour line and quarter from the substyle . where i take for example a north plain declining east 72. d. 45. min. in lat . 52. 30. min. imcl . mer . hou . quart . 76 = 30. 12 00 20 = 15 3 ... 16 = 30 4 00 12 = 45 4 . 09 = 00 4 .. 05 = 15 4 ... 01 = 30 5 00 02 = 15 5 . 06 = 00 5 .. 09 = 45 5 ... 13 = 30 6 00 17 = 15 6 . 21 = 00 6 .. 24 = 45 6 ... 28 = 30 7 00 32 = 15 7 . 36 = 00 7 .. 39 = 45 7 ... 43 = 30 8 00 47 = 15 8 . 51 = 00 8 .. 54 = 45 8 ... 58 = 30 9 00 62 = 15 9 . 66 = 00 9 .. 69 = 45 9 ... 73 = 30 10 00 the manner of drawing this table is thus . the inclination of meridians ( which because its a north decliner , is twelve at midnight . ) i finde 76. d. 30. min. now considering the sun never riseth till more than half an hour after three in this latitude , i know that one quarter before four is the first line proper for this plain : therefore reckoning 15. d. for an hour , or 3. d. 45. m. for a quarter of an hour , i finde three hours , three quarters ( the distance of a quarter before four from midnight ) to answer 56. d. 15. min. which being subtracted from 76. d. 30. min. the inclination of meridians there remains 20. d. 15. m. for the distance of one quarter before four from the substyle . again , from 20. d. 15. min. subtract 3. d. 45. min. ( the quantity of degrees for one quarter of an hour ) and there remains 16. d. 30. min. for the distance of the next line from the substyle , which is the hour of four in the morning . thus for every quarter of an hour continue subtracting 3. d. 45. min. until your residue or remain be less than 3. d. 45. min. and then first subtracting that residue out of 3. d. 45. min. this new residue gives the quantity of degrees for that line on the other side the substyle . now when you are passed to the other side of the substyle , continue adding 3. d. 45. min. to this last remain for every quarter of an hour , and so make up the table for what hours are proper to the plain . 3. draw the square abcd. of what quantity the plain will admit , and make the angle cag equal to the angle of the substyle with twelve . again , cross the line ag. in any two convenient points , as e. and f. at right angles by the lines kl . and cm . 4. take the distance from the center unto 45. the radius to the lesser lines of tangents , which is continued to 76. on the sector side , enter this distance in 45. on the larger lines of tangents , and keeping the sector at that gage , take out the tang . 20. d. 15. min. ( which is the distance of the first line from the substyle ) set this from 73. d. 30. min. ( the distance of your last hour line from the substyle , as you see by the table ) toward the end upon the lesser line of tangents , and where it toucheth as here at 75. 05. call that the gage tangent . 5. enter the whole line kl . in the gage tangent , which in this example is 75. d. 05. min. and keeping the sector at that gage , take out the tangent 73. 30. min. which is your last hour , and set from l. on the line kl . unto v. again , take out the tangent 20. d. 15. min. which is your first line , and set it from k. towards v. and if it meet in v. it proves the truth of your work , and a line drawn through v. paralel unto ag. is the true substyle line . then keeping the sector at its former gage , set off the tangents of the hours , and quarters ( as you finde them in the table ) from v. towards k. and from v. towards l. making points for them in the line kl . lastly , enter the radius of your tangents to these hour points in the radius of secants , and set off the secant of the styles height from v. to t. thus have you the hour points and style on one line of contingence . to mark them out upon the other line do thus . set the radius to the hour points upon the former line of contingence , from h. to p. on the line chm. and entring hv. as radius in the line of tangents , take out the tangent of the styles height , and set from p. to r. again , enter hr. radius in your line of tangents , and keeping the sector at that gage , take out the tangents for each hour , and quarter , according to the table , and lay them down from h. to the proper side of the substyle toward c. or m. and applying a ruler to the respective points on kl . and cm . draw the lines for the hours and quarters . lastly , enter hr. radius on the lines of secants , and taking out the secant of the styles height , set it from h. to s. and draw the line st . for the style . sect. 6. to describe a direct polar dial. see fig. 15. draw bcde . a rectangle paralellogram , from the middle of bc. to the middle of de. draw the line 12. or substyle , appoint what place you please in bc. or cd . for the hour point of 7. in the morning , and 5. afternoon . then , enter 12. 7 = 12. 5. in the tangent 75. and set off 12. 1. = 12. 11. = tang . 15. and 12. 2. = 12. 10 = tang . 30. and 12. 3. = 12. 9. = tang . 45. lastly , 12. 4. = 12. 8. = tang . 60. from these points draw the hour lines of 7. 8. 9. 10. 11. 12. 1. 2. 3. 4. 5. which are all the hours proper for these plains . sect. 7. to draw a declining polar . see fig. 16. 1. by chap. 2. sect. 6. par. 4. finde the inclination of meridians , and distance of the substyle from the horizontal . 2. by chap. 3. sect. 5. par. 2. make a table for the distance of the hour points from the substyle . 3. draw the square bcde . set off the angle cag . for the substyle , and cross that substyle line at right angles in any two convenient places , as at h. and k. with the lines phs. and rkt. for contingent lines . 4. take any convenient length for your styles height , and enter it radius in your line of tangents , keeping the sector at that gage , prick off the hours from the substyle ( by your table ) upon both the contingent lines . draw lines by the points in both contingents , and you have the hours : for all other declining reclining inclining plains , it would be needless ( i presume ) to insist upon the description of them : sith so much hath been already mentioned , chap. 2. s. 6. that there can scarcely be any mistake , unless through meer wilfulness , or grandnegligence . chap. iv. to determine what hour lines are proper for any plain superficles . by projection of the sphere . see fig. 17. draw the circle nesw . representing the horizon , and crossing it into quadrants n. is north. s. south , e. east , w. west , ns . the meridian ( which let be infinitely produced ) z. the center represents the zenith . to finde the pole set half the co-tangent of the latitude from z. toward n. it gives the point p. for the pole or the point through which all the hour lines must pass . the suns declination in cancer subtracted from the latitude , and the tangent of half the remain set from z. to π. gives the intersection of cancer with the meridian . again , adde the complement of the suns delineation in cancer , unto the complement of your latitude , and the tangent of half that sum set from z. to ψ , gives the diameter of that tropick , half ψ , is the radius to describe it . half the tangent of your latitude set from z. to q. gives that point for the intersection of the equator with the meridian ; and the co-secant of the latitude set from oe . toward n. gives the point ζ. the center of the equator . adde the suns greatest declination ( or his declination in capricorn ) to the latitude , and the tangent of half that sum set from z. toward s. gives the point φ , where capricorn intersects the meridian . subtract the suns declination in capricorn from your latitude , and that remain subtract from 180. the tangent of half this last remain , set from z. toward n. gives the point t. the diameter of capricorn , and half the distance t φ. is the radius to describe it . set the secant of the latitude from p. towards s. it gives the point h. the center of the hour line of six cross the line zsh. at right angles in the point h. then entring ph. radius on the lines of tangents , set off the hour centers both wayes from h. reckoning 15. d. for an hour . lastly , setting one point of your compasses in these center points , extend the other to p. and with that radius describe the hour lines . thus have you the sphere projected , the following sections will determine the hours for all plains . sect. 1. to determine the hour lines for erect direct plains . fig. 17. the line ns . represents an erect east , and west plain . that side next w. is west , the other side next e. is east , where you may see that the sun shines upon the east until twelve , or noon , and at that time comes upon the west . the fine we . represents a direct north and south plain , the side next n. is north , the other next s. is south , where the north cuts the tropick of cancer ( which in the hour lines you finde 'twixt 7. and 8. in the morning ; and again 'twixt 4. and 5. afternoon ) is the time of the suns going off , and coming on that plain . where the south cuts the equator , which is in the points of six , and six is the time of the suns going off , and coming on that plain . sect. 2. to determin the hour lines for direct reclining inclining plains . fig. 17. nbs . on the convex side is a west incliner , where it cuts capricorn , is the time of the suns coming on that plain , afternoon . on the concave side its an east recliner , where it cuts cancer , is the time of the suns going off that plain , afternoon . ncs. on the convex side is an east incliner , where it cus capricorn , is the time of the suns going off in the morning . on the concave side it is a west recliner , where it cuts cancer , is the time of the suns coming on in the morning . wde. on the convex side is a south incliner , where until d. reach below oe . it hath all hours from six to six , and until d. reacheth below φ. it may have the twelve a clock line . but when d. reacheth below oe . draw a paralel of declination to pass through the point d. and the intersection of that paralel with the limb of the circle ne sw . doth among the hour lines , shew the time of the suns coming upon that plain in the morning , and going off again afternoon , when d. reacheth below φ. the intersection of the ark wde. with the tropick of capricorn , shews the time of the suns going off that plain before noon , and coming on again , afternoon . and the intersection of the tropick of capricorn with the limb shews the first hour in the morning the sun comes on , and the last hour afternoon , that it staves upon that plain . the convex side of wde. is a north recliner , where it cuts cancer , is the time of the suns going off in the morning , and coming on again afternoon . wfe. on the convex side is a north incliner , where it cuts cancer , is the time of the suns going off in the morning , and coming on afternoon . on the concave side is a south recliner , where until f. reach beyond p. it enjoyes the sun only from six to six . when f. reacheth beyond p. where the ark cuts cancer , you finde how much before six in the morning the sun comes on , or after six at evening it goes off . to draw any of these arks , ex. gr . the ark nbs . do thus . set the tangent of half the reclination inclination from z. on the line zw . and it gives the point b. produce ze. and set the co-secant of the reclination inclination from b. towards e. which reacheth to g. then g. is the center . & gb . the radius to draw that ark . note . the semidiameter of the circle senw . is radius to all the tangents , and secants , which you make use of for placing any oblique plain upon the scheme . sect. 3. to determin the hour lines for erect declining plains . fig. 18. for south-east , or north-west plains . by a line of chords set the angle of declination on the limb from w. toward s. as h. lay a ruler to hz . and draw hzk. which on that side next nw . represents the north-west , and where the line cuts cancer , you have the time of the suns coming on afternoon , and staying until sun set . but if it cut cancer twice , then in the morning hours it shews what time the sun goes off this plain , having all hours from sun rise to that time , and in the evening hours you have the time of the suns coming on again , and staying till sun set . that side hzk. next se. represents a south-east . where the line cuts the equator in the evening hours , is the time of the suns going off , where it cuts cancer , in the morning hours , is the time of the suns coming on that plain . for north-east , or south-west , set the declination by a line of chords from e. towards s. as l. lay a ruler to zl . and draw the line lzr. which on that side next ne. is north-east , where it cuts cancer , is the time of the suns going off in the morning . if it cuts cancer twice , you have in the evening hours the time of the suns coming on again , and staying until sun-set . on that side next sw . is the south-west . where it cuts the equator , or capricorn in the morning hours , is the time of the suns coming on , where it cuts cancer in the evening hours , is the time of the suns going off . sect. 4. to determin the hours of declining reclining plains inclining plains . fig. 18. first , set in the plain according to its declination . by sect. 3. ex. gr . lzr. a north-east , or south-west declining 50. d. 00. min. this done cross the line lzr. representing the declination of the plain , at right angles in the point z. as czbg . then for north-east incliners , or south-west recliners , set half the tangent of the reclination inclination from z. toward c. 〈◊〉 t. and set the co-secant of the reclination inclination from t. toward b. as tg . then g. is the center , and gt . the radius to describe the ark rtl. whose convex side represents a north-east incliner , where it cuts libra or capricorn , is the time of the suns going off in the morning ; if it cuts cancer twice , the intersection of cancer with the evening hours shews what time the sun comes again upon such a plain afternoon , and continues till sun setting . the concave side is a south-west recliner , where it cuts cancer in the morning hours , is the time of the suns coming on , in case it intersects cancer twice in the evening hours , you have the time that the sun goes off . for a north-east recliner , or south-west incliner , set the point t. from z. toward b. and the point g. from t. ( so placed ) toward c. and draw the ark on that side rzl. toward b. whose convex side will represent a south-west incliner , and where the ark cuts the equator or capricorn , you have the time of the suns coming on that plain . the concave side is a north-east recliner , where the ark cuts cancer , is the time for the suns going off that plain . when the ark cuts cancer twice , the sun comes on again before it sets . for a north-west recliner , or south-east incliner . enter the declination by sect. 3. as hzk. cross it in the point z. at right angles , as ozd. set half the tangent of the reclination inclination from z. toward o. as v. and the co-secant of the reclination inclination from v. toward d. as f. then is f. the center , and fv. the radius to draw the ark hvk. where it cuts cancer the hour lines , tell you the time of the suns going off in the morning , and entring again afternoon , upon the north-west recliner . where it cuts the equator you have the time of the suns going off the south-east , where it cuts cancer in the morning hours is the time of the suns coming on that plain . for a north-west incliner , or south-east recliner , set the point v. from z. toward d. and the point f. set from v. ( so placed ) toward o. and draw the ark on that side z. next d. then where the convex side cuts cancer , you have the time of the suns going off in the morning ; and coming on again afternoon upon the north-west incliner . where the concave side cuts the equator , you have the time of the suns going off the south-east recliner ; where it intersects cancer , is the time of his coming on that plain in the morning . note . all the precedent rules about plains are appropriated to us that live in northern hemisphere , in case any one would apply them to the south hemisphere : what is here called north , there name south , and what we here term south , there call north , and the rules are the same . — si quid novisti plenius istis , promptius istis , rectius istis , candidus imperti : sinon , his u●ere mecum . finis . geometrical dyalling, or, dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned mr. samuel foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by john collins ... collins, john, 1625-1683. this text is an enriched version of the tcp digital transcription a33999 of text r17003 in the english short title catalog (wing c5373). textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. the text has been tokenized and linguistically annotated with morphadorner. the annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). textual changes aim at restoring the text the author or stationer meant to publish. this text has not been fully proofread approx. 180 kb of xml-encoded text transcribed from 64 1-bit group-iv tiff page images. earlyprint project evanston,il, notre dame, in, st. louis, mo 2017 a33999 wing c5373 estc r17003 12394868 ocm 12394868 61099 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a33999) transcribed from: (early english books online ; image set 61099) images scanned from microfilm: (early english books, 1641-1700 ; 958:23) geometrical dyalling, or, dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned mr. samuel foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by john collins ... collins, john, 1625-1683. [6], 82, (i.e. 80) p. : ill., plates. printed by thomas johnson for francis cossines and are to be sold at his shop ... : also to be sold by henry sutton ..., london : 1659. reproduction of original in the university of illinois (urbana-champaign campus). library. errata: p. 82. eng foster, samuel, d. 1652. dialing -early works to 1800. a33999 r17003 (wing c5373). civilwar no geometrical dyalling: or, dyalling performed by a line of chords onely, or by the plain scale. wherein is contained two several methods of i collins, john 1659 31860 3 0 0 0 0 0 1 b the rate of 1 defects per 10,000 words puts this text in the b category of texts with fewer than 10 defects per 10,000 words. 2005-07 tcp assigned for keying and markup 2005-08 aptara keyed and coded from proquest page images 2005-09 andrew kuster sampled and proofread 2005-09 andrew kuster text and markup reviewed and edited 2005-10 pfs batch review (qc) and xml conversion geometricall dyalling or , dyalling by a line of chords onely by iohn collins accomptant philomath london printed for francis cossinet in tower street and for h : sutton behind exchang generall scheme projection of ye sphere anno 1659 in a circle in a parralellogram chords fitted to ye schemes in ye booke ruler chords in time fitted to the booke ruler geometricall dyalling : or , dyalling performed by a line of chords onely , or by the plain scale . wherein is contained two several methods of inscribing the hour-lines in all plains , with the substile , stile and meridian , in their proper coasts and quantities ; being a full explication and demonstration of divers difficulties in the works of learned mr. samuel foster deceased , late professor of astronomy in gresham colledge , ; also a collection of divers things from the works of clavius and others . whereto is added four new methods of calculation , for finding the requisites in all leaning plains , with full directions suited to each method for placing them in their proper coasts , without the help of any dclinations . also how by projecting the sphere , to measure off all the arks found by calculation , and to determine what hours are proper to all kinde of plains , omitting superfluity . lastly , the making of dyals from three shadows of a gnomon placed in a wall at random , with a method of calculation suited thereto , and divers ways from three shadows , to finde a meridian-line . written by john collins of london accomptant , philomath . london , printed by thomas johnson for francis cossinet , and are to be sold at his shop at the anchor and mariner in tower-street , at the end of mincing-lane , with other mathematical books ; also to be sold by henry sutton mathematical instrument-maker , living in threed-needle street behinde the exchange . 1659. to the reader . being in conference with my loving friend mr. thomas rice , one of the gunners of the tower , much exercised in making of dyals in many eminent places of the city , he was pleased to communicate unto me the knowledge of a general scheme , for inscribing the requisites in all oblique leaning plains , which he added , was the useful invention of the learned master samuel foster , professor of astronomy in gresham colledge london , deceased , from whom he received instructions concerning the same , in the year 1640. and the said mr. rice not having his papers about him , did dictate to me from his memory the construction and practice of the said scheme , which i afterwards methodized just as it is delivered in page 25 and 26. at the same time also i received directions for inscribing the substile and stile in upright decliners , and east or west reclining , or inclining plains , but they were of another mans invention , and did not seem to be derived from the former general scheme , and therefore i have not used them in this treatise , but derived the performance thereof from the said general scheme : moreover , mr. rice added , that in regard of the death of the author , and since of his executor , who had the care and inspection of his papers , i should do well to study out the demonstration of the former scheme , and make it publick , the rather , because it hath been neglected . the manner of inscribing the hour-lines being already published in a treatise of the authors , intituled , posthumi posteri ; this desire of his , which was also furthered by mr. sutton and others , i am confident is fully effected in the following treatise , and much more then was in his request , and here let me adde , that i have had no other light from the endeavors of the learned author , then what was as above communicated unto me , all which might be expressed in half a page of paper , or little more , to which i have made this large access and collection , nam facile est inventis adore ; not that i would hereby any thing endeavor to ecclipse the authors work , the excellency of whose inventions in this and other kindes , will speak forth his renown to all posterity . and though in probability i have not performed so much , nor so well , as was obvious to the knowledge of the learned author , yet i am confident when the reader understands what is written , he would be as loath to be without the knowledge thereof , as my self ; and i am induced to believe , that the author left nothing written about many particulars in this treatise ; throughout which , we suppose the reader , furnished with the common rudiments of geometry , that he can raise perpendiculars , draw parallels , describe a parralellogram , bring three points into a circle , understands definitions and tearms of art , knows what a line of chords is , can prick off an arch thereby ; all which , with delineations for all the usual cases of triangles , both from projection and proportions , the reader will meet with in my treatise , called , the mariners plain scale new plain'd , now in the press . vale & fruere . i remain thy friend , and a well-willer to the publique advancement of knowledge , john collins . the contents : dyals distinguished page 1 , 2 to take the suns altitude without instrument p. 3 to finde the reclination of a plain p. 4 also the declination thereof p. 5 a general proportion and scheam for finding the suns azimuth or true coast p. 6 to draw a horizontal dyal p. 7 also a south dyal p. 8 a new way to divide a tangent liue into five hours and their quarters p. 10 , 11 a direct south polar dyal p. 12 to prick off the requisites of upright decliners p. 13 to prick off an arch or angle by sines or tangents p. 14 the scheam for placing the requisites of upright decliners demonstrated p. 15 , 16 to inscribe the hour-lines in an upright decliner p. 17 the demonstration thereof p. 18 to 21 an east dyal p. 22 requisites placed in east or west leaning plains p. 23 the demonstration thereof p. 25 the construction of the general scheam for placing the requisites in declining re-inclining plains p. 25 to 27 the first method of calculation for oblique plains p. 28 to 30 and directions for the true placing the requisites suited thereto p. 30 , 31 the general scheam demonstrated p. 32 to 34 the hour-lines inscribed in an oblique plain ibid. the general scheam fitted for latitudes under forty five degrees p. 35 to draw the hour-lines in a declining polar plain p. 36 also how to delineate the hour-lines in plains having small height of stile p. 37 , 38 another way to perform the same p. 39 to 41 a second method of calculation for oblique plains p. 42 proportions for upright decliners p. 43 a third method of calculation for oblique plains p. 44 directions for placing the requisites suited thereto p. 46 a fourth method of calculation for oblique plains p. 47 through any two points assigned within a circle to draw an arch of a circle that shall divide the primitive circle into two semicircles . p. 49 to measure the arks of upright decliners by projection p. 50 also the arks of leaning east or west plains thereby p. 51 to project the sphere for oblique plains — to measure off all the arks that can be found by calculation with the demonstration of all the former proportions — from p. 52 to 59 to determine what hours are proper to all plains p. 60 to 61 another method of inscribing the hour-lines in all plains by a parallelogram p. 62 to draw the tangent scheme suited thereto p. 63. the hour-lines so inscribed in a horizontal and south dyal p. 63 , 64 as also in an upright decliner p. 65 with another tangent scheme suited thereto for pricking them down without the use of compasses p. 66 , 67 a general method without proportional work , for fitting the parallelogram into oblique plains that have the requisites first placed p. 69 , 70 by help of three shadows to finde a meridian-line p. 70 , 71 another scheme suited to that purpose p. 73 a method of calculation for finding the azimuth , latitude , amplitude , &c. by three shadows p. 75 from three shadows to inscribe the requisites and hour-lines in any plain p. 77 which is to be performed by calculation also p. 97 characters used in this book . + plus , for more , or addition . minus less , or substraction . = equal . q for square . □ square . ▭ the rectangle or product of two terms . ∷ for proportion . if the brass prints in this book be thought troublesome to binde up , they may be placed at the end thereof , for the pages to which they relate , are graved upon them . a quadrant of a circle being divided into 90d , all that is required in this treatise , is to prick off any number of the said degrees with their sub-divisions , which may be easily done anywhere from a quadrant drawn and divided into nine equal parts , and one of those parts into ten sub-divisions , called degrees , but for readiness the equal divisions of a whole . quadrant are transferred into a right line ( as in the frontispiece ) called a line of chords , serving more expeditly to prick off any number of degrees or minutes in the arch of a circle . an advertisement . the reader may possibly desire to be furnished with such scales to any radius , these and all manner of other mathematical instruments either for sea or land , are exactly made in brass or wood , by heury sutton in threed-needle-street , behinde the exchange , or by william sutton in upper shadwell , a little beyond the church , mathematical instrument-makers . a direct south diall reclining 60d lat 51d 32′ an erect direct north diall lat 51d 32′ a south diall declin 40d east reclining 60d lat 51d 32′ a north diall declin 40d west reclining 75d latitude 52d 32′ the distinction of dyals . though much of the subject of dyalling hath been wrote already , by divers in diverse languages , notwithstanding the reader will meet with that in this following treatise , which will abundantly satisfie his expectation , as to the particulars in the preface , not yet divulged in any treatise of this nature . in this treatise , that we might not be too large , divers definitions are passed over , supposing that the reader understands what a dyal is , what hour lines are , that that part of the stile that shews the hour , ought to be in the axis of the world , that the hour lines being projected on any regular flat , will become straight lines . dyals are by clavius in the second chapter of his gnomonicks distinguished into seventeen kindes . 1. the horizontal , being parallel to the horizon . 2. south and north direct , by some called a vertical dyal , because parallel to the prime vertical or circle of east and west . 3. 4. south direct , reclining or north inclining less more then the pole . 5. 6. south direct , inclining or north reclining less more then the equinoctial 7. south direct reclining , or north inclining to the pole , called a direct polar plain , because parallel thereto 8. south direct inclining , or north reclining to the equator , called an equinoctial plain , because parallel thereto . 9. south and north , declining east or west . 10. east and west direct . 11. east and west direct , reclining or inclining . 12. a south north plain declining east or west reclining inclining to the pole , called a polar decliner . 13. a south north plain declining east or west inclining reclining to the equinoctial , called an equinoctial decliner . 14. 15. a south north plain declining east or west reclining inclining whereof are two sorts , the one passing above , the other beneath the pole 16. 17. a. south north plain declining east or west inclining reclining whereof are 2. sorts , the one passing above the equinoctial , the other beneath it . of each of these we shall say something , but before we proceed , it will be necessary in the first place to shew how to finde the scituation of any plain . 1. to finde whether a plain be level or horizontal . the performance hereof is already shewed by mr. stirrup , in his compleat dyallist , page 58. where he hath a scheme to this purpose , i shall onely mention it : get a smooth board , let it have one right edge , and near to that edge , let a hole be cut in it for a plummet to play in ; draw a line cross the board perpendicular , to the streight edge thereof passing through the former hole ; if then setting the board on its smooth edge , and holding it perpendicularly , so that the plummet may play in the hole , if which way soever the board be turned , the threed will fall ( being held upright with the plummet at the end of it , playing in the former hole ) directly on the perpendicular line drawn cross the board , the said line no wayes reclining from it , the plain is horizontal , otherwise not . 3. to take the altitude of the sun without instrument . upon any smooth board , draw two lines at right angles , as ab and ac , and upon a as a center with 60d of a line of chords describe the arch bc , and into the center thereof , at a drive in a pin or steel needle , upon which hang a threed and plummet , then when you would observe an altitude , hold the board so to the sun , that the shaddow of the pin or needle may fall on the line ac , and where the threed intersects the arch bc , set a mark , suppose at e then measure the ark be on your line of chords , and it shews the altitude sought , for want of a line of chords , you may first divide the arch bc into 9 parts , and the first of those divisions into 10 smaller parts . 3. to draw a horizontal and vertical line upon a plain . the readyest and most certainest way to do this , especially if the plain lean downwards from the zenith , will be by help of a threed and plummet held steadily , to make two pricks at a competent distance in the shadow of the threed on the plain , projected by the eye , and a line drawn through those , or parallel to those two points , shall be the plains perpendicular or vertical line , and a line drawn perpendicular to the said line , shall be the plains horizontal line . to finde the reclination of a plain . the reclination of a plain , is the angle comprehended between the plains perpendicular , and the axis of the horizon , and the same definition may serve for the inclination , onely the upper face of a plain , leaning from the zenith , is said to recline , and the under face to incline . in this sense mr. oughtred in his circles of proportion , mr. wells in his dyalling , mr. newton in his institution , and mr. foster in his late writings , understand it , and so it is to be taken throughout this treatise ; but here it will not be amiss to intimate that mr. gunter , mr. wingate , and mr. foster in a treatise of a quadrant , published in anno 1638. account the inclination from the horizon ; the complement of the angle here defined , and accordingly have suited their proportions thereto , in which sense , the reclining side is said to be the upper face of an incliner ; but for the future it will be inconvenient to take it any more in that acception . now to finde the reclination of a plain , apply the streight edge of the board we used for trying of horizontal plains , to the plains perpendicular , holding the threed and plummet so in the grooved hole , that it may intersect the line , parallel to the streight edge of the board , and make two pricks upon the board where the threed passeth , and it will make an angle with the said line , equal to the plains reclination , to be measured with chords , according as was directed for taking of altitudes ; and if need require , a line may be drawn parallel to that drawn on the board , which represented the threed , which will make the same angle with the plains perpendicular , the former line did . the inclination will be easily got , by applying the streight edge of the board , to the plains perpendicular , and holding the threed and plummet so at liberty , as that it may cross the line parallel to the boards streight edge , and it will make an angle therewith , equal to the plains inclination . of declining plains . a plain hath its denominations from the scituation of its poles : by the poles of a plain , is meant a line imagined to pass through the plain at right angles thereto , the extremities of which line on both sides the plain , are called its poles . if a plain look full south , without swerving , either to the east or west ; it is said to be a direct south plain , if it swerve towards east or west , it is said to decline thereto ; if it stand upright without leaning , it is said to be erect now a south plain that declines eastward , the opposite face thereto , is said to be a north plain declining as much westward , by which means the declination of a plain will never exceed 90d , and to which sense the following directions toroughout this treatise are suited . now we may define the declination of a plain to be the arch of the horizon , contained between the true points of east or west , and the plain , equal whereto is the arch of the horizon between the true north or south , and that vertical circle or azimuth that passeth through the plains poles . to finde the declination of a plain . upon a board , having one streight smooth edge , draw a line parallel thereto , and thereon describe a semicircle , with the radius of the chords , which distinguish into two quadrants by a perpendicular from the center ; then holding the board parallel to the horizon , and applying the streight edge to the wall , hold up a threed and plummet , so that the shadow thereof may pass through the center , and in the shadow , make a mark near the limb , from which a line drawn into the center shal represent the shadow of the thread . in this scheam , let ☉ c represent the shadow of the threed passing thorow the center ; at the same instant , take the suns altitude , and by the following directions , finde the suns true azimuth . admit the sun be 75d to the eastwards of the south , prick it from ☉ to s , then the distance between p and s shews the quantity of the plains declination , and the coast is also shewn , for if p fall to the eastwards of s , the declination is eastwards , if to the westwards of it , it is westward ; in this example it is 30d eastwards . we may suppose that the reader knoweth which way from the line of shadow to set of the suns true coast , that he may easily do by observing the coast of his rising or setting , or whether his altitude increase or decrease , and needs no directions . a general proportion for finding the azimuth . as the cosine of the altitude , is to the secant of the latitude , or as the cosine of the latitude : is to the secant of the altitude ; so is the difference of the versed sines of the suns distance from the elevated pole , and of the ark of difference between the latitude and altitude . to the versed sine of the azimuth from the north in this hemisphere . declination 23d 31′ , north . latitude 51d 32′ altitude 41d : 34′ azimuth 105d from the north upon c as a center , describe the semicircle ebm , and draw ecm the diameter , and cb from the center perpendicular thereto , from b set off the declination to d , the latitude to l , the altitude to a ; and draw the line ca continued . the nearest distance from d to bc when the declination is north , place from c towards m when south towards e , and thereto set k , assuming the diameter em to represent the secant of the latitude , the radius of the said secant shall be the cosine of the latitude doubled ; wherefore the nearest distance from l to cm doubled , shall be the radius to the said secant , which prick on the line of altitude from c to z , then the nearest distance from z to cm shall be the cosine of the altitude to that radius , which extent prick from e to f , and draw the lines ef and fm ; then for the third term of the proportion , take the distance between a and l , and prick it from m to g , from which point take the nearest distance to bc , and place it from c to x , so is the distance kx , the difference of the versed sines sought , which prick from f to q , and draw qr parallel to fm , so is rm the versed sine of the suns azimuth from the north sought , and cr the sine of it to the southwards of the east or west , and bp the measure thereof in the limbe , found by drawing a line from r parallel to gb ; in this example the suns true goast is 15d to the southwards of the east or west . note also , that the point r may be found without drawing the lines ef , qr by entring the extent kx , so that one foot resting on the diameter , the other turned about , may but just touch fm . to make a horizontal dyal . a plain is then said to be horizontal , when it is parallel to the horizontal circle of the sphere proper to the place of habitation ; these are such kinde of dyals as stand commonly upon posts in a garden . by the former directions , a true meridian line may be found , let it here be represented by the line fm . a horizontal dyal for the latitude of london , 51 degrees , 32 minutes . cross the same with a perpendicular , to wit , the line of six , and draw an oblong or square figure as here is done , wherein to write the hours , from m to c set off such a radius as you intend to draw a circle withall , which for convenience may be as big as the plain will admit , and therewith upon c as a center describe a circle , and set off the radius of the said circle from f to d , from m to k , and draw the right line fd , then prick the poles height or latitude of the place from f to l , and from l to s , and through the point s draw the line ms , and it shall represent the cock or stiles height from the point l , take the nearest distance to fm , and prick that extent from f to n on the line fd , then a ruler laid from n to k , where it intersects the meridian is the regulating point , or point ☉ , then from m divide the circle into twelve equal parts for the whole hours , setting the letters of the alphabet thereto , and lay a ruler from the regulating point to each of those divisions , and it will intersect the circle on the opposite side , from which intersections , lines drawn into the center at m , shall be the hours lines required , to be produced beyond the center , as many as are needful , which shall be the hours before or after 6 in the summer half year , the halfs and quarters are after the same manner to be inscribed , by dividing each equal division of the circle into halfs and quarters . this circular dyalling , was in effect , published and invented by mr. foster in his book of a quadrant , in anno 1638. for the demonstration of this work will demonstrate the truth of those circular performances , which he operates on the back of that quadrant , but is more expresly hinted in his posthuma , the demonstration whereof shall follow . those that have plain tables with a frame , may have tangent lines put on the sides of their frame ; and then if a center be found upon the paper under the frame by the intersection of a ruler laid over those tangents , the requisite divisions of a circle to any radius that can be described upon the paper , will be most readily given without dividing any circle or setting of marks or letters thereto , the frame keeping fast that paper , on which the draught of the dyal is made , which may also be supplyed from a circle divided on pastboard cut out , which is to be laced upon a board over the paper whereon the draught is to be made . an upright full south dyal . this is no other then a horizontal dyal , in that latitude which is equal to the complement of the latitude of the place , you are in , onely the hours must not be continued beyond the center , and the delineation requires no other directions then the former . on a direct north dyal , in this latitude , there will not be above two hours above , and two hours beneath each end of the horizontal line to be expressed , and the stile will have the same elevation and point upward . horizontall diall an vpright south diall in all upright north and south plains , the meridian or hour line of 12 is perpendicular to the plains horizontal line , if the plain be direct the height of the stile above the substile , is equal to the complement of the latitude . if the plain be south reclining , or north inclining , the height of the stile above the substile , is equal to the ark of difference between the complement of the latitude and the ark of re-inclination , and if this latter ark be greater then the former , the contrary pole is elevated . if the plain be south inclining , or north reclining , the height of the stile above the substile , is equal to the sum of the colatitude , and of the re-inclination , and when this latter ark is greater then the latitude , the stiles height will be greater then a quadrant . such horizontal or direct south dyals , both upright and leaning , whereon the stile hath but small elevation , are to be drawn with a double tangent line without a center , wherein the following directions for direct polar plains , and those for other oblique plains , whereon the stile hath but small elevation , will fully direct you . all plains that cut the axis of the world , have a center ; but if they be parallel thereto , the stile hath no elevation . such direct south reclining , or north inclining plains , whose arch of reclination is equal to the complement of the latitude , are parallel to the axis , and are called polar plains ; in these the hour lines will be tangent lines of any assumed radius , and the parallel height of the stile above the plain , must be made equal to the radius of the tangent , by which the hours were set off . first draw the tangent line cg , and perpendicular thereto acb , upon c as a center , with any radius describe the quadrant of a circle hb , and prick off the radius from c to a , from h to e , from b to d , from d to f , from f twice to g. a ruler laid from a to e findes the point 1 on the tangent cg for the hour line of one , and laid from a to d findes the point 2 for the hour line of two , the hour line of three is at the point h , of four at the point f , of five at the point g ; thus are the whole hours easily inserted . now to insert the halfs and quarters . divide the arches be , ed , dh into halfs , and a ruler laid from a to those divisions wil finde points upon ch , where all the half hours under 45d are to be graduated ; and if a ruler be laid from b to those respective divisions of the quadrant bh , it will finde points on the tangent cg , where the whole hours and halfs are to be graduated above 45d , after the same manner are the quarters to be inserted ; but in regard the halfs and quarters above 45d will by this direction be found with much uncertainty , i have added the help following . first , divide the halfs and quarters under 45d as now directed , then for those above make use of this table . in a direct south polar dyal . the distances of these hour lines . are equal to the distances of these hour lines . 3 and 3 ¼ 3 ½ and 4 2 ¼ and 3 ¾ 4 : and 4 ¼ 4 ¼ and 4 ½ 4 ¼ and 4 ¼ ¾ and 1 ¼ 1 ¼ and 2 ½ 1 ½ and doubled ¾ and 1 ¾ 3 ½ and 3 ¾ doubled : otherwise , 1 ¾ and 2 ¾ 2 and 3 ¾ this i do not assert to be absolutely true , but so neer the truth , that there will not arive above one thousandth part of the radius difference in the greatest dyal that is made , and will be more certain then any sector , though of a vast radius , or then they can with convenience be prickt down the common way , by a contingent line , the meaning of the table will be illustrated by one or two examples . the distance between the hour lines of two and a quarter , and three and three quarters will be equal to twice the distance between the hour lines of twelve , and one and a half . also the distance between four and a quarter , and four and a half , is equal to twice the distance between three and a half , and three and three quarters ; or it is equal to the distance between one and three quarters , and two and three quarters , but the former is nearer the truth . it will be inconvenient in such plains , as also in direct east or west dyals , to express any hour line from the substile beyond 75d or 5 hours . to these i may add some other observations which were communicated by doctor richard sterne , to mr. sutton , which may be of use to try the truth of these kinde of plains . 1. the distance between the hours of 3 and 4 , is equal to the distance between the hours of 1 and 3. 2. the distance between the hours of 2 and 4 , is double to the distance between the hours of 12 and 2. 3. the distance between the hours of 11 and 4 , is doubte to the distance between 12 and 3 , or to that between 9 and 12 , and so equal to the distance between 9 and 3 , also equal to the distance between 4 and 5. 4. the distance between the hours of 11 and 5 , is double to the distance between the hours of 11 and 4 , as also to the distance between the hours of 4 and 5 , and between 9 and 3 ; and quadruple to the distance between 12 and 9 , or between 12 and 3. these are absolutely true , as may be found by comparing the differences of the respective tangents from the natural tables . to draw a polar direct south dyal . having drawn the plains perpendicular in the middle of the plain , let that be the hour line of 12 , then assuming the stile to be of any convenient parallel height , that will suit the plain , making that radius , divide a tangent line into hours and quarters by the former directions , and prick them down on the plain , upon a line drawn perpendicular to the meridian or hour line of 12 on each side thereof , and through the points so prickt off , draw lines parallel to the meridian line , and they shall be the hour lines required , as in the example . to describe an equinoctial dyal . such direct north recliners , or south incliners , whose re-inclination is equal to the latitude , are parallel to the equinoctial circle , and are therefore called equinoctial dyals , there is no difficulty in describing of these : divide a circle into 24 equal parts for the whole hours , & afterwards into halfs & quarters , and place the meridian line in the plains perpendicular , assuming as many of the former hours as the sun can shine upon for either face , and then placing a round wyre in the center for the stile , perpendicular to the plain , and the dyal is finished . a south plaine declining 30d east latitude 51-32′ page 13 page 17 a south diall declin 30d east latitude 51-32′ to prick off the substile and stiles height on upright decliners in their true coast and quantity . on such plains draw a horizontal line , and cross the same with a perpendicular or vertical , at the intersection set v at the upper end of the vertical line set s , at the lower end n , at the east end of the horizontal line set e , at the west end w , prick off the declination of the plain in its proper coast from s or n to d , and draw dv through the center , the same way count off the latitude to l , and from it draw a line into the center ; in the same quarter make a geometrical square of any proportion at pleasure , so that two sides thereof may be parallel to the horizontal and vertical line , at the intersection of one of the sides thereof , with the vertical set a , and of the other side , with the horizontal line , set b , and where the latitude line intersects the side of the square , set f. to prick off the substilar . for latitudes above 45d take bf , and prick it on the line of declination beyond the center from v to o , and from the point o , draw the line oc parallel to the vertical line , and produced beyond o , if need require , and thereon from c to i prick the side of the square , and a line drawn into the center , shall be the substilar line ; but for latitudes under 45d prick the side of the square from v to o , and draw oc as before , and make ci equal to af , and a line from i drawn into the center , shall be the substilar line . stiles height . from the point i erect the extent oc perpendicularly to the substilar oi , at the extremity thereof set k , from whence draw a line into the center , and the angle ivk will shew how much the stile is to be elevated above the substilar line . in order to the demonstration hereof , let it be observed that an angle may be prickt off by sines or tangents in stead of chords . to prick off an angle by sines or tangents . as in the scheme annexed , let bc be radius , and let there be an arch prickt off with chords , as be ; i say , if the tangent of the said ark ba be taken out of a line of tangents to the same radius , and be erected perpendicular to the end of the radius , as ba , a line drawn from a into the center , shall include the same angle as was prickt off by chords , as is evident from the definition of a tangent . in like manner an angle may be prickt off by sines , the nearest distance from e to bc is the sine of the arch be , so in like manner the nearest distance from b to ec , is the sine of the same arch ; wherefore if with the sine of an arch from the end of the radius be described another ark , as f , and from the center or other extremitie of the radius , a line drawn just touching the same , the angle included between the said line , and the radius shall be an ark equal to the ark belonging to the said sine ; and what is here done by a line of natural sines or tangents , by help of a decimal line of equal parts , equal to the radius ' may be done by help of the natural tables without them . any proportion relating to the sixteen cases of sphoerical triangles , amongst which the radius is always ingredient , may be so varyed , that the radius may be in the third place , and a tangent or a sine in the fourth place , and then if the ark belonging to the fourth proportional be known , an angle equal thereto may be prickt off by tangents or sines according to the nature of the fourth term , as beforth ; if it be unknown , notwithstanding an angle equal thereto may be prickt off with sines or tangents according to the nature of the fourth term , from the two first terms of the proportion , because they are in such proportion , the first to the second , as the radius or third term is to the fourth ; upon this basis follows the demonstration of this scheme . the demonstration of the former scheme for upright decliners . 1. that the substilar line is true prickt off . assuming vb equal to va to be radius , then bf becomes the co-tangent of the latitude , whereto vo is made equal , which becoming radius , oc is the cosine of the declination , and vc the sine . now one of the proportions for finding the substile distance from the meridian is this following : as the sine of the plains declination , is to the tangent of the latitude : so is the radius to the tangent of the substiles distance from the horizontal line . from whence it follows , that if the sine of the declination be pricked on the horizontal line from the center , as vc , and the tangent of the latitude , erected perpendicularly thereto , as ci equal to va , it shall give a point from whence a line drawn into the center , shall be the substilar line . now i am to prove that va is the tangent of the latitude , thus it is made good : in a tangent line of 45d , if the tangent of any part or ark of it be assumed to be radius ( as here fb the cotangent of the latitude equal to vo , which before was made radius ) then doth the said whole tangent line ( which in this case is any side of the square ) become the tangent of that arks complement : thus , va becomes the tangent of the latitude , because the radius is a mean proportional between the tangent of an ark , and the tangent of that arks complement ; for , as the cotangent of the latitude bf , is to the radius bg , so is the radius bf , to the tangent of the latitude bg , there being the like proportion between the two latter , as the two former terms . 2. that the stiles height is true prickt off . to perform this , the cosine of the declination oc , to that lesser radius , was erected perpendicularly from the point i , in the substilar 〈◊〉 , and ik made equal thereto . now one of the proportions for calculating the stiles height is : as the secant of the latitude , is to the cosine of the declination : so is the radius , to the sine of the stiles height . consequently the stiles height may be prickt off from the two first terms of the proportion , if it can be proved that vk is the secant of the latitude ; here let it be remembred that vp is the tangent of the latitude , ip the sine of the declination , and ik the cosine . by construction the angles vpi and vik are right angles , therefore the square of vi is equal to the two squares of vp , and ip by 47 prop. 1 euclid ; again the square of vk is equal to the two squares of ik and iv , wherefore the square of vk is equal to the sum of the three squares of ik , ip and vp ; but the squares of pi a sine , and ik its cosine are equal to the square of the radius . but the square of the radius , and the square of vp , the tangent of the latitude , is equal to the square of vk , therefore vk is the secant of the latitude , because the square of the radius , more the square of the tangent of any ark , is equal to the square of the secant of the said ark . 3. that vk is equal to vf . let vf be an assumed radius , so will fb be the cosine of the latitude . again observe that in any line of sines if the sine of any ark be made radius , the whole line becomes the secant of that arks complement , so here vk being the secant of the latitude is equal to vf the radius , for let vt be made equal to vf , i say it holds as vo the cosine of the latitude , is to the radius vt , so is vo the radius to vt the secant of the latitude , which is therefore equal to vk . whence the trouble of raising a perpendicular in this scheme , because the side of the square passeth through the complement of the latitude , is shunned the point k falling in the outward circumference . lastly , from all this it unavoidably follows that vi is the cosine , and ik the sine of the stiles height to the radius vf , which was to be proved . an upright south dyal , declining 30 degrees eastwards , latitude 51 degrees , 32 minutes . to draw the hour lines of the former upright decliner . first draw the plains perpendicular co , which for upright plains is the meridian line , and with the radius vf of the former scheme , draw an occult ark upon c as a center , and therein set off nv equal to tn of the former scheme , and draw the line ivc for the substile , and with the radius vc describe a circle thereon . 1. the first work will be to finde the regulating point in the substile , called the point sol. prick the radius of the said circle from i to p , and from c to q , and draw the line ip , and therein make ik equal to ik in the former scheme , then a ruler said from k to q , where it intersects the substilar is the regulating point ☉ . 2. to finde a point from whence the circle is to be divided into 12 equal parts . lay a ruler from o to the point ☉ , and it intersects the circle on the opposite side , at it set m. 3. to divide the circle . from the point m before found , lay a ruler through the center v , and it will finde a point on the other side the circle , at it set f , the points mf divide the circle into halfs , and each half is to be divided into six parts , the radius vm will easily divide each half into three parts , and then it will be easie to divide each of those parts into halfs , and so the whole circle will be divided into 12 equal parts , and if it be desired to inscribe the half hours and quarters , then must each of those parts be divided into halfs and quarters . having thus divided the circle into 12 parts , distinguish them with the letters of the alphabet . 4. to draw the hour lines , and to number them . lay a ruler to each of those letter divisions , and upon the regulating point ☉ in the substile , and it will finde points on the opposite side of the circle from which if lines be drawn into the center at c ' they shall be the hour lines required . 5. an inconvenience shunned . when the points through which the hour lines are to be drawn , fall near the center , the hour lines cannot be drawn from the said points with certainty . in this case let any hour line near the said point be produced , if need require , and upon c as a center with the radius cv describe the ark of a circle from the produced hour-line , take half the distance between the point , where the produced hour line , cuts the circle , and the point through which the hour line proposed is to be drawn , and prick that extent off in the arch swept , setting one foot where the arch swept cuts the hour line already drawn , from whence half the distance was measured , and the other foot will finde a point in the said arch , through which the hour line desired is to pass : thus we inscribed the hour line of three in the former dyal . when the north pole is elevated , the center of the dyal must be below , but when the south pole is elevated , as in this example , it must be above , and the point ☉ must be alwayes found in the other part of the diameter most remote from the center . 6. the stiles height may be transferred from the former scheme into this , by pricking the arch tk twice in this circle , and it findes a point , from whence a line drawn into the center , shall represent the stile . that the hour lines are true delineated . the point sol found in the substile , divides the diameter of the circle drawn on the said substile in such proportion , as the radius is to the sineof the stiles height , by reason of the equiangled triangles , whose bases bear such proportion as their perpendiculars . the next work performed by dividing the circle into 12 equal parts , and finding points on the opposite side , by laying a ruler over those equal parts through the point sol , carrieth on this proportion . as the radius is to the tangent of the hour from the substile , so is the sine of the stiles height to the tangent of the hour line , from the substile , being the proportion used in all dyals , whereby to set off the hours . in the third figure following , let ba represent the substile , and let the regulating point be at m , so that bm bears such proportion to ma , as the radius doth to the sine of the stiles height , let the perpendicular bhe represent a line of tangents , whose radius is equal to bc , and the perpendicular ad , another line of tangents to the same radius both infinitely produced , and let be represent the tangent of some particular arch or hour to be drawn , from e through the point m draw a right line to d , and the proportion lies evident in right lines . as the radius bm : is to the tangent of the hour line from the subtile be ∷ so is the sine of the stiles height ma to the tangent of the hour line from the substile ad . if from every degree of a tangent line , lines be drawn into the center of the circle proper to that tangent , they shall divide that quadrant to which they belong into 90 equal parts , this follows from the definition of tangents , but if the lines be drawn from every degree of the said tangent to the extreamity of the diamer , they shall divide a semicircle into 90 equal parts by 20 prop. 3. euclid . because an angle in the circumference , is but half so much as it is in the center . thus the semicircle bga is supposed to be divided into 90 equal parts from the tangent be , by lines drawn to a , as is the line ega , so is the other semicircle from the other tangent ad , by lines drawn to b , as is bfd ; now if it can be demonstrated that the points gmf are in a right line , it follows that the same proportions may be carried on from the equal divisions of these semicircles , as were done in the right tangent lines be and ad , this proposition being very well geometrically demonstrated by my loving friend mr. thomas harvy ; his demonstration thereof shall hereafter follow . having found the distance of the hour line , from the subsistle ab , if b be made the center of the dyal , the right line bf drawn into the center , shall represent the hour line proposed , making the same angle with the substile , as was found by the proportion ; note that no hour can be further distant on one side or other of the substile then 90d , and the said ark of distance will be found in one of the semicircles , and if the center were not placed in the circumference , the angle found would be extended beyond its due quantity , in all upright decliners or leaning plains : the first proportion carried on , is this ; as the sine of the stiles height : is to the tangent of the distance between the meridian and substile : so is the radius : to the tangent of the inclination of meridians : whereby is found the point from whence the circle is to begin to be divided into 12 parts for the whole hours with their halfs and quarters , and those equal divisions give the angles between the meridian of the plain , and the respective hours , called by some the angles at the pole , and then the work is the same as before : now to the proposition . construction . the right lines ad , be are parallel , and touch the circle in the extreams of the diameter ab , de is drawn at pleasure , cutting the tangents in d and e , and the diameter in m , bd and ae cutteth the circumference in f and g , it is required to be proved that the points f , m , and g , are in a right line . draw the lines mf , mg , first if ad be = be then am is = mb , for the triangles adm and bem are equiangeld , therefore m is the center . again ad being = be and the angle dab = eba & ab common to both triangles dab , eba therefore the angles dba , eab are equal , and the double of them fma , gmb the angles at the center , are also equal , wherefore fm and mg is one and the same right line which was to be proved . secondly , if ad be not equal to be , as in the second , third , & fourth scheams let be be greater , and make bh = ad and bi = am , and draw the right line ah cutting the circumference in k ; draw likewise ik , ih , gk , and extend gk infinitely both wayes , which shall be either parallel to ab ( as in the second figure ) or cut it in the point p one side or the other , as in the third and fourth figures , in each from the points a and b , and the center c at right angles to gk , draw the lines ao , bn , cl lastly draw bg , from which construction it willfollow , that 1. hi is parallel to ed for hb is = ad and bi = am and the angles hbi and dam are equal , therefore the angle amd = bme is = bih wherefore hi is parallel to ed. 2. ik is parallel to mf for hb is = ad and the angle hba = dab , and ab common to both the triangles hba and dab , therefore the angles hab and dba are equal , wherefore ak = bf , but ak being = bf and ia = mb , and the angle kai = mbf , therefore ki is parallel to mf . 3. the triangles ahe and agk , are equiangled , so also are the triangles ahb and ago , because the angle akg = abg is = aeb and eah common to both the triangles ahe and agk , therefore they are equiangled ; again the angles ahb and and ago being equal , the right angled triangles abh and aog are also equiangled . 4. nl is equal to lo , because bc is equal to ca and nk is = go , because kl is = lg ; now in the equiangled triangles abh , aog as bh : ha ∷ og : ga and in the triangles ahe , agk as ha : he ∷ ga to gk , therefore ( ex equo ) it will be as bh : he ∷ og : gk , but as bh : he ∷ bi : im for ih is parallel to em , therefore as bi : im ∷ og : gk but nk is = og , therefore bi : im ∷ nk : kg , but ic is half of im and kl half of kg , therefore bi : ic ∷ nk : kl , therefore by composition of proportion , as bc : ic ∷ nl : kl , alternately , as bc : nl ∷ ic : kl . in the second figure bc is = nl , therefore ic is = kl , wherefore ( because ic is also parallel to kl ) ik is parallel to cl , by like reason gm is also parallel to cl , but in the third and fourth figures the sides pc , pl of the triangle pcl are cut proportionally by the parallel bn , and as before , as bc : nl ∷ ic : kl , and as pc : pl ∷ bc : nl , therefore pc : pl ∷ ic : kl , wherefore ik is parallel to cl . again as before , bc : nl ∷ ic : kl , but cm is = ic and ig is = kl , therefore as bc : nl ∷ cm : lg , but as hath been said , as bc : nl ∷ pc : pl , therefore as pc : pl ∷ cm : lg , wherefore because the sides pm : pg , of the triangle pmg are cut proportionally in c and l , cl and mg are parallel one to another . now in all the three figures it hath been proved that ik and mg are either of them parallel to cl , therefore they are parallel one to another , wherefore the angle gmi is equal to the angle kib which before was proved to be equal to amf , therefore the angles gmi and amf are equal one to another , wherefore fm and mg is one and the same right line which was to be proved . to draw an east or west dyal . let the hour line of six , which is also the substilar line , make an angle equal to the latitude of the place , with the plains horizontal line , above that end of it that points to the coast of the elevated pole , then draw a line perpendicular to the substilar line , which some call a contingent line , and with such a radius as you determine the stile shall have parallel height above the substile : divide a tangent line of hours and quarters according to the direction for direct polar recliners , then through those divisions draw lines parallel to the substile and they shall be the hour lines required ; thus the hours of 5 and 7 , are each of them tangents of 15d from six , and so for the rest , let them be numbred on each side of six ( being the substile ) for an east dyal with the morning hours , for a west dyal with the afternoon hours , the one being the complement of the other to 12 hours , and therefore we have but one example , namely an east dyal for the latitude of london . how to fill this or any other plain , with any determined number of hours , shall afterwards be handled . a west diall reclining 50d latitude 51-32′ an east diall inclining 40d latitude 50d to prick off the requisites of an east or west reclining or inclining dyal in their true scituation and quantity . in these plains , first draw the plains perpendicular , and cross it with a horizontal line , which is also the meridian line ; in any one of the quarters , make a geometrical square , as before directed for upright decliners , and from the plains vertical in the said quarter , count off the latitude to l , and from the horizontal line in the said quarter count off the reclination or inclination to r , and from l and r , draw lines into the center , and where the latitude line intersects the side of the square , set f. 1. to prick off the substile . for latitudes above 45d place bf from v the center in the horizontal line for recliners northwards , for incliners southwards , and thereto set o , and through the said point , draw a line parallel to the vertical , and place the nearest distance from a to rv on the former parallel , from o for recliners upwards to i , but for incliners downwards , and a line thence drawn into the center , shall be the substile . 2. the stiles height . place the nearest distance from b to rv on a perdendicular raised from the point i in the substile , and it findes the point k , whence a line drawn into the center , shall represent the stile . in the other hemisphere the words northwards and southwards , must be mutually changed . for latitudes under 45′ the side of the square must be placed from v to o and af must be placed on the line of reclination or inclination from v to c , the nearest distance from c to va placed on the perpendicular , passing through o for recliners upwards , but for incliners downwards , findes the point i , through which the substile is to pass , and the nearest distance from c to vb raised perpendicularly on the point i in the substile , findes the point k for the stile , as before . demonstration . 1. for the substile . in these plains vb or va being radius fb is the contangent of the latitude and the nearest distance from a to rv , is the cosine of reclination or inclination to the same radius , and the nearest distance from b to rv , the sine thereof ; this for latitudes above 45 , but for lesser latitudes . af the tangent of the latitude was made radius , and thereupon the radius or side of the square be-became the cotangent of the latitude , and the nearest distance from c to va was the cosine , and from c to vb the sine of the reclination or inclination , so that the prescribed construction in both cases erects the cosine of the re-inclination on the cotangent of the latitude , which is made good from this proportion . as the cotangent of the latitude is to the cosine of the re-inclination from the zenith so is the radius : to the tangent of the substilar from the meridian . 2. for the stile . a proportion that will serve to prick it off , is , as the cosecant of the latitude , is to the sine of the re-inclination , so is the radius : to the sine of the stiles height . page 25 a west diall reclining 50d latitude 51 32′ a south plaine declining 40d east inclining 15 deg a west plain , reclining 50 degrees , latitude 51 degrees , 32 minutes . the point sol , substile , stile , and hour-lines , are all found after the same manner as in an upright decliner , the substile being set off from the meridian line here , after the same manner as it was from the meridian there . a south plain declining east 40 degrees , inclining 20 degrees , latitude , 51 degrees , 32 minutes . to prick off the requisites in all declining , reclining , and incliing plains in their true coast and quantity . now followeth those directions inlarged , which as i said in the epistle , i received from mr. thomas rice . upon any plain first draw a true horizontal line , at the east end thereof set e , and at the west end w , cross the said line with a perpendicular , which may be called the plains perpendicular , by some termed the plains vertical line , or line of reclination , at the intersection of these two lines set v , and upon it as a center describe a circle ( the radius whereof may be equal to 60d of a line of chords ) at the upper end of the vertical line set s , and at the lower end n. as the declination is , set it off with chords from s or n towards the true coast , and at it set d , from whence draw a line through the center ; also set off the latitude of the place the same way and in the same quarter , and at it set l , from which draw a line to the center . from that end of the horizontal line , towards which the declination was counted , set the inclination ( which as well as the reclinais reckoned from the zenith , the former being the denomination of the under , the latter of the upper face ) upwards towards s , and the reclination downwards towards n , and at it set r , from whence draw a line through the center to the other side , of the circle . in the same quarter of declination , draw ha parallel to the horizontal line , and fg parallel to the vertical line , in a geometrical square , of like and of any convenient distance from the center at pleasure , and where the latitude line intersects the side of the square , let the letter f be placed . on the line of declination beyond the center , make vo equal to fg , and draw ob parallel to the horizontal line , continued till it meet with the side of the square fg produced , at the point of concurrence , set b , and where it intersects the plains perpendicular , set p , and draw oc parallel to the vertical line cutting we at c , and make ha equal to oc or bg , now by help of the three points a , b , c thus found , the requisites will be easily prickt off . 1. the substilar line . the nearest distance from a to rv , set on the line co ( produced if need be ) from c to i the same way the distance was taken from a , that is , if downward or upward , the other must be so too , will shew where vi the substilar line is to be drawn . 2. the stiles height . the least distance from b to rv , set on a perpendicular raised upon the substile from the point i will finde the point k , from whence draw a line into the center at v , and the angle ivk will shew how much the stile is elevated above the substile , and if the work be true , vk and vf will be equal , whence it follows , that the trouble of raising the mentioned perpendicular may be shunned . 3. the meridian line . the least distance from c to rv , set upon the line opb , from p on that side , which is farthest from the line rv , will finde the point m , from whence a line drawn into the center , shall be the meridian line . and i adde that on all north recliners in the northern hemisphere , the meridian line must be drawn through the center on the other side ; and then the construction of the scheam will place it below the plains horizontal line , which is its proper scituation for the said upper face , and for the under face the scheam placeth it true without caution . 4. a polar plain how known . if the line rv fall just into b , the plain is a polar plain , in such a plain the stile hath no height , but is parallel to the axis , in this case the inclination of meridians must be known , directions for such plains must afterwards follow . but if the line rv fall between the points b and p , then must the substile stile and meridian be all drawn through the center , and stand beyond on the other side . annotations on the former scheam . 1. that for latitudes under 45d this construction of the scheam , supposeth the sides of the square produced , which will therefore be lyable to large excursions or other inconveniences , wherefore for such latitudes , i shall somewhat vary from the construction prescribed . 2. in finding the substilar line , in stead of erecting ci upon vc , you may prick the same on the vertical line vn , and thereto erect vc , and get the point i possibly with more certainty by finding the intersection of two arks where the said point is to pass . 3. in pricking off the meridian line , the distance of c from the center may be doubled or tripled , but so must likewise vp , and the nearest distance from c to rv erected on a line drawn parallel to we , passing through the point p so found , and in stead of drawing such a parallel , the point m may be found by the intersection of two arks . 4. that this scheam placeth the requisites of all dyals in their true coast and quantity , yet notwithstanding if this scheam be held before a looking-glass , the effigies thereof in the glass shews how the scheam would happen and place the requisites , namely , the stile , substile and meridian , for a plain of the same denomination , but declining to the contrary coast . and if the face of the said scheam be laid upon a window , and the substile , stile and meridian be continued through the center on the backside thereof , it shews you how these requisites are to be placed on the opposite side of the plain , which being done , may be held before a looking-glass as before , and will be represented for the contrary declination of that opposite face : the truth of all which will be confirmed from the scheam it self . this scheam for declining , reclining , or inclining plains , useth a new method of calculation , derived from an oblique triangle in the sphere , wherein there is two sides with the angle comprehended , given to finde both the other angles , which is reduced by a perpendicular to two right angled triangles , from which the following proportions are derived . i shall therefore first deliver the said method , then demonstrate that the said proportions are carried on in the scheam ; and lastly , from the sphere , shew how those proportions do arise . 1. to finde a polar plains reclination or inclination . as the radius , is to the cosine of the plains declination , so is the cotangent of the latitude , to the tangent of the reclination or inclination sought . 2. to finde the distance of the substile or meridian line , from the plains perpendicular for a polar plain . as the radius , is to the sine of a polar plains reclination , so is the tangent of the declination , to the tangent of the substilar line from the plains perpendicular . 3. the inclination of meridians . as the radius , is to the sine of the latitude , so is the tangent of the declination , to the tangent of the inclination of meridians . affections of a polar plain . the substilar on the upper face , lies above that end of the horizontal line , towards the coast of declination , and the meridian lyes parallel to the substile beyond it , towards that end of the horizontal line that is towards the coast of declination . for declining , reclining , or inclining plains . first finde a polar plains reclination for the same declination . then for south recliners and north incliners , get the difference , but for north recliners and south incliners , the sum of a polar plains reclination , and of the re inclination of the plain proposed , and then it holds . 1. for the substile . as the cosine of the said ark of difference or sum according as the plain leans northward or southwards , is to the sine of the polar plains reclination , so is the tangent of the declination , to the tangent of the substilar from the plains perpendicular . 2. for the stiles height . as the radius , is to the cosine of the substiles distance from the plains perpendicular , so is the tangent of the sum or difference of reclinations , as before limited , to the tangent of the stiles height . 3. meridians distance from the plains perpendicular . as the radius , is to the sine of the re inclination , so is the tangent of the declination , to the tangent of the meridian from the plains perpendicular . 4. inclination of meridians . as the sine of the stiles height , is to the tangent of the distance between the meridian and substile , so is the radius , to the tangent of the inclination of meridians . for south recliners or north incliners , the difference between the substiles distance from the plains perpendicular , and the meridians distance therefrom , is equal to the distance between the meridian and substile ; the like for such north recliners or south incliners , as recline or incline more then an equinoctial plain , having the same declination , but if they lean above it , or have a lesser reclination , the sum is the distance between the meridian and the substile . the three first proportions , besides the finding of a polar reclination , are used in the scheam for the placing of the requisites , and the latter proportion in the circular scheam for drawing the hours . another proportion for finding the inclination of meridians by calculation , is : as the cosine of the latitude , is to the sine of the substiles distance from the plains perpendicular , so is the cosine of the re inclination of the plain , to the sine of the inclination of meridians . the reclination of an equinoctial plain to any assigned declination , is necessary for the determining of divers affections : the proportion to finde it , is : as the radius : is to the cosine of the plains declination : so is the tangent of the latitude , to the tangent of the reclination sought . the upper face of an equinoctial plain , is called a north recliner , the meridian descends from the end of the horizontal line opposite to the coast of declination , the substilar line is the hour-line of six , and maketh right angles with the meridian line . directions for the true scituating of the meridian and substile suited to the former method of calculation . 1. for plains leaning northwards . if a south plain recline more then a polar plain , having the same declination , the plain passeth beneath the pole of the world , the north pole is elevated upon the upper face , the substile and meridian line lye above that end of the plains horizontal line , towards the coast of declination , the substilar line being next the plains perpendicular . for the under face being a north incliner , the south pole is elevated , the lines lye in the same position below the plains horizontal line , and on the contrary side of the plains perpendicular . if a south plains reclination be less then the polar plains reclination , the plain passeth above the pole , and the north pole is elevated on the under face , being the inclining side . the substile and meridian lye above that end of the plains horizontal line that is opposite to the coast of declination , the meridian being nearest the plains perpendicular , for the upper face being a south recliner , the south pole is elevated , and the lines lye in the same position below the plains horizontal line , but on the contrary side of the plains perpendicular descending below that end of the horizontal line , opposite to the coast of declination . 2. for plains leaning southwards generally on the upper face the north pole is elevated on the under face the south pole . to place the substile . such north recliners whose reclination is less then the complement of a polar plains reclination , the substile is elevated above the end of the horizontal line contrary to the coast of declination , and on the under face , being a south incliner , the substilar is depressed below the end of the horizontal line , opposite to the coast of declination . but when the reclination is more then the complement of the reclination of a polar plain , the substile is to lye below the plains horizontal line , from that end opposite to the coast of declination . but for south incliners , being the under face , the substile is elevated above the end of the horizontal line , opposite to the coast of declination . to place the meridian . on all north recliners the meridian lies below the horizontal line from that end thereof , opposite to the coast of declination , because at noon the sun being south , casts the shadow of the stile to the northwards . on the under face , being a south incliner , it must always be placed below the horizontal line , below that end of it toward the coast of declination . these directions suppose the declination to be denominated from the scituation of that face of the plain on which the dyal is to be made , and the horizontal line for all dyals that have centers , is supposed to pass through the same . now to the demonstration of the former scheam . 1. 't is asserted that if rv fall into the point b , the plain is a polar plain , in which case the stile is parallel to the axis of the world . demonstration . every declining plain may have such a reclination found thereto , as shall make the said plain become a polar plain , and the proportion to finde it , may be thus : as the tangent of the latitude , is to the cosine of the declination , so is the radius , to the tangent of the reclination sought . in the former scheam . if we make fg the cotangent of the latitude radius , the side of the square will be the tangent of the latitude , now vo equal to fg , being radius , oc equal to gb , is the cosine of the declination ; wherefore a line drawn into the center from b , shall include the angle of a polar plains reclination agreeable to the two first terms of the proportion , and to the directions for pricking off an angle by tangents . 2. that the substile is true prickt off . upon v as a center with the radius vb , imagine or describe a circle then is bg equal to vp , the sine of a polar plains reclination , which is equal to ha , and the ark comprehended between a and b , will be a quadrant . but in a quadrant any line being drawn from the limbe passing through the center , the nearest distance from the end of one of the radij will be the sine of the ark thence counted , and the nearest distance from the other radius thence counted , will be the sine of the former arks complement ; so in this scheam the nearest distance from b to rv , when a plain reclines , is the sine of the ark of difference , but when it inclines of the sum of the reclination of the plain proposed , and of the reclination of a polar plain , and the nearest distance from a to rv , is the cosine of the said ark . make vq on the plains perpendicular equal to ic , which is equal to the nearest distance from a to rv , and from the point q erect the perpendicular qt , whereto the line po will be parallel , and consequently there will be a proportion wrought : thus it lyes , as vq the cosine of the sum of the polar plains reclination , and of the reclination of the plain proposed , is to pv equal to bg , the sine of a polar plains reclination . these two terms are of one radius , namely vb , so is the tangent of the declination qt , to the tangent of the substiles distance from the plains perpendicular po , these two terms are to another radius , namely qv , then if the tangent of an ark be erected on its own radius , as here is qi equal to cv , and a line be drawn from the extreamity into the center , the angle belonging to that tangent shall be prickt off agreeable to the general direction . 3. that the stiles height is true prickt off . the proportion altered to bring the radius in the third place will be , as the secant of the substiles distance from the plains perpendicular , is to the tangent of the sum or difference of reclinations , as before limited , so is the radius , to the tangent of the stiles height . in the scheam making vq radius , vi becomes the secant of the substiles distance from the plains perpendicular , and the nearstdiestance from b to rv , is the tangent of the difference or sum of the reclinations , which when vb was radius , was but the sine thereof , the reason why it now becomes a tangent , is because the consine of the said ark vq , is made radius : but , as the cosine of any ark , is to the radius , so is the sine of the said ark , to the tangent of the said ark . therefore the nearest distance from b to rv equal to ik , being erected thereon , and a line from the extreamity drawn into the center , shall prick off the stiles height suitable to the two first terms of the former proportion , and to the general direction for pricking off an angle by tangents . 4. that the meridian is true prickt off , the proportion to effect it : is , as the cotangent of the plains declination , is to the sine of the re inclination , so is the radius , to the tangent of the meridian from the plains perpendicular . if vc be made radius , then is co equal to vp the tangent of the complement of the plains declination , and the nearest distance from c to rv , is the sine of the re inclination to the same radius , which is erected perpendicularly on vp suitable to the two first terms of the proportion and the general direction . lastly , that vk is equal to vf , these symbols are used , q signifieth square , + for more or addition , = equal . vbq = vqq + ikq , the reason is because vb being radius , vq is the cosine of an ark to that radius , and ik the sine by construction . vbq = vgq + gbq , therefore these two squares are equal to the two before . if to the latter part of each of these equations , we adde qiq or rather its equal poq , the sum shall be equal to vkq . i say then vqq + ikq + poq = vkq , this will be granted from the former demonstration for upright decliners . again , vgq + gbq + poq = vfq , therefore vf is equal to vk , this cannot be denyed , because , vgq + fgq = vfq . and the two squares gbq , or rather vpq + poq are equal to gfq , which is equal to voq by construction . to draw the hour-lines for the former south plain , declining eastwards 40 degrees , inclining 15 degrees latitude , 51 degrees , 32 minutes . first having assigned the center of the dyal , through the same draw the plains perpendicular , represented by the prickt line cn , and with the radius of the former scheam upon c as a center , describe an occult ark , and therein set off nv equal to the substiles distance from the plains perpendicular , and through the point v draw the substilar , and upon v as a center , describe the circle , and prick off the meridians distance from the the substile in the former scheam , namely yx twice in this circle from i to o , and draw co for the meridian , after the same manner set off the stile , then finde the regulating point sol , divide the circle , and draw the hour-lines according to former directions , and when hour-lines are to lye both above and below the center , they are to be drawn through . to fit the dyalling scheam , for latitudes under 45 degrees . a south plain declining 50 degrees east , inclining 20 degrees , latitude 30 degrees . the former construction would serve , if the sides of the square were produced far enough , but to shun any such excursion , make vo equal to the side of the square , and through the point f , draw a parallel to the plains perpendicular , and where the parallel op produced interesects it , is the point b , upon v as a center , with the extent page 34 a south diall declin 40d east inclin 15′ latit 52-32 a south plain declin 50d east inclining 20d lat 30d a south plain declin 30d east reclining 34-31 lat 51-32 page 35 a south plaine declin 30d east reclining 25d lat 51-32 page 37 vb , draw the arch zb , and prick off a quadrant thereof from the point b , and it will finde the point a , the point c is found no otherwise then before ; and now having these three points , the whole work is to be finished according to those directions , to which when the stile hath a competent height , nothing need more be added , unless it be some examples . how to draw such dyals whereon the stile hath no elevation as polar plains , or but very small elevation , as in upright far decliners , and many leaning plains . these plains are known easily , for if the re-inclination pass through the point b , the plain is a polar plain , and the substile is to be found by the former construction , which the scheam makes the same with the meridian : moreover , another ark is to be found , called the inclination of meridians : the proportion to finde it , is : as the radius , is to the sine of the latitude , so is the tangent of the declination , to the tangent of the inclination of meridians . if co be made radius , cv will be the tangent of the declination , which enter on vl from v to k , and the nearest distance from k to sv shall be the tangent of the inclination of meridians , which is to be prickt on its own radius from p to m , and draw a line from the center , passing through m to the limbe , whereto set f , so is the arch nf the inclination of meridians sought , to wit , the arch of time between the substile and meridian . to draw the hour-lines . first draw a perpendicular on the plain vn , and upon v as a center , describe the arch of a circle , and from the dyalling scheam prick off the substiles distance , and draw iv which shall represent the same , as also the inclination of meridians from i to f , then upon v as a center , describe as great a circle as the plain will admit , and finde the point f , therein also , by laying a ruler over v the center , and f in the former circle , and from the said point divide the circle into twenty four equal parts for the whole hours ( but we shall not need above half of them ) then determine what shall be the parallel height of the stile above the substile , and prick the same on the substilar line , from v the center to i , through which point draw a contingent line at right angles to the substile , and laying a ruler from the center over each of the divisions of the circle , through the points where it intersects the contingent line , if lines be drawn parallel to the substile , they shall be the hour-lines required , the hour-line that belongs to the point f being the meridian-line or hour-line of 12 , after the same manner are the halfs and quarters to be inscribed , if the contingent line be too high , the center v may be placed lower , if it be required , to fit so many hours precise to the plain ; first draw it very large upon some floor , and then it may be proportioned out for a lesser plain at pleasure , as was mentioned for east or west plains . a south plain , declining eastwards , 30 degrees , reclining 34 degrees , 31 minutes . the scheam placeth all things right for equinoctial reclining plains , without any further caution . to draw the hour-lines on such plains , where the stile hath but small elevation . a south plain declining 30 degrees eastwards , reclining 25 degrees , latitude 51 degrees 32 minutes . in these plains , because the hour-lines will run close together , the dyal must be drawn without a center , by help of two contingent lines , and first of all the inclination of meridians must be known , thereby is meant the arch of time between the substile and the meridian line or hour line of 12 , and that may be found several ways ; here i shall follow the proportion in sines before delivered . making vi radius prick the same on the latitude line vl , and from the point found , take the nearest distance to the horizontal line , place this extent from v to f , and draw cf ; then take the nearest distance from r to the plains vertical sn , which place from v to q , then draw qt parallel to fc , so is vt the sine of the inclination of meridians , which may be easily measured in the limbe by the arch sy , by drawing a line parallel to the plains perpendicular from the point t , or by pricking the same on ha produced , and laying a ruler thereto , or by drawing the touch of an arch with vt upon s as a center , then a ruler laid from v touching the outward extremity of that arch , findes the point y. moreover , we need not make vi radius , but prick the nearest distance from l to ve , from v upwards , then the nearest distance to the plains perpendicular from the intersection of the substile , with the limbe must be placed from v towards w , and from the two points thus found a line drawn , and the rest of the work , as before . to draw the hour-lines . first draw the plains perpendicular cn , and draw an occult arch , wherein prick down ny and nk , and draw the substile and stile as before , making the same angles . through any two points in the substile , as at a and b , draw two right lines continued , making right angles therewith . draw a line parallel to the stile at any convenient distance , which is to represent the new stile , as here de . take the nearest distance from b to de , and set it on the substile from b to v , also the nearest distance from a to de , and set it from v to c , through which point draw another line perpendicular to the substile . upon v as a center , describe the arch of a circle of as large a radius as the plain will admit , and from the substile on the same side thereof the meridian happened in the former scheam , set off the inclination of meridians , and it findes the point m , from whence divide the circle into 24 equal parts , and draw lines from the center v through those parts , cutting both the contingent lines b and c , the respective divisions of the contingent line c , must be transferred into the contingent line a , and there be made of the like distance from the substile as in the said line c , then lines drawn through the divisions of the two contingent lines a and b , shall be the respective hour-lines required . a south plain declining eastwards 30 degrees , reclining 25 degrees , latitude 51 degrees , 32 minutes . after the same manner must such east and west recliners or incliners , that have small elevation of the stile , and upright far decliners be pricked down , and in these plains the meridian many times must be left out , the proportion to finde the inclination of meridians for upright decliners : is , as the radius , is to the sine of the latitude , so is the cotangent of the plains declination , to the cotangent of the inclination of meridians . in the dyalling scheam , making cv radius , co is the cotangent of the declination , which enter on the latitude line vl , and take the nearest distance to sv , which extent prickt upon co , and it findes a point , through which a line drawn from the center to the limbe , shall shew the inclination of meridians to be measured from n. otherwise : that we may not transfer large divisions on the contingent line from a small circle , and that the plain may be filled with any determined number of hours , such as by after directions shall be found meet , draw any right line on a board or floor that shall represent the plains perpendicular , as cn , and from the same set off the substile and stiles height from the general scheam as before , drawing a line parallel to the stile as de , also a line perpendicular to the substile , which i call the floor contingent , and that it may be large , let it be of a good distance from the center ; from the point of intersection at y , take the nearest distance to the parallel stile , which prick from y to v , and upon v as a center describe as large a circle as may be with convenience , and from the substile set off the inclination of meridians therein to m ( which ark refers to v as its center ) and from the said point divide it into 24 equal hours ( or fewer , no more then are required ) and laying a ruler over v , and those respective divisions graduate them on the large contingent line , i say from this large contingent line thus drawn and divided , we may proportion out the divisions of two ( or many ) contingent lines that are lesser , and thereby fill the plain with any proper number of hours required . then in order to drawing the hour-lines on the plain . the first work will be to draw the larger contingent line on the plain , which may be drawn anywhere at pleasure : for performing whereof , note , that what angle the substile makes with the plains perpendicular , the contingent line is to make the same with the horizontal line , and the complement thereof , with the vertical line ; also draw another contingent line above this , parallel thereto at any convenient distance . in the bigger contingent line assume any two points to limit the outward most hours that are intended to be drawn on the plain , as admit on the former plain , i would bring on hours from six in the morning , to two in the afternoon , between the space a and b of the greater contingent , take the said extent ab , and upon the point b at 2 of the floor contingent , describe an ark therewith , to wit , l , then from a draw the line al , just touching the outward extreamity of the said ark . i say the nearest distance from d to al , being pricked on the plains greater contingent from a to d , findes a point therein through which the stile is to pass . also the nearest distance from y to al , findes the space ay on the plains greater contingent , and through the point y a line drawn perpendicular to ab , shall be the substilar line . the stile is to be drawn through the point d , making an angle with the plains contingent line equal to the complement of its height above the substile in this example 80d 57′ , to wit , the arch ny. the respective nearest distances to al , from each hour-point , in the floor contingent , being pricked on the plains greater contingent from a towards b , findes points therein , through which the hour-lines are to pass . the next work will be to limit one of the extream hours on the plains lesser contingent , and that must be done by proportion . as the distance between the parallel stile and substile on the floor contingent , is to the distance between the stile and substile on the plains lesser contingent , so is the distance between the substile and either of the extream or outward hours on the floor contingent , to the distance between the substile , and the said outward hour on the plains lesser contingent . a south diall declining 30d east reclining 25d lat 51-32 this proportion is to be carryed on in the draught on the floor , place dy from a to g on the floor contingent , and with the extent eg taken from the plains lesser contingent , upon g on the floor contingent , draw the arch o , and from a draw a line touching the outward extreamity of the said arch , and let it be produced . the nearest distance from y to ao , being prickt on the plains lesser contingent , reaches from g to c , the point limiting the outward hour of six . then if the nearest distances to the line ao , be taken from all the respective hours on the floor contingent , and placed on the plains lesser contingent from c towards f , you will finde all the hours points required , through which and the like points on the plains greater contingent , the hour-lines are to be drawn . here note , that the extent dy on the floor contingent , may be doubled or tripled ; if it be tripled it reacheth to h , also the extent eg on the plains lesser contingent , is to be encreased after the same manner , and an ark therewith described on h before found , as q , and by this means the line ao will be drawn and produced with more certainty , then by the ark o near the center . and what is here done by help of the stiles distance from the substile , may be done by help of the outward hour , if the distance of the said hour-line from the substile be found geometrically or by calculation , for the said hour-line will make an angle with the plains contingent lines , equal to the complement of the ark of its distance from the substile . this plain is capable of more hours which cannot conveniently be brought on . after the same manner are upright far decliners to be dealt withall , and all other plains having small height of stile . but to limit the outward hours on polar plains , and east or west plains , the trouble will not be half so much . a south plain declining eastwards 30 degrees , reclining 25 degrees , latitude 51 degrees , 32 minutes . a second method of calculation for oblique plains . by the former method the meridians distance from the plains perpendicular is to be found , and the polar reclination calculated . then for south recliners , or north incliners , get the difference , but for north recliners or south incliners the sum , of the polar plains reclination , and of the reclination of the plain proposed , and it holds . as the cosine of the polar plains reclination , is to the sine of the former sum or difference , so is the sine of the latitude , to the sine of the stiles height . which pole is elevated is elevated is easily determined , by comparing the reclination of the proposed plain with the polar reclination , and all other affections are to be determined , as in the first method . then for the substile , and inclination of meridians . as the cosine of the stiles height , is to the sine of the plains declination , so is the cosine of the latitude , to the sine of the substiles distance from the plains perpendicular : and so is the cosine of the reclination , to the sine of the inclination of meridians . this method ariseth from the aforementioned oblique triangle in the sphere , in which by help of two sides , and the angle comprehended , the third side is first found , and the other requisites by the proportions for opposite sides and angles . proportions for upright decliners . 1. to finde the substiles distance from the meridian . as the radius , is to the sine of the declination , so is the cotangent of the latitude , to the tangent of the substile from the meridian . 2. angle of 12 , and 6. as the radius , is to the sine of the plains declination , so is the tangent of the latitude , to the tangent of the angle between the horizontal line and six . 3. inclination of meridians . as the sine of the latitude , is to the radius , so is the tangent of the declination , to the tangent of the inclination of meridians : 4. stiles height . as the radius , is to the cosine of the latitude , so is the cosine of the plains declination , to the sine of the stiles height . these arks are largely defined in my treatise , the sector on a quadrant . for east and west re-incliners , the complement of the latitude of the place , is such a new latitude : in which they shall stand as upright plains , and the complement of their re-inclination is their new declination in that new latitude , having thus made them upright decliners , the former proportions will serve to calculate all the requisites . in all upright plains , the meridian lyeth in the plains perpendicular , and if they decline from the south ( in this hemisphere ) it is to descend or run downward ; if from the north it ascends , and the substile lyeth on that side thereof opposite to the coast of declination . in east or west re-incliners , it lyeth in the plains horizontal line , on the inclining side the south pole is elevated , but on the upper side the north pole , and the substile lyeth above or below that end of the meridian line , which points to the pole elevated above the plain . on all plains whatsoever to calculate the hour distances . as the radius , is to the sine of the stiles height above the substile , so is the tangent of the angle at the pole , to the tangent of the hour-lines distance from the substilar line . by the angle at the pole , is meant the ark of difference between the ark called the inclination of meridians , and the distance of any hour from the meridian for all hours on the same side of the meridian the substile falls , and the sum of these two arks for all hours on the other side the meridian . all hours on any plain go to the contrary coast of their scituation in the sphere , thus all the morning or eastern hours , go to the western coast of the plain , and all the evening or western hours , go to the eastern coast of the plain . a third method of calculation for leaning plains , that is , for all sorts of plains that do both decline , and also incline or recline . they may be referred to a new latitude , in which they shall stand as upright plains , and then they will have a new declination in that new latitude ; which two things being found , the former proportions for upright decliners will serve to calculate all the arks required . how this may be done on a globe , is not difficult to apprehend , having set the globe to your latitude , let one of the meridians of the ecliptick or longitude in the heavens , represent a declining reclining plain , this circle intersects the meridian of the place in two points , the one above , the other beneath the horizon : imagine the globe to be so fixed , that it cannot move upon its poles , then elevate or depress the globe so in the meridian that the point of intersection above the horizon may come under the zenith , then will the pole of the world be elevated above the horizon to the new latitude sought , and where the meridian of longitude that represents the plain intersects the horizon it shews the new declination . or it may be thus apprehended : the distance between the pole of the world , and that point of intersection that represents the zenith of the new latitude , is the complement of the said new latitude , and the distance between that point , and the equinoctial is the new latitude it self ; the new declination is the complement of the angle between the plain and the meridian of the place , an ark usually found in calculation under this denomination . to finde these arks by calculation . as the radius , is to the cosine of the plains declination , so is the cotangent of the re-inclination from the zenith , to the tangent of the meridional ark , namely the ark of the meridian between the plain and the horizon . and this is the first thing master gunter and others finde ; for south recliners north incliners the one being the upper , the other the under face , get the difference between this ark and the latitude of the place , the complement of the said residue , remainder , or difference , is the new latitude sought ; but for north recliners or south incliners , the difference between this fourth arch and the complement of the old latitude is the new latitude . to finde the new declination : as the radius , is to the cosine of the re-inclination , so is the sine of the old declination , to the sine of the new . this method is hinted to us in mr. fosters posthuma , also in his book of dyalling in anno 1638 , where he refers leaning plains to such a latitude wherein they may become east or west recliners , but that method is to be deserted , as multiplying more proportions then this , and doth not afford that instrumental ease for pricking down the hours that this doth . affections determined . such south recliners , whose meridional arch is less then the latitude , pass beneath the pole , and have the north pole elevated above them , but if the meridional ark be greater then the latitude , they pass above the pole , the north pole is elevated on the under face , all other affections are before determined . if the meridional arch be equal to the latitude , the plain is a polar plain ; for plains leaning southwards , if the meridional arch be equal to the complement of the latitude , the plain is an equinoctial plain , if it be more , the plain hath less reclination then an equinoctial plain , if it be less it hath more , and all affections necessary for placing ( and calculating ) the meridian line were before determined . this method of calculation findes the substiles distance from the meridian , not from the plains perpetdicular , wherefore it must be shewed how to place it in plains leaning southwards , for plains leaning northwards use the former directions . to place the substile in north recliners . in these plains the meridian and substilar are to meet at the center , and not being drawn through , will make sometimes an acute , sometimes an obtuse angle . when the plains meridional ark is greater then the colatitude , they make an obtuse angle , in this case , having first placed the meridian line , above it prick off the complement of the distance of the substile from meridian to a semicircle . but when the meridional ark is less then the colatitude , prick off the said distance it self above the meridian line . in south incliners . when the plains meridional ark is greater then the colatitude , the substile and meridian make an acute angle , when it is equal to the colatitude , they make a right angle , when it is less then the colatitude they make an obtuse angle , and must be prickt off by the complement of their distance to a semicircle , the substile always lying on that side of the meridian , opposite to the coast of declination . a fourth method of calculation for leaning plains . an advertisement . in this method of calculation for all plains leaning northward , both upper and under side their declination is the arch of the horizon between the north and the azimuth of the plains south pole , so that their declination is always greater then a quadrant ; but for all plains leaning southwards , both upper and under face , their declination is the arch of the horizon between the north and the plains north pole , wherefore it is always less then a quadrant ; in this sense declination is used in the following proportions . as the sine of half the sum of the complements , both of the latitude and of the reclination , is to the sine of half their difference , so is the contagent of half the declination , to the tangent of a fourth arch. again , as the cosine of half the sum of the former complements , is to the cosine of half their difference , so is the contangent of half the declination , to the tangent of a seventh arch. get the sum and difference of the fourth and seventh arch , then if the colatitude be greater then the complement of the reclination , the sum is the substiles distance from the plains perpendicular , and the difference the inclination of meridians . but if it be less , the difference is the substiles distance from the plains perpendicular , and the sum the inclinations of meridians . to place the substile . for plains leaning southwards , when the angle of the substile from the plains perpendicul is less then a quadrant , it will on the upperface lye above that end of the horizontal line that is opposite to the coast of declination , and on the under face lye beneath it , but when it is greater , it will lye below the said end , on the upper face , and above it on the under face , but this will not be till the reclination be more then the complement of the reclination of a polar plain that hath the same declination ; for plains leaning northwards the directions of the first method suffice . to place the meridian . either calculate it , and place it according to the directions of the first and second method , or else calculate it by this proportion . as the radius , is to the sine of the stiles height , so is the tangent of the inclination of meridians ( when it is obtuse , take its complement to a semicircle ) to the tangent of the meridian line from the substilar . for plains leaning northward , the first directions must serve , but for southern plains the second , because the distance of the meridian is calculated from the substile supposed to be placed , and here the work is converse to that , for in that we supposed the meridian placed , and not the substile . for the stiles height . as the sine of the fourth arch , is to the sine of the seventh arch , so is the tangent of half the difference of the complements both of the latitude and reclination , to the tangent of an arch sought . how much the said ark being doubled wants or exceeds 90d , is the stiles height . in south recliners , if the said ark being doubled , is less then 90d , its complement is the elevation of the north pole , and the plain falls below the pole . but if the said arch exceed 90d the plain passeth above the pole , and the excess is the elevation of the north pole on the under face of a south recliner , called a north incliner , and the affections were determined in the first method where the declination hath its denomination from that coast of the meridian to which the plain looketh . these methods of calculation may not precisely agree one with another , though all true , unless the parts proportional be exactly calculated from large tables in every operation , which to do as to the examples in this book , my leisure would not permit ; this last method is derived also from the former oblique triangle , the proportions here applyed , being demonstrated in trigonometria brittanica by mr. newton . the demonstration of the former proportions in projecting the sphere , it is frequently required to draw an arch through any two different points within a circle , that shall divide the said circle into two equal semicircles construction . 1. draw a line from one of the given points through the center , for conveniency through that point which is most remote . 2. from the center raise a radius perpendicular to that line . 3. and from the said point draw a line to the end of the radius . 4. from the end of the radius raise a line perpendicular to the line last drawn , and where it intersects the former line drawn through the first point and center , is a third point given , describe a circle through these three points , and the proposition will be effected . example . let it be required to draw the arch of a circle through the two points e and f that shall divide the circle bd into two equal parts . operation . from e draw eg , through the center a , make ad perpendicular thereto , joyn ed , and make dg , perpendicular to ed cutting eg in g , through e , f , and g , draw the arch of a circle which will divide the circumference bdc into two equal parts in b and c , that is , if ca be drawn , it will pass through b , if not , let it pass above or below , as let it pass below and cut bfe in h. demonstration . by construction edg and dag are right angles ; therefore □ ad = ▭ eag by 13 prop. 6 euclid . because edg being a right angle , ad is a mean proportional between ea and ag , but ▭ eag should be = ▭ cah by 35. prop. of 3 euclid . therefore ▭ cah = □ ad . but □ ad = □ ai that is = ▭ cai , therefore ▭ cah = ▭ cai which is absurd , therefore ci cannot pass below b , the same absurdity will follow if it be thought to pass above it , therefore ca produced , will fall in the point b , wherefore bdc is a semicircle , which was to be proved : and hereof i acknowledge i have seen a demonstration by the learned teacher of the mathematicks , mr. john leak , to this effect . to project the sphere and measure off the arks of an upright decliner . upon z as a center , describe the arch of a circle , and cross it with two diameters at right angles in the center , whereto set nesw to represent the north , east , south and west . prick off the latitude from n to l , and lay a ruler to it from e , and where it cuts nz , set p to represent the pole . prick off the declination of the plain from e to a , and from s to d , and draw the diameter azb , which represents the plain , and dzc , which represents the poles thereof . through the three points cpd , draw the arch of a circle , and there will be framed aright angled triangle zhp right angled at h , in which there will be given the side zp the complement of the latitude , with the angle pzh the complement of the declination . whereby may be found the stiles height represented by the side ph , the substiles distance from the plains perpendicular represented by zh , and the angle between that meridian which makes right angles with the plain , and the meridian of the place represented by the angle zph , shewing the arch of time between the substile and meridian , called the inclination of meridians , from which triangle are educed those proportions delivered for upright decliners . to measure off these arks . 1. the substile . a ruler laid from d to h , findes the point f in the limbe , and the arch cf is the measure of the substiles distance from the meridian , to wit , 21d 41′ . 2. the stiles height . set off a quadrant from f to g , lay a ruler from g to d and where it intersects bz , set ☉ which is the pole of the circle cpd , lay a ruler from ☉ to p , and it intersects the limbe at i , so is the arch ai the measure of the stiles height , to wit , 32d 32′ 3. inclination of meridian . lay a ruler from p to ☉ , and it intersects the limbe at t , and a south plaine declin 30d east latitud 51d 32′ page 51 a west plaine reclining 50d latitude 51d 32′ the arch wt is the measure of the inclination of meridians , to wit , 36d 25′ . 4. angle of 12 and 6. in like manner we may draw a circle passing through the points wpe , as the prickt arch pe doth , then in the triangle zpq right angled at p , we have the side zp given , and the angle pzq to finde the side zq , lay a ruler from d to q , and you will finde a point in the limbe , the distance whereof from c is the measure of the arch sought , to wit , 57d 49′ , to be measured by projection as the inclination of meridians . lastly the hour-lines , these are represented by meridians drawn through the poles of the world , as in the triangle hpq there will be given the stiles height ph , and the angle hpq , to wit , the ark of difference between the inclination of meridians and the hour from noon , for all hours on that side of the meridian the substile falls , but on the other side the sum of these two arks , and this angle is called the angle at the pole ; the side required is hq , the distance between the substile and the hour line proposed , which must be any hour , though in this scheam it represents the horary distance of six from the substile . to project the sphere for an east or west reclining or inclining plain , latitude 51 degrees , 32 minutes , a west plain , reclining 50 degrees . having drawn the fundamental scheam , and therein set off the latitude as before , count the reclination from n to r , and by laying a ruler , finde the point a , through it and the north and south points draw the arch of a circle which shall represent the plain , finde the pole thereof by setting off a quadrant from r to g , then through the pole of that circle ☉ , and the pole of the world p , draw the arch ☉ pr , then in the triangle nhp right angled at h , we have given the side np the latitude , and the angle pnh the reclination , to finde ph the stiles height , and nh the substiles distance from the meridian , and the angle nph the inclination of meridians , which is also represented by the angle bps . 1. to measure the stiles height . set off a quadrant from b to c , and draw a line through the center , and where it intersects the plain at f , is the pole of the plains meridian , lay a ruler from f to h , and it cuts the limbe at i. also lay it thence to p , and it cuts the limbe at k , the arch ik is the measure of the stiles height , to wit , 36d 50′ 2. the substile . a ruler laid from ☉ to h , cuts the limbe at m , and the arch nm is the measure of the substiles distance from the meridian , to wit , 38d 59′ . 3. the inclination of meridians . set off a quadrant from k to o , and lay a ruler from it to f , and it intersects the meridian of the plain at q , then a ruler laid from p to q , findes the point t in the limbe , and the arch st is the measure of the inclination of meridians , to wit , 53d 26′ . otherwise with less trouble lay a ruler from p to f , and it intersects the limbe at v , and the arch ev is the measure of the inclination of meridians , as before . to project the sphere to represent a declining reclining plain . a south plain declining 40 degrees east , reclining 60 degrees , latitude 51 degrees , 32 minutes . having drawn the fundamental circle , prickt off the declination , and found the pole point as before , prick the reclination from b to i , and laying a ruler to it from a , finde the point r , and through the three points bra describe a circle , representing the plain , also from k finde the pole thereof ☉ , and through the two points p and ☉ draw the arch of a circle fg , representing the plains meridian , at the intersection of the plain with the meridian set z , and draw the arch of a polar plain through p to s , and there will be several triangles constituted , from which were derived the several methods of calculation . in the right angled triangle anz there is given na the complement of the declination , and the angle naz the complement of the reclination , whereby may be found zn the plains meridional ark , which taken from np rests zp the complement of the new latitude ; also the angle nza which is the complement of the new declination , and hence were derived the proportions for the third method , likewise in the same triangle may be found za the meridians distance from the horizon . in the oblique angled triangle cp ☉ , there is given the side cp , the complement of the latitude , the side c ☉ , the complement of the reclination with the angle pc ☉ , the complement of the declination from the south to a semicircle , whereby may be found the angle cp ☉ , the inclination of meridians , and the angle c ☉ p whereof the measure is rh the distance , of the substile from the plains perpendicular , and the third side p ☉ , the complement whereof is ph the stiles height , and from hence was derived the third method of calculation suited to proportions for finding both the unknown angles of an oblique spherical triangle at two operations , when there is given two sides with the angle comprehended between them . the first method of calculation is built upon the perpendicular trigonometrie , for the perpendicular ps reduceth the former oblique triangle , to two right angled triangles , to wit , the right angled triangle , psc , and the right angled triangle ps ☉ , both right angled at s. in the right angled triangle psc , we have cp given the colatitude , and the angle pcs the declination to finde sc a polar plains reclination thereto . again , in oblique spherical triangles , reduced to two right angled triangles by the demission of a perpendicular , it is a common inference in every book of trigonometry , when two sides with the angle comprehended are given , to finde one of the other angles : that , as the sine of the side between the angle sought and perpendicular , is to the tangent of the given angle , so is the sine of the side between the angle given , and perpendicular , to the tangent of the angle sought . and so in that oblique triangle , the difference between the reclination of the plain proposed , and the polar plain is rs , then because r ☉ is a quadrant , s ☉ is the complement of the former ark , therefore it holds : as s so : t scp ∷ s sc : t s ☉ p which is the very proportion delivered delivered in the said method for finding the substiles distance . then in the right angled triangled ps ☉ , we have s ☉ , and the angle s ☉ p to finde the side p ☉ , whereby is got the stiles height ; the inclination of meridians is found in the oblique spherical triangle by the proportion of opposite sides and angles . lastly , in the right angled triangle zrc , there is given rc , and rcz to finde rz the distance of the meridian from the plains perpendicular . to measure the respective arks abovesaid . 1. the new latitude . a ruler laid from w to z , and p will give you the arch mo in the limbe , the complement of the new latitude , to wit , 27d 40′ . 2. the new declination . the ruler laid from the plains zenith at z , to its pole at ☉ , findes the point q in the limbe , and the arch sq is the new declination , to wit , 18d 56′ . 3. the substiles distance from the plains perpendicular . a ruler laid from ☉ to h , findes the point t in the limbe , and the arch ct being 26d 26′ is the substiles distance from the plains perpendicular . 4. the meridians distance from the plains perpendicular . a ruler laid from ☉ to z , findes the point v in the limbe , and the arch dv being 35d 56′ is the meridians distance from the plains perpendicular . 5. the stiles height . set off a quadrant from g to x , and draw xc , where it intersects the plain as at y , is the pole of the arch fg , then laying a ruler from y to h and p , you shall finde the stiles height in the limbe to be the arch 2 , 3 , namely , 26d 6′ . a south plaine declin 40d east reclining 60d lat 51-32′ page 55 a south plaine declin 30d east reclining 25d lat 51 32′ 6. the inclination of meridians . a ruler laid from p to y , intersects the limbe at ¶ , the arch w ¶ is 20d 58′ , and so much is the inclination of meridians . the polar reclination cs is 31d 19′ . the scheam determineth all the affections of the plain . 1. it shews that h the point of the substilar lies on that side the plains perpendicular , that is towards the coast of declination . 2. that z the point for the place of the meridian , lyes towards the same coast as before , but below the substilar line . 3. the arch ph shews you that the north pole is elevated above the upper or reclining face . after the same manner may all the requisite arks be measured , and affections determined for all plains whatsoever . a south plain declining 30 degrees eastwards , reclining 25 degrees , or a north plain declining 30 degrees westwards , inclining ●● degrees . in this scheam we have the same oblique triangle pc ☉ reduced to two right angled triangles psc and ps ☉ , sc is the inclination of a polar plain , and rc the inclination of the plain proposed , the difference is sr , and the complement of it , is the complement of s ☉ to a semicircle , because s ☉ is greater then a quadrant , and the proportions are wholly the same , though the triangle have sides greater then a quadrant . the north pole is elevated on the inclining face , the meridian z lyes from the plains perpendicular towards that end of the horizontal line , opposite to the coast of declination , the same way and beneath it lyeth the substilar . the complement of the new latitude zp is 10d 10′ the new declination , viz. the complement of nza is 26d 57′ the meridians distance from the plains perpendicular rz 13d 43′ the substiles distance therefrom rh 18d 22′ the stiles height ph is — 9d — 3′ the inclination of meridians , to wit , the angle cp ☉ 27d 18′ the polar reclination — cs — 34d 31′ a south plain declining 40 degrees east , inclining 15 degrees , or rather a north plain declining 40 degrees west , reclining 15 degrees . here again the oblique triangle cp ☉ is reduced to two right angled triangles psc and ps ☉ , and sr is the sum of the polar reclination sc , and the re inclination of the plain proposed cr , and s ☉ is the complement hereof , because ☉ r is a quadrant , finde the equinoctial point ae . the polar reclination cs 31d 19′ . the new latitude zae is 32d 15′ . the new declination being the complement of nza is 38d 23′ the meridians distance from the plains perpendicular zr 12d 33′ the substiles distance from the plains perpendicular rh 32d 16′ the stiles height ph is — 41d 30′ the inclination of meridians ☉ pc rather the acute angle ☉ pn is 55d 58′ the meridian z lies from the plains perpendicular towards the coast of declination on the reclining side , but must be drawn through the center , because the sun at noon casts his shadow northwards , unless in the torrid or frozen zone , and the substile hlyes on the other side the plains perpendicular . a north plain declining 40 degrees eastwards , reclining 75 degrees . in this plain likewise the oblique triangle c ☉ p is reduced to two right angled triangles ps ☉ and psc by the perpendicular ps which is part of the arch of a polar plain , here cr more cs is equal to the sum of the plains reclination proposed , and of the polar plains reclination , which is greater then a quadrant for the arch r ☉ is a quadrant ; now the cosine of an ark greater then a quadrant is the sine of that arks excess above a quadrant , wherefore the sine of s ☉ is the cosine of the sum of both the reclinations , and the case the same as before . a south plaine declin 40d east inclining 15d lat 51d 72′ page 56 a north plain declin 40d east reclin 75d lat 51d 32′ prick off a quadrant from g to x , and draw xc , it cuts the plain at y , a ruler laid from y to h , and p findes the points 2 , 3 in the limbe , and the arch 2 , 3 being 61d 31′ , is the stiles height ; orther complement thereof to a semicircle might be found by measuring the arch pf . a ruler laid from ☉ to z , findes the point v in the limbe for the meridian line , from which draw a line through the center on the other side , and it will be placed in its true coast and quantity from the plains perpendicular at a , to wit , 39d 2′ . the inclination of meridians , to wit , the angle cp ☉ is 20d 30′ , the new latitude zae 26d 38′ . the new declination is 9d 35′ to wit , the complement of krb. the polar reclination cs is 31d 19′ . the truth of this stereographick projection is fully handled by aguilonius in his opticks , and how to determine the affection of any angle of an oblique spherical triangle , i have fully shewed in a treatise , called the sector on a quadrant . for the resolution of spherical propositions , delineations from proportions or the analemma , will be more speedy and certain ( though they may also be thus resolved ) which i have handled at large in the mariners plain scale new plain'd . to determine what hours are proper to all kinde of plains . to do this it will be necessary to project upon the plain of the horizon , the summer and winter tropicks . get the sum and difference of the colatitude , and of the suns greatest declination , so we shall obtain his greatest and least meridian altitudes . the depression of the tropick of cancer under the horizon , is equal to the least meridian altitude , and the depression of the tropick of capricorn to the greatest . example : 38d 28′ colatitude 23d 31′ 61d 59′ greatest 14d 57′ least meridian altitude , having drawn the primitive circle , &c. as before . prick 14d 15′ from s to c , and 61d 59′ from s towards w , a ruler laid from the points found , will intersect the meridian zs at the point l for the winter tropick , and k for the summer tropick , through which the circles that represent them are to pass , to finde the semidiameters whereof , set off their depression from n towards e , thus 14d 57′ the depression of the summer tropick terminates at o , a ruler laid from e to ☉ , findes the point x in the meridian szn produced , so is xk the diameter of the summer tropick , which being divided into halfs , will finde the center thereof whereon to describe it . in like manner is the diameter of the winter tropick to be found ; or if the amplitude be given ( or found as elsewhere is shewed ) which at london is 39d 54′ , and set off both ways from g and e we shall have three points given through which to draw each tropick , and the centers falling in the meridian line will be found with half the trouble , as to finde a center to three points . also project the pole point p as before , being thus prepared fkg will represent the summer and hli the winter tropick . let it be required to know what hours are proper for a south plain declining 30d eastwards , through the three points bpa describe the the arch of a circle bqp , then laying a ruler from b to q , finde the point r in the limbe , and from it set off a quadrant to m , then a ruler laid from b to m findes the point ☉ the pole of the hour circle bqp , then laying a ruler from p to ☉ , it findes the point t in the limbe , and the arch et being 65d 40′ is the measure of the angle bps , which turned into time is 4 ho . 23′ prope , and sheweth that at no time of the year the sun will shine longer on the south side of this plain , then 23 minutes past 4 in the afternoon . in like manner if the arch of a circle be drawn through the two points pv , we may finde the time when the sun will soonest in the morning begin to shine on the south side of this plain . a south plaine declin 30d east lat 51d 32′ page 61 a south plaine declin 60d east reclining 40d lat 51-32′ so if there were a south plain declining 60d eastwards , reclining 40 degrees here represented by bra , if it were required to know what hours are proper for the upper , and what for the under face , then where the plain intersects the tropicks as at i and k , draw two meridians into the pole at p , to wit , ip and kp , and first finde the angle ipz , as was before shewed , to wit , 53d 14′ which in time is 3 hours 33 minuutes , shewing that the sun never shines longer on the upper face of the plain , then 33 minutes past 3 in the afternoon , which is capable of receiving all hours from sun rising to that period of time , and the angle kpz , to wit , 39d 50′ in time 2 hours 39 minutes , shews that the sun never begins to shine sooner on the under face then 39 minutes past 2 in the afternoon after , which all the hours to sun-set may be expressed . to finde these arks of time by calculation , there must be given the stiles height above the plain 15d 22′ , ph , and the complement of the inclination of meridians to a semicircle , 136d 32′ , to wit , hps , then in the right angled triangle phi there is given ph the stiles height , pi the complement of the declination , besides the right angle at h , to finde the angle iph 83d 18′ which taken from the complement of the inclination of meridians hps , there rests the angle ips the arch of time sought , to wit , 53d 14′ . the ascensional difference may be found by drawing the arch of a circle through the three points tpf , and thereby the length of the longest day determined that no hours be expressed , on which the sun can never shine . another manner of inscribing the hour-lines in all plains having centers . the method here intended , is to do it in a parallelogram from the meridian line , whence the hour-lines may be prickt down by a tangent of three hours , with their halfs and quarters from a sector , without collecting angles at the pole , or by help of a scheam which i call the tangent scheam , the foundation of this dyalling supposeth the axis of the world to be inscribed in a parallelipiped on continued about the axis , the sides whereof are by the plains of the respective hour circles in the sphere divided into tangent-lines , that is to say , each side is divided into a double tangent of 45d set together in the middle , and the said parallelipiped on being cut by any plain , the end thereof supposed to be intersected , shal be either a right or oblique angled parallelogram , and then if from the opposite tangent hour points on the sides of the intersected parallelipipedon , lines be drawn on the plain , they shall cross one another in a center , and be the hour-lines proper to the said plain , but of the demonstration hereof , i shall say no more at present , the inquisitive reader will finde it in the works of clavius . to draw the tangent scheam . i have before in page 10 shewed how to divide a tangent line into hours and quarters , which in part must be here repeated ; draw any right line , as mabh , from any point therein as at b , raise a perpendicular , and upon b as a center , describe the quadrant ch , and prick the radius from b to a , from c to g , from h to f , and laying a ruler from a to f , and g , you will finde the points d and e upon the perpendicular cb , i say the said perpendicular is divided into a tangent line of three hours , and the halfs and quarters may be also divided thereon , by dividing the arches cf , fg , and gh into halfs and quarters , then from those subdivisions , laying a ruler to a , the halfs and quarters may be divided on , as were the whole hours . being thus prepared , draw the lines mb , le , kd , and ic , all parallel one to another , passing through the points b , 1 , 2 , 3. in this scheam they are perpendicular to bc , but that is not material , provided they pass through the same points , and are parallel one to another , yet notwithstanding the points a and h must be in a right line perpendicular to cb. this scheam thus prepared , i call the tangent scheam , because a line ruled any way over it , shall be divided also into a tangent of the like hours and quarters , whence it follows that one of these scheams may serve to inscribe the hour-lines into many dyals , which i shall next handle . to inscribe the hour lines in a horizontal dyal . having drawn the meridian line m , xii , and perpendicular thereto the hour-line of six , let it be observed that the sides of the horizontal dyal in page 8. to wit vi , ix , and vi , iii and ix , iii , are a right angled parallelogram , the one side whereof being the diameter of the circle being radius , the other side thereof must be made equal to the sine of the latitude , in that scheam the nearest distance from l to mf was the sine of the latitude or stiles height , the semidiameter of the inward circle , being radius , and that extent being doubled and pricked from m to vi on each side , as also from f twice to ix and iii , by those extents the parallelogram was bounded , the side iii , f ix , being parallel to the horizontal line . then take the extent mf from the horizontal dyal , and place it in the tangent scheam from m to o , and draw the line mo , and the respective divisions of the said line being cut by the parallels of the tangent scheam are the same with the divisions of the hour-lines on the inward sides of the horizontal dyal vi , ix , and vi , iii , and from the tangent scheam they are to be transferred thither with compasses . also place f , ix or f , iii from the horizontal dyal into the tangent scheam from m to n , and draw the line mn , which being cut by the parallels of the tangent scheam , the distances of those divisions from m are to be pricked down in the horizontal dyal , the first from f to xi , and i the second from f to x and ii , &c. to inscribe the hour-lines in a direct erect south dyal . the diameter of the circle in the south dyal in page 8 , is the same as in the horizontal , and the nearest distance from l to fm was the consine of the latitude , and was pricked twice on the horizontal line from m to vi on each side , whereby that inward parallelogram was limited . wherefore the divisions of the line mo in the tangent scheam , are the same with the hour distances in this dyal on the sides vi , iii , and vi , ix ; then for the divisions of the hours on each side of xii , take the extent xii , ix or xii , iii , and because it is less then the outward parallel distance of the sides of the tangent scheam , having therein made mi perpendicular to bm , place this extent from i to p , and draw the line mp , then prick the extent rl from the tangent scheam , from f to xi and i , and the extent qk , from f in the dyal , to x , and ii , and from the hour points so found , draw lines into the center at m , and they shall be the hour-lines required . to delineate an upright decliner in an oblique parallelogram . an upright south dyal declining 30 degrees west , latitude 51 degrees , 32 minutes . first draw the meridian or plains perpendicular cn , and upon c as a center , with the radius of the dyalling scheam , describe a south plained declin 40d east reclining 60d lat 51d 32′ page 64 a north plaine declin 60 west inclining 60d lat 51-32′ place this anywhere page 65 a south plaine declin 30d west latitude 51d 32′ a south diall declin 40d east inclin 15d lat 51d 32′ in latitudes under 45d the side of the square av must be assumed to be the cotangent of the latitude , the radius whereto will be af the tangent of the latitude , to which radius being prickt on the side of the square from the center , the sine of the declination must be taken out as before , and erected on the cotangent of the latitude , and this work must be performed on that side of the center on which the substile lyes . to fit in the parallelogram . produce the lines we , vd , and vl , in the dyalling scheam far enough , then assuming any extent to be radius , enter it on the lines vd and vl from the center to y and z , the nearest distance from y to ve is the cofine of the latitude to that radius which enter on the line of 6 or gc , so that one foot resting thereon , the other turned about may just touch cn , at the point found set h , and make ci on the other side equal to ch. the nearest distance in the dyalling scheam from z to ve , is the cosine of the declination to the former radius , which prick on the meridian line from c to l. and draw a line through l parallel to hi , and therein make lp , lq each equal to ch , and draw hp and iq , and there will be an oblique parallelogram constituted , the sides whereof will be tangent lines . nota , we might assume the point g in the hour-line of six , for the parallelogram to pass through , and the nearest distance from d to ve in the dyalling scheam , would be the cosine of the declination to the same radius to be prickt on the meridian line as before . to inscribe the hour-lines . in the following tangent scheam made as the former , produce cb , and make bq equal to bc , then take the extent pq on the dyal , and upon q as a center describe the ark y therewith , and draw the line yc just touching the extreamity , then you may proportion out the hours in this manner . if a line be drawn in the dyal from h to l , and from l to i , the hour-lines being drawn shall divide each of these lines into a double tangent , and consequently the hour-lines may also be prickt off on the said lines , after the method now prescribed . for upright far decliners and such plains as have small height of stile , recourse must be had to former directions for drawing them with a double contingent line , each at right angles to the substile . the foundation whereof is this , any point being assumed in the substilar line of a dyal , the nearest distance from that point to the stile , is the sine of the stiles height , the radius to which sine is the distance of the assumed point in the substilar line from the center of the dyal ; then in all dyals the hour distances from the substilar line are tangents of the angle of the pole , the sine of the stiles height being made the radius thereto , having finished the delineation of the dyal , the stile is to be placed directly over the substilar line , without inclining to either side of the plain , making an angle therewith equal to its height above the same , for the substilar line is elsewhere defined to be such a line over which the stile is to be placed in its nearest distance from the plain , therefore if the stile incline on either side , it will be nearer to some other part of the plain then the substilar line , whence it comes to pass in places near the equinoctial , if an upright plain decline but very little , the substile is immediately cast very remote from the meridian . another way to prick down the hour-lines in declining leaning plains . every such plain in some latitude or other will become an upright decliner : first therefore by the former directions prick off the substile , stile and meridian , in their true coast and quantity , and perpendicular to the meridian , draw a line passing through the center , and a south diall declin 40d east inclin 15d lat 51d 32′ a north diall declin 40d east reclining 75d lat 51d 32′ it shall represent the horizontal line of the plain in that new latitude as here vs ; from any point in the stile as k , let fall a perpendicular to the substile at i , and from the point i , in the substile let fall a perpendicular to the meridian at p. to finde the new declination . prick ip on the substilar line from i to r , and draw rk , so shall the angle irk be the complement of the new declination , and the angle ikr the new declination it self . to finde the new latitude . upon the center v with the radius vk , describe a circle , i say then that vp is the sine of the new latitude to that radius which may be measured in the limbe of the said circle , by a line drawn parallel to vs , which will intersect the circle at f , so is the arch sf the measure of the new latitude . to prick off the hour-line of six . this must be prickt off below the horizontal line , the same way that the substilar lyes : the proportion , is , as the cotangent of the latitude , is to the sine of the declination , so is the radius , to the tangent of the angle between the horizon and six . if rk be radius , then is vp the tangent of the new latitude , but if we make vp radius , then is rk the cotangent of the new latitude . wherefore prick the extent rk on the horizontal line from v to n , and thereon erect the sine of the declination to the same radius perpendicularly , as is na , and a line drawn into the center shall be the hour-line of six ; the proportioning out of the sine of the declination to the same radius , will be easily done , enter the radius vp from k to d , and the nearest distance from d to ik , shall be the sine of the new declination to that radius . to fit in the parallelogram . this is to be done as in upright decliners , for having drawn a line from the new latitude at f into the center , if any radius be entred on the said line from v , the center towards the limbe , the nearest distance from that point to vp the meridian line , shall be the cosine of the new latitude to that radius . again if the same radius be entred on rk produced if need be , the nearest distance to vr ( produced when need requires ) shall be the cosine of the new declination , and then the hour-lines are to be drawn as for upright decliners , nothing will be doubted concerning the truth of what is here delivered , if the demonstration for inscribing the requisites in upright decliners be well understood , it being granted that oblique plains in some latitude or other will become upright decliners . there are two examples for the latitude of london suited to these directions , in both which the letters are alike , the one for a south plain declining 40d eastwards , inclining 15d , the other for a north plain declining 40d east , reclining 75d . to finde a true meridian line . for the true placing of an horizontal dyal , as also for other good uses it will be requisite to draw a true meridian line , which proposition may be performed several ways , amongst others the learned mathematician francis van schooten in his late miscellanies demonstrates one , performed by help of three shadows of an upright stile on a horizontal plain , published first without demonstration in an italian book of dyalling by mutio oddi . but if all three be unequal , as let ac be the least , erect three lines from the point a , perpendicular to ap , ac , ad as is af , ag , and ah equal to the stiles height ae , and draw lines from the extreamities of the three shadows to these three points as are fb , gc , and hd ; then because ac is less then ab , therefore gc will be less then fb , by the like reason gc will be less then hd , wherefore from fb and hd cut off or substract fi and hk equal to gc , and from the points i and k let fall the perpendiculars il , km , upon the bases ab , ad , afterwards draw a line joyning the two points m , l , and from the said points let fall the perpendiculars ln equal to li , and mo equal to mk . then because the two shadows ab and ad are unequal , in like manner fb and hd will be unequal ; but forasmuch as fi and hk , are equal by construction , it follows that li , km , or ln and mo will be unequal , and forasmuch as these latter lines are parallel a right line that connects the points o and n , being produced will meet with the right line that joyns m , l produced , as let them meet in the point p , from whence draw a line to c , and it shall be a true line of east and west , and any line perpendicular thereto shall be a meridian line , thus the perpendicular aq let fall thereon , is a true meridian line passing through the point a , the place of the stile or wyre . whereto i adde that if mo & ln retaining their due quantities be made parallel it matters not whether they are perpendicular to ml or no , also for the more exact finding the point p , the lines mo and ln , or any other line drawn parallel to them , may be multipyled or increased both of them the like number of times from the points m and l upwards , as also from the points o and n downwards , and lines drawn through the points thus discovered , shal meet at p without producing either ml or on . see 15 prop. of 5 euclid . and the fourth of the sixth book . the greater part of van schootens demonstration is spent in proving that ml and on produced will meet somewhere , this for the reasons delivered in the construction i shall assume as granted , then understand that the three triangles abf , acg and adh stand perpendicularly erect on the plain of the horizon beneath them , upon the right lines ab , ac and ad , whence it will come to pass that the three points f , g , and h meet in one point , as in e the top of the stile ae , and that the right lines fb , gc and hd are in the conique surface of the shadow which the sun describes the same day by his motion , the top of which cone being the point e. wherefore if from those right lines we substract or cut off the right lines fi , gc and hk being each of them equal to one another , then will the points i , c and k , fall in the circumference of a circle , the plain whereof is parallel to the plain of the equator ; and therefore if through the points k and i such a position a right line be imagined to pass , and be produced to the plain of the horizon , it will meet with ml produced in the point p , where on being produced , will also meet with it ; so that the point p being in the plain of the circle , as also in the plain beneath it , as also the point c being in each plain , a right line drawn through the points p and c will be the common intersection of each plain , and the line pc will be parallel to the plain of the equator , and is therefore a true line of east and west , which was to be proved . on all plains though they decline , and recline , or incline , after the same manner may be found the line pc , which will represent the contingent line of any dyal , and a perpendicular raised upon the line cp shall be the substilar line , which in oblique plains is the meridian of the plain , but not of the place , unless they are both coincident from this manner of finding a meridian line on a horizontal plain , nothing else can be deduced without more scheams : from the three shadows , may be had the three altitudes , and the meridian line being given , the azimuths to those three shadows are likewise given , which is more then need be required in order to the finding of the latitude of the place , and the declination and amplitude of the sun , which because this scheam doth not perform of it self , i shall adde another to that purpose . by three altitudes of the sun , and three shadows of an index on an horizontal plain , to finde a true meridian line , and consequently the azimuths of those shadows , the latitude of the place , the suns amplitude and declination . let the three shadows be ca , the altitude whereto is af 22d , 28′ the second shadow cb , the altitude whereto is bg 59d , 21′ both these in the morning , the third shadow in the afternoon cd the — altitude whereto is — dh 18d , 20′ let the angle acb be 70d , and the angle bcd 135d , by following operations we shall finde that the shadow ca , is 10d to southwards of the west , that the shadow cb is 60d to northwards of the west , that the shadow cd is 15d southwards of the east . having from the three shadows prickt off the three altitudes to f , g , and h , from those points to the shadows belonging to them , let fall the perpendiculars fi , gk , hl , which shall be the sines of those altitudes , and the bases ic , kc , and lc shall be the cosines , from the point k in the greater altitude , draw lines to the points i and l in the lesser altitudes and produce those lines . from the points k and i , the sines of the two altitudes , are to be erected perpendicularly , thus km is made equal to kg , and in is made equal to if , then producing mn and ki , where they meet as at w , is one point , where the plain of the suns parallel of declination intersects the plain of the horizon ; in like manner on the base kl , the sines of two altitudes kg , and lh , ought to stand perpendicularly from the points k and l in their common base , but if they retain the same height , and are made parallel to one another , a line joyning the points of the tops of those sines produced , shall meet with the line joyning the points of their bases produced , in the same point as if they were perpendicular . thus kp and lq are drawn at pleasure through the points k , and l parallel one to another , and kg is the sine of the greater altitude , and lo is equal to the sine of the lesser altitude , and these two points being joyned with the line og produced , meets with the line kl produced , in the point e , another point where the plain of the suns parallel intersects the plain of the horizon . or you may double kg , and finde , the point p , as also double lo and finde the point q , a line drawn through p and q , findes the point e , as before , but with more certainty . if you joyn ew it shall be a true line of east and west , and a perpendicular let fall thereon from c the center , shall be a true meridian line to the perpendicular stile or wyer , as is cs . the arch sr is the suns amplitude from the north 50d 6′ the shadow being contrary to the sun , casts his parallel towards the south . draw kt parallel to sc , and it shall be the perpendicular distance between the sine of the suns greatest altitude , and the intersection of his parallel with the horizon . upon k erect kv equal to the sine of the suns greatest altitude of the three kg , and the angle ktv shall be equal to the complement of the latitude , for the angle between the plain of any parallel of declination and the plain of the horizon , is always equal to the complement of the latitude , and if upon t as a center with the radius cs , you describe the ark kx , the said ark shall measure the complement of the latitude in this example 38d 28′ which being given together with the sine of the amplitude cy , it will be easie to draw a scheme of the analemma , whereby to finde the suns declination , the time of rising , and setting , his height at six the azimuth thereto , the vertical altitude and hour thereto , &c. and many other propositions depending on the suns motion , as i have elsewhere shewed . the whole ground hereof is , that a right line extended through the tops of the sines of any two altitudes of the sun taken the same day before his declination very , shall meet with the plain of the horizon in such a point where the plain of the suns parallel intersects the plain of the horizon , and finding of two such points , a line drawn through them , must needs represent the intersection of those two plain ; in the former scheme the sines of the two altitudes are km and in , a line drawn through the bottomes of those sines , as ki extended shall be in the horizontal plain , and the line mn extended through the tops of those sines is in the plain of the suns parallel , as also in the horizontal plain ; now whether these lines stand erect or no is not material , provided they retain their parallelisme and due length , and pass through the points of their bases i , k , for the proportion of the perpendiculars to their bases , will be the same notwithstanding they incline to the horizon . to calculate the latitude , &c. from three shadows . a usual and one of the most troublesome propositions in spherical trigonometry , is from three shadows to finde the latitude of the place : thus maetius propounds it , and with many operations both in plain and spherical triangles resolves it . the first operations are to finde the suns 3 altitudes to those shadows , and that will be performed by this proportion , as the length of the shadow , is to the perpendicular height of the gnomon , so is the radius , to the tangent of the suns altitude above a horizontal plain , which proportion on other plains will finde the angle between the sun and the plain or wall . next maetius gives the distances between the points of the three shadows , and then by having three sides of a plain triangle , he findes an angle , to wit , the differences of azimuth between the respective shadows , which angle may be measured off the plain with chords . but propounding it thus , three altitudes of the sun above a horizontal plain , with the differences of azimuth between the three shadows belonging to those altitudes , being given , let it be required to finde the suns true azimuth , the latitude of the place , and the suns amplitude . and how this may be calculated from the former scheme , i shall now shew . in the triangle ick of the former scheme , the two sides ic and ck , represent two shadows , and the angle ick is the angle or difference of azimuth between them , and the said sides ic and ck , are the cosines of the altitude proper to those shadows ; now by seven operations in right lined triangles , we may finde the proper azimuth or true coast of any of those shadows . 1. in the right lined triangle ick , having the two sides ic and ck , with the angle between them ick , at one operation may be found both the other angles cik and ikc . 2. in the same triangle by another operation , may be found the side ik . then to proceed , draw nf parallel to ik , and fm will be the difference of the sines of both the altitudes belonging to those shadows , whereby may be found kw . 3. the proportion lyes , as fm the difference of the sines of both those altitudes , is to fn equal to ik before found , so is km the sine of the greater altitude , to kw sought . 4. 5. 6. by three like operations may be found in the other shadow triangle , the angles ckl , and klc , with the side ke . having proceeded thus far to the angle ikc , adde the angle ckl , the sum is equal to the angle wke . 7. then in the triangle wke , we have the two sides thereof wk , and ke given , and the angle comprehended by them , and at one operation we may finde both the other angles ewk , and kew , the complement of the angle twk , is the angle wkt . the difference between the angles wkt , and ikc , is the angle ckz , which shews the suns azimuth from the meridian proper to that shadow , which may be otherways found , for the difference between ekt and tkc , also shews it . by two other operations the latitude may be found . 1. as the radius : is to wk before found ∷ so is the sine of twk to kt : 2. as vk the sine of the greater altititude , is to kt before found , so is the radius , to the tangent of the latitude . by another operation may be found the amplitude cy , having found the angle zkc , the angle ckp is the complement thereof , then it holds : as the radius , is to ck the cosine of the greater altitude , so is the sine of the angle ckp to cp , the difference between which , and yp equal to tk , is yc the sine of the amplitude sought . to make any dyal from three shadows . the geometrical performance of the former proposition , is insisted upon by clavius in his book of the astrolable , but ▪ he mentioneth no method of calculation as derivable from it : from this proposition monsieur vaulezard a french mathematician educeth a general method for making of dyals , from three shadows of a gnomon stuck into a wall at randome , whereof he doth not so much as mention any demonstration ; i shall endeavour to deliver the method thereof with as much perspicuity as i can . every plain in some place or other , is an horizontal plain ; admit an oblique plain in our latitude , the substilar line represents the meridian of that place , and any contingent line drawn at right angles , thereto will represent a true line of east and west in reference to that horizon , and if the hours did commence at 12 , from each side the substile , the dyal here would shew the true time of the day there . in every oblique plain assuming any point in the stile , and crossing the stile with a perpendicular to that point , which shall meet with the substile , the point so found in the substile , is the equinoctial point , in respect of the assumed point in the stile ; hence we may inferre that if these two points and the substilar line were given the center of the dial might be easily found , now the former construction applyed to an oblique plain , assumed to be an horizontal plain , in respect of some unknown place , will find the equinoctial point and the substilar line , the stile point being assumed in the extremity of a gnomon any wayes placed or stuck in a wall at randome . from the former scheme it may be observed that the sine of any of the three altitudes being erected on the perpendicular between the foot of that sine and the suns parallel ▪ gave an angle equal to the elevation of the equinoctial above the horizon , which is the thing sought in the following work , but the sine of the greatest altitude performs the proposition best . if a stick or pin be stuck into a wall for this purpose , and doth not make right angles therewith , a perpendicular must be let fall from the extremity thereof into the wall , which is called the perpendicular stile , and the distances of the shadowes ; from the foot thereof must be measured thence , in respect of the tip of this perpendicular stile the equinoctial point must be found , wherefore it will be convenient to assume the said perpendicular stile to be the sine of the greatest of the three altitudes of the sun above the plain , which properly in respect of us are angles between the wall and the sun . up on this assumption it will follow , that the shortest shadow will be the cosine of that angle , and the distance between the tip of the stile , and the extremity of that shadow will be the radius : now from the lengths of the three shadows , and the height of the perpendicular stile , it will be easie to find the sines and the cosines of the angles between the wall and the sun . fig. 1a fig. 2a fig. 3a fig. 4a being thus prepared , draw three lines elsewhere as in the third figure meeting in a center , and making the like angles as the shadows did , and let them be produced beyond that center , and have the same letters set to them : make ch cg in this scheme , equal to the nearest distances from h i to cb , and make cl equal to ec , in the former scheme . draw the lines ig , ih produced ; and make hp go equal to nearest distances from i h to bg , and parallel to hp and go , draw the lines in , im , each of them made equal to cb , then draw the hipotenusals mo , np , meeting with the bases at k and l , draw the line kl , and it shall represent the plains equinoctial or contingent line . from i , let fall the perpendicular iq produced , and it shall be the substilar line on the plain and the point q is the equinoctial point sought . to place the substile . then repair to the plain whereon you would make the dyal represented by the first scheme , and place cq therein , so that it may make the same angle with the shadow ce , as it doth with the line ic in the third scheme , and it shall be the substilar line , which is to be produced , also prick iq from the third scheme from c to q on the plain in the substilar line , and upon the point c , perpendicular to the substile , raise the line ac , and make it equal to cb , drawing the line aq , then will the angle aqc be equal to the complement of the stiles height ; for it is the angle between the plain and the equinoctial ; just as before in the horizontal plain , the sine of the greatest altitude was erected on a line falling perpendicular from the foot of the said sine , to the intersection of the plain of the suns parallel with the plain of the horizon , and thereby gave the angle between the suns parrallel and the horizon , equal to the complement of the latitude . to find the center of the dyal and stiles height . if from the point a you raise the line av perpendicularly to aq , where it cuts the substilar line , as at v , is the center of the dial , and he line av represents the stile . 3. to draw the meridian line . if the plain recline , a thread and plummet hanging at liberty from the point b , and touching the plain , will find a point therein , suppose k a line drawn from k into the center of the dial , shall be the meridian line of the place . a broad ruler with a sharp pin at the bottome of it , being in a right line , with a line traced through the length of that ruler , whereto the plummet is to hang , having a hole cut therein for the bullet to play in , will find this point in a reclining plain . on an inclining plain , if the thread and plummet hang upon the stile at liberty , to some part of the stile more remote from the center , fasten another thread , which being extended thence to the plain just touching the former thread and plummet hanging at liberty , will find many points upon the plain , from any of which , if a line be drawn into the center it shall be the meridian line required ; see the fourth figure : or in either of these cases hold a thread and plummet so at liberty that it may just touch the stile , and bringing your sight so , as to cast the said thread upon the center at the same by the interposition of the thread ; the eye will project a true meridian-line on the plain , for the thread represents the axis of the horizon , and the plain of the meridian , is in the said axis . if the center fall inconvenient upon the plain , it will be necessary to draw the plains perpendicular , passing through the foot of the perpendicular stile c , and measure the angle of some of the shadowes from it , and accordingly so place it in the third figure or draught on the floor , on which find the center , and afterwards assign the center on the plain where it may happen convenient , and from the said draught , by help of the plains perpendicular , set off the substile and stiles height ; on plains on which probably the stile hath but small height , the perpendicular stile in this work must be assumed the shorter . 4. to draw the hour lines . these may be set off in a paralellogram after the substile , stile ' and meridian are placed which i have handled before . or they may be inscribed by the circular work , if we assume the perpendicular stile ac , to be the sine of the stiles height , the radius thereto will be av , prick the said radius on the substile from v the center to , z , and upon z as a center , describe the circle as in the first figure , and find the regulating point ☉ by former directions , over which from o the point where the meridian of the place cuts the said circle , lay a ruler , and it finds m , on the opposite side , being the point from whence the circle is to be divided into 12. parts , for the houres with subdivisions for the halfes & quarters . here note that the arch mv , is the inclination of meridians , the whole semicircle being divided but into 90d , and if that be first given ( as hereafter ) we may thereby find the point o , whence the meridian line is to be drawn into the center . 5. a method of calculation suited hereto . this is altogether the same as for finding the azimuth of the shadowes on the horizontal plain and as easie , whereby in the third figure , the angle of the substile ciq , must be set off from the shadow ice , just as the meridian line cp , might be set off from the shaddow ck on the horizontal . and the complement of the stiles height aqv , in the first figure here , as the complement of the latitude ktv , was found there . lastly , for placing the meridian line of the place by calculation , the substiles distance from the plains perpendicular , and the reclination of the plain must be found , which may be got easily at any time without dependence on the sun , and then in the often-mentioned oblique triangle in the sphere , we have two sides with the angle comprehended given , to wit , the complement of the stiles height , the complement of the reclination , and the substiles distance from the plaines perpendicular whereby may be found the inclination of meridians : and consequently the meridian line of the place , also the latitude thereof , with the plains declination , if they be required . to perform all this other wise geometrically and instrumentally , ( but not by calculation ) there was an entire considerable quarto treatise , with many excellent prints from brass-plates thereto belonging , printed at paris in france , in an. 1643. by monsieur desargues of lions , which treatise i have seen but not perused , a year after was published the small treatise of monsieur vaulezard before mentioned . the inscribing of the signes , azimuths , parallels of the longest day , &c. are lately handled by mr. leybourn in his appendix to mr. stirrups dyalling , as also by mr. gibson in his algebra , who thinks in many cases that they deform a plain , and are seldome understood by the vulgar , wherefore it will not be necessary to treat thereof . the directions throughout this book are suited to the northern hemisphere and are the same in the southern hemisphere , if the words south for north , and north for south be mutually changed . since the printing of this treatise , i have not had time to revise it with the copy , and so cannot give thee a full account of what faults may have escaped , which i think are not many ; these few following be pleased to correct . errata . page 17 line 3 upon m read upon c. p. 15 l. 34 for substilar sine r. substilar line . p. 25. l. 12 for 20d r. 15d . p. 36 l. 17 for 51′ . r. 31′ . finis . the description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of gunters quadrant applyed thereunto ... / contriv'd & written by j. brown, philomath. brown, john, philomath. 1661 approx. 191 kb of xml-encoded text transcribed from 105 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2004-08 (eebo-tcp phase 1). 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(eebo-tcp ; phase 1, no. a29756) transcribed from: (early english books online ; image set 97766) images scanned from microfilm: (early english books, 1641-1700 ; 1545:10) the description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of gunters quadrant applyed thereunto ... / contriv'd & written by j. brown, philomath. brown, john, philomath. [24], 168 p., [8] leaves of plates : ill., charts. printed by t.j. for j. brown and h. sutton, and sold at their houses, london : 1661. woodcut illustration of man sighting with sextant: t.p. verso. errata: p. 168. imperfect: pages stained and tightly bound with slight loss of print. reproduction of original in the british library. created by converting tcp files to tei p5 using tcp2tei.xsl, tei @ oxford. re-processed by university of nebraska-lincoln and northwestern, with changes to facilitate morpho-syntactic tagging. gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors. eebo-tcp is a partnership between the universities of michigan and oxford and the publisher proquest to create accurately transcribed and encoded texts based on the image sets published by proquest via their early english books online (eebo) database (http://eebo.chadwyck.com). the general aim of eebo-tcp is to encode one 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from proquest page images 2004-04 mona logarbo sampled and proofread 2004-04 mona logarbo text and markup reviewed and edited 2004-07 pfs batch review (qc) and xml conversion the description and use of a joynt-rule : fitted with lines for the finding the hour of the day , and azimuth of the sun , to any particular latitude ; or to apply the same generally to any latitude together with all the uses of gunters quadrant applyed thereunto , as sun-rising , declination , amplitude , true place , right ascension , and the hour of the night by the moon , or fixed stars ; a speedy and easie way of finding of altitudes at one or two stations ; also the way of making any kinde of erect sun-dial to any latitude or declination , by the same rule : with the description and use of several lines for the mensuration of superficies , and solids , and of other lines usually put on carpenters rules : also the use of mr. whites rule for measuring of board and timber , round and square ; with the manner of using the serpentine-line of numbers , sines , tangents , and versed sines . contriv'd & written by j. brown , philom . london , printed by t. i. for i. brown , and h. sutton , and sold at their houses in the minories , & thredneedle-street . 1661. to the reader . courteous reader , among the multitude of books which are printed and published , in this scribling age , some serious , some seditious ; some discovering or savouring of art , others of ignorance , possibly every one endeavoring to bring their male : among the rest of the crowd , i , like the widow , throw in my mite . if it be ( or seem to be ) little , it . is like the giver , and therefore i presume will of some be accepted , as little as it is ; and as little worth as it is , it is like enough to be challenged : but i shall endeavor to prevent prejudice , by the following discourse . having for some time been enquiring to find out a way , whereby work-men might on their rules ( their constant companions ) have a way easily and exactly to finde the hour of the day , and suns altitude and azimuth , and the like ; and have at several times for several men , at their request , used one and the other contrivance , to finde the hour : as that of the cillender , quadrant , or the like , as the altitude by a tangent on the inside of a square , or joynt-rule , and the line of sines on the flat side ; but still one inconvenience or other of trouble in adding of complements , or difficulty of taking of aititudes , or trouble to the memory , did accrue to the work ; or else the radius was small , and so much the more short of exactness : at last there came to my sight a quadrant made by mr. thomson , and as i was informed , was first drawn or contrived to that form , by mr samuel foster , that ingenious ●rtist , and laborious student , and reader of the mathematicks in gresham colledge : and considering of the ease and speed in the using thereof , i set my self to the contriving thereof to a more portable form , at last took some pains in delineating one , and another in several forms , and enquired after the uses thereof , and in effect have done , as mr. gunter with stofflers astrolabe , and nepeirs logarithms , and as mr. oughtred with gunters rule , to a sliding and circular form ; and as my father thomas brown into a serpentine form ; or as mr. windgate in his rule of froportion , and as of late mr. collins with mr. gunters sector on a quadrant , so may this not unfitly be called , the quadrant on a sector . and in fine , the invention will be valued for the learned authors sake , and never a whit the worse for the new contrivers sake : for first , hereby it is made large in little room , and as well on wood as on brass , which is an incommunicable property to broad quadrants , though of never so good matter , as experienced workmen know right well ; and by a tangent of 30 degrees laid together , is gotten all kinde of angles or altitudes under 90 degrees , and to be afforded for a low price , in comparison of other instruments , which will not perform the same operations any better . having made some joynt-rules in the manner following , and exposing them to sale , i have been many times solicited to write somewhat of the use , and now at last after near a years suspence , have committed the following discourse to publique view , partly to save the labor of tedious transcribings , and also to make so useful , cheap , and exact an instrument , ( if it be truly made ) to be more known or occupied . in which business , i desire to disclaim all vain-glorious os●entation , and therefore have nakedly and plainly asserted the manner how , and why it comes to be published to the world by me . it is a mechanical thing , and mechanically applied , and of mechanical men will be humanically accepted , i doubt not . having begun to write , i could not break off so short and abruptly , as at first i did intend to do ; therefore have added this short discourse of ordinary dyalling , the exact method of which i finde in no other author that ever i met with , ( and indeed i have not time to read many ) yet i dare presume , that for speed , ease , convenience , and exactness , inferior to none , especially the way of making far declining dyals ; as for other declining reclining dyals , i referre you to other authors , or to a discourse thereof by it self : if i finde encouragement , and ability to perform the same , a copy whereof i have had a long time by me , written by a very ingenious artist ) the demonstration of which dyals is most excellently and easily shewed by the figure inserted , page 77. as for the other part for taking of altitudes and angles , it may also be very conveniently done , if the rule be fitted to a three-leg staff , with a small ball-socket to set it level , or upright , as other surveighing-instruments be , as will be amply found , if a tryal be made thereof . that of ma●er white 's rule is a thing that hath given very good content to several gentlemen in the counties of essex , suffolk , and norfolk , and indeed is a very neat and accurate way of operation , well becoming a gentleman ; for while a workman shall take measure , his rule keeps the count of length , or breadth ; and having the length first given , the girt or squareness is no sooner agreed on , but you have the content without pen or compasses . as for the other lines , as decimal-board , and timber-measure , inches , and foot , in the way of reduction , girt-measure , circles , diameter , circumference , squares inscribed and equal : the use of them will be very grateful to many a learner . lastly , this brief touch of the serpentine-line i made bold to assert , to see if i could draw out a performance of that promise , that hath been so long unperformed by the promisers thereof . these collections , courteous reader , i have printed at my friends and my own proper charges , and if they prove to be ( as i do hope they will ) of publique benefit , i shall enjoy my expectation , and be ready at all times to serve you further , as i may , in these or other mathematical instruments , at my house at the sun-dyal in the minories , and remain to you much obliged , february 8 ▪ 1660. iohn brown. a table of the things contained in this book . chap. i. page the description of the rule for the hour onely 1. 2. 3 chap. ii. to rectifie or set the rule to his true angle for observation 4 to finde the suns altitude 5 to finde the hour of the day 6 to finde the suns rising and setting 7 to finde a level or perpendicular 7 chap. iii. a further description of the rule for hour and azimuth generally 10 , 11 , 12 chap. iiii. to finde the suns declination 13 to finde the suns true place , and right ascention 14 to finde the suns amplitude 14 chap. v. to finde the suns azimuth at any altitude and declination , in this particular latitude in summer 16 to finde the azimuth in winter , and equinoctial 17 , 18 chap. vi. to finde the hour of the night by the moon . 19 to finde the moons age 20 to finde the moons place 21 to finde the moons hour by the 11 chap. 2 and 3 proposition 23 to finde the true hour of the night thereby 24 chap. vii . to finde the hour of the night by the sixed stars 25 three examples thereof 26 , 27 , 28 chap. viii to finde the amplitude , azimuth , rising and so●●hing of the fixed stars , and examples thereof in page 29 , 30 , 31 , 32 chap. ix . to finde the hour and azimuth , &c. in any latitude 33 to finde the suns rising , setting , and ascentional difference ibid. to finde his amplitude in any latitude 34 to finde the suns altitude at six in any latitude . 35 to finde the hour when the sun is in the equinoctial 36 to find the hour in any l●titude , altitude , and declination 37 to finde the suns azimuth in any latitude , at any declination , and altitude in summer 38 to finde the same in winter 39 chap. x. to finde the inclination of meridians , substile , stile , and angle , between 12 and 6 for erect decliners three ways , one particular , and two general 40 to finde the substile , stiles elevation , inclination of meridians 41 to finde the angle between 12 and 6 for a particular latitude 42 to perform the same in general for any latitude by the general scale of altitudes 43 , 44 five canons to finde the same by the artificial sines and ta●gents , and how to work them on the rule 45 , 46 chap. xi . to draw a horizontal dyal to any latitude 47 to draw a vertical direct north or south dial to any latitude 48 , 49 to draw a direct erect east or west dial 50 , 51 chap. xiii . to finde the declination of a plain 52 to do it by the needle 53 , 54 to finde the quantity of an angle the sector or rule stands at 55 an example of the work 56 to finde a declination by the sun at any time 57 , 58 some precepts and examples for the same 59 , 60 , 61 to finde a declination at two choice particular times , viz. when the sun is in the meridian of place , or plain 62 , 63 how to supply a deficiency in one line of the rule , by another line on the rule 64 chap. xiiii . to draw a vertical declining dial to any declination and latitude . 65 to perform it another way 66 , 67 to supply a defect on the parallel contingent 68 chap. xv. to draw the hour-lines on an upright declining dial , declining above 60 degrees 96 to make the table for the hours 70 to finde the substile , stiles augmentation , and radius , to fit and fill the plain with any certain number of hours 71 , 72 , 73 , 74 , 75 an advertisement relating to declining reclining plains 76 , 77 , 78 , 79 chap. xvi . to finde a perpendicular altitude at one or two stationi 80 to do it at one station 81 to perform it at two stations 82 , 83 , 84 to work it by the sector lines 85 , 86 chap. xvii . the use of several lines inserted on rules for the use of several workmen , for the mensuration of superficial , and solid measure , and reduction , &c. 87 the use of inches , and foot measure laid together , in giving the price of one , to know the price of a hundred , or the contrary 88 the use in buying of timber , knowing the price of a load , or 50 foot , to know the price of one foot , or the contrary 89 , 90. the like work for the great hundred or 112 l. to the c. 90 , 91 the use of the line of decimal board measure 92 the use of the line of decimal timber measure 93 the use of the line of decimal yard measure 94 or that which agreeth with feet and inches 95 the use of the line of decimal round or girt measure 96 the use of the line of decimal sollid measure by the diameter 97 the table of decimal superficial under measure 98 , 99 the table for decimal sollid under measure 100 , 101 the table of under-yard measure for foot measure or inches 102 , 103 the table of under girt measure to inches 104 the table of under diameter measure to inches and quarters 105 a table of brick measure 106 the use of the lines of circumference , diameter , and squares equal and inscribed 108 chap. xviii . the use of mr. whites rule or the sliding rule in arithmetique , and measuring superficial and sollid measure from 110 to 118 chap xix . to finde hour and azimnth by that sector to finde the azimuth by having latitude , suns declination and altitudes , cemplements , and the hour from noon 119 to finde the hour , by having the same complements , and the azimuth from south 120 having the complements of the latitude and altitude , and the suns distance from the pole , to finde the suns azimuth 121 having the same complements , to finde the hour by the sector 122 having latitude , declination , meridian , and present altitude , to finde the hour of the day . 123 having latitude , declination , and altitude , to finde the suns azimuth at one operation by the sector 124 having the length of the shadow , to find the altitude , and the contrary 126 having latitude and declination , to find the suns rising and setting 127 to finde the suns altitade at any hour generally 129 to finde when the sun shall be due east or west 130 to finde the quesita in erect dials 131 , 132. chap. xx. the use of the serpentine line . the description thereof . 134 , 135 , 136 , 137 some observations in the use thereof 138. 139 , 140 to finde the hour of the day thereby , according to mr. gunter 142 to finde the hour thereby . according to mr. collins 145 to finde the azimuth thereby mr. collins his way 146 to finde the azimuth mr. gunters way 148 to finde the hour and azimnth at one operation , by help of the natural sines and versed sines 149 to 154 to finde the suns altitude at any hour or azimuth 155 , 156 to finde the hour by having the azimuth , and the contrary 157 five other useful propositions 159 to square and cube a number , and to finde the square , and cube root of a number 162 , 163 to work questions of interest and annuities 164 the use of the everlasting almanack 167 the right ascension and declination of 12 principal fixed stars in the heavens ; most of which are inserted on the rule : or if room will allow , all of them .   r. asc. declina . stars names h. m. deg. m. pleiades , or 7 stars 03 24 23 20 bulls-eye 04 16 15 48 orions girdle 05 18 01 195 little dog 07 20 06 08 lyons heart 09 50 13 40 lyons tayl 11 30 16 30 arcturus 14 00 21 04 vultures heart 19 33 08 00 dolphins head 20 30 14 52 pegasus mouth 21 27 08 19 fomahant 22 39 31 17s pegasus lower wing 23 55 13 19   1   3   5 7 4     6   8       moneths 9   11   2 10 12   1 2 3 4 5 6 7   8 9 10 11 12 13 14 days 15 16 17 18 19 20 21   22 23 24 25 26 27 28   29 30 31         week-days s m t w t f sat dom. letter d c b a g f e leap years 68 80 64 76 60 72 84 e●acts 26 9 12 25 28 11 23 the description and use of a joynt-rule . chap. i. the description of the lines on the rule , as it is made onely for one latitude , and for the finding the hour of the day onely . first open the ( joynt of the ) rule , then upon the head-leg , being next to your right hand , you have a line beginning at the hole , which is the center of the quadrantal lines , and divided from thence downward toward the head , into as many degrees as the suns greatest altitude in that latitude will be , which with us at london is to 62 degrees ; which line i call the scale of altitudes , divided to whole , halfs , and sometime quarters of degrees . 2. secondly , on the other leg , and next to the inside is the line of hours , usually divided into hours , quarters , and every fifth minute , beginning at the head with 4 , and so proceeding to 5 , 6 , 7 , 8 , 9 , 10 , 11 , and 12 at the end , and then back again with 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , for the morning and afternoon hours . 3. next to this is a kalendar of moneths and days in two lines ; the uppermost contains that half year the days lengthen in , and the lowermost the shortning days , as by the names of the moneths may appear ; the name of every moneth standing in the moneth , and at the beginning of the moneth : and all but the two moneths that have the longest , and the shortest days , viz. iune and december , are divided into single days , the tenth day having a figure 10 , or a point or prick on the head of the stroke , and the fifth onely a longer stroke without a prick , and the beginning of every moneth a long stroke , and every single day all alike of one shortness , according to the usual manner of distinguishing on lines . 4. and lastly you have a line of degrees , for so they be most properly called , and they are the same with the equal limb on quadrants , and serve for the same use , viz. for taking of altitudes , or horizontal angles , and are divided usually to whole , and half degrees of the quadrant , and figured with 30 , 40 , 50 , 600 , 7010 , 8020 , and 90 , just on the head , cutting the center or point , where the scale of altitudes and the line of hours meet ; which point , for distinction sake , i call the rectifying point . and the reckoning on this line , as to taking of altitudes , is thus : at the number 600 is the beginning , then towards the head count 10 , 20 , 30 , where the 90 is ; then begin at the end again , & count as the figures shew you to 90 at the head , as before . chap ii. the uses of the rule follow . 1. to rectifie or set the rule to his true angle . open the rule to 60 degrees , which is done thus , ( indifferently : ) make the lines on the head , and the lines on the other leg , meet in a streight line ; then is the scale of altitudes and the line of hours set to an angle of 60 degrees , the rectifying point , being the center of that angle ; or to do it more exactly , do thus : put one point of a pair of compasses into the rectifying point , then open the other to 10 , 20 , 30 , or 40 , on the scale of altitudes , the compasses so opened , and the point yet remaining in the rectifying point , turn the other to that margenal line in the line of hours , that cuts the rectifying point , and there stay it ; then remove the point that was fixed in the rectifying point , and open or shut the rule , till the point of the compasses will touch 10 , 20 , 30 , or 40 , being the point you set the compasses too in the scale of altitudes , in the innermost line that cuts the center , and the rectifying point , then is it set exactly to 60 degrees , and fitted for observation . 2. to finde the suns altitude at any time . put a pin in the center hole , at the upper end of the scale of altitudes , and on the pin hang a thread and plummet ; then if the sun be low , that is to say , under 25 degrees high , as in the winter it will always be , then lift up the moveable leg , where the moneths and the degrees be , till the shadow of the end fall just on the meeting of that leg with the head , then the thread shall shew the suns altitude , counting from 600 towards the head , either 10 , 20 , 25 , or any degree between . but if the sun be above 25 or 30 degrees high , lift up the head leg till the shadow of that play as before , or make the shadow of the pin in the center hole play on the innermost line of the scale of altitudes where the pin standeth , then the thread will fall on the degree , and part of a degree that his true altitude shall be . but if the sun be in a cloud , and can not be seen so as to give a shadow , then look up along by the head-leg , or moveable leg , just against the middle of the round body of the sun , and the thread playing evenly by the degrees , shall show the true altitude required . the like must you do for a star , or any other object , whose altitude you would find . 3. having found the suns altitude , and the day of the moneth , to finde the hour of the day . whatsoever you finde the altitude to be , take the same off from the line of altitudes , from the center downwards with a pair of compasses , then lay the thread ( being put over the pin ) on the day of the moneth , then put one foot of the compasses in the line of hours , in that line that cuts the rectifying point , and carry it further off , or nigher , till the other foot of the compass being turned about , will just touch the thred , at the nearest distance , then the point of the compasses on the line of hours , shall shew the true hour and minute of the day required . example on the 2. of july . 1. i observe the altitude in the morning , and i finde it to be 30 degrees high , then laying the thread on the day of the moneth , and taking 30 degrees from the scale of altitudes , and putting one point in the line of hours , till the other point turned about , will but just touch the th●ead , and i finde it to 23 minutes past 7 , but if it had been in the afternoon , it would have been 37 minutes past 4. 2. again , on the tenth of august in the afternoon , at 20 degrees high , i take 20 degrees from the scale of altudes , and laying the thread on the day of the moneth , viz. the tenth aforesaid , counting from the name at the beginning of august , toward september , and carrying the compasses in the line of hours , till the other point doth but just touch the thread , and you shall finde it to be 54 minutes past 4 a clock . 3. again , on the 11. of december at 15 degrees high , work as before , and you shall finde it to be just 12 a clock ; but to work this , you must lay the rule down on something , and extend the thread beyond the rule , for the nighest distance will happen on the out-side of the rule . 4. again , on the 11 of iune at noon i finde the altitude to be 62 degrees high , then laying the thread on the 10 th or 11 th of iune , for then a day is unsensible , and working as before , you shal finde the point of the compasses to stay at just 12 a clock , the time required for that altitude . 4. to finde the suns rising any day in the year . lay the thread on the day of the month , and in the line of hours it sheweth the true hour and minute of the suns rising or setting ; for the rising , count the morning hours ; and for the setting , count the evening hours . 5. to finde if any place lye level , or nor . open the rule to his true angle of 60 degrees , then set the moveable leg upon the place you would make level , and if the thread play just on 60 degrees , it is a true level place , or else not . 6. to try if any thing be upright or not . hang a thread and plummet on the center , then aply the head leg of the rule to the wall or post , and if it be upright , the thread will play just on the innermost line of the scale of altitudes , or else not . chap. iii. a further description of the rule , to make it to shew the suns azimuth , declination ; true place , right ascention , and the hour of day or night , in this , or any other lattitude . 1. first in stead of the scale of altitudes to 62 degrees , there is one put to 90 degrees in that place , and that of 62 is put by in some other place where it may serve as well 2. the line of hours hath a double margent , viz , one for hours , and the other for azimuths , & then every 5 th minute is more properly made 4 , or else every 2 minutes , and in a large rule to every quarter of a degree of azimuth , or to every single minute of time . 3. the degrees ought to be reckoned after 3 maner of wayes : first as before is exprest ; secondly from 60 toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , &c. to be so accounted in finding the azimuch for a particular latitude ; and and thirdly from the head or 90 , toward the end , with 10 , 20 , 30 , 40 , 50 , 60 , 70 , 80 , &c. for the general finding of hour and azimuth in any latitude , and many other problems of the sphere besides ; to which may be added , where room will alow , a line of hours , beginning at 6 at the head , and 12 at the end , but reckoning 15 degrees for an hour , and 4 minutes for every degree , it may do as well without it . 4. to the kalender of moneths and days , is added a line of the suns true place in the zodiack , or where room fails , the characters of the twelve signs put on that day of the moneth the sun enters into it , and counting every day for a degree , may indifferently serve for the use it is chiefly intended for . 5. under that is a line of the suns right ascension , to hours and quarters at least , or rather every fifth minute , numbred thus : 12 and 24 right under ♈ and ♎ , or the tenth of march , and so forward to the tenth of iune , or ♋ , where stands 6 , then backwards to 12 where you began , then backwards still to the eleventh of december , with 13 , 14 , 15 , 16 , 17 , 18 , to ♑ , then from thence forward to 24 where you first began : but when you are streightned for room , as on most ordinary rules you will be , then it may very well suffice to have a point or stroke , shewing when the sun shall gradually get an hour of right ascension , and from that for every day count four minutes of time , till it hath increased to an hour more , and this computation will serve very well ; and in stead of saying 13 , 14 , 15 hours of right ascension , say 1 , 2 , 3 , &c. which will perform the work as well , and reduce the time to more proper terms . 6. there is fitted two lines , one containing 24 houres , and the other 29 days , and about 13 hours , and they serve to finde the time of the moons coming to the south , before or after the sun , and by that , the time of high-water at london-bridge , or any other place , as is ordinary . chap. iv. the uses follow in order . 1. to finde the suns declination . lay the thread on the day of the moneth , then in the line of degrees you have the declination . from march the tenth toward the head is the declination northward , the other way is southward , as by the time of the year is discovered . example : on the tenth of april it is 11d 48 ' toward the north ; but on the tenth of october it is 10d 30 ' toward the south . 2. as the thread is so laid on the day of the moneth , in the line of the suns place , it sheweth that ; and in the line of the suns right ascension , his right ascension also , onely you must give it its due order of reckoning , as thus : it begins at ♈ aries , and so proceeds to ♋ , then back again to ♑ at the eleventh of december , then forwards again to ♈ aries , where you began . 3. to finde the suns right ascension in hours and minutes . lay the thread as before , on the day of the moneth , and in the line of right ascension you have the hour and minute required , computing right according to the time of the year , that is , begin at the tenth of march , or ♈ aries , and so reckon forwards and backwards as the moneths go . example . on the tenth of april the suns place is 1 degree in ♉ taurus , and the suns right ascension 1 hour 55 minutes : on the tenth of october 27d 1 / 4 in ♎ libra , and his right ascension is 13 hours and 42 minutes . 4. to finde the suns amplitude at rising or setting . take the suns declination out of the particular scale of altitudes , and lay it the same way as the declination is , from 90 in the azimuth scale , and it shall shew the amplitude from the east or west , counting from 90. example : may the tenth it is 33. 37. chap. v. having the suns declination , or day of the moneth , to finde the azimuth at any altitude required for that day . first finde the suns declination , by the first proposition of the fourth chapter , then take that out of the particular scale of altitudes , or scale to 62 degrees ; then whatsoever the altitude shall happen to be , count the same on the degrees from 60 toward the end of the rule , according to the second maner of counting , in the third proposition of the third chapter , and thereunto lay the thred , then the compasses set to the declination , carry one point along the line of hours on the same side of the thread the declination is ; that is to say , if the day of the moneth , or declination be on the right side the aequinoctial , then carry the compasses on the right side ; but if the declination be on the south side , that is , toward the end , ( counting from the tenth of march , or aries or libra , then carry the compasses along the line of hours and azimuths on the left side of the th●ead , as all win●er time it will be , and having set the compasses to the least distance to the thread , it sh●ll stay at the suns true azimuth from the south required , counting as the figures are numbred ; or from east or west , counting from 90. example 1. on the tenth of iuly i desire to finde the suns azimuth at any altitude , first on that day i finde the suns declination to be 20. 45 , which number count from the beginning of the particular scale of altitudes toward 62 , and that distance take between your compasses , then are they set for all that day ; then supposing the suns height to be ten degrees , lay the thread on 10 , counted from 60 toward the left end , then carrying the compasses on the right side of the thread , ( because it is summer or north declination ) on the line of azimuths , it shall shew 110. 40 , the azimuth from the south required ; but if you count from 90 , it is but 20. 40. from the east , or west point northward , according to the time of the day , either morning or evening . example 2. again , on the 14. of november , or the 6. of ianuary , when the sun hath the same declination south-ward , and the same altitude , to work this you must lay the rule down on something , then lay the thread on the altitude , counted from 60 toward the end ( as before ) and carrying the compasses on the south-side of the aequinoctial , along the azimuth-line , till the other point do but just touch the th●ead , and it shall stay at 36. 45 , the azimuth from south required ; if it be morning , it wants of coming to south ; if it be after-noon , it is past the south . example 3. but if the sun be in the aequinoctial , and have no declination , then it is but laying the thread to the altitude , and in the line of azimuths the thread shall shew the true azimuth required . as for instance : at 00 degrees of altitude , the azimuth is 90 , at 10 degrees it is 77. 15 , at 20 degrees 62. 45 , at 30 degrees high 43. 15 , at 35 degrees high 28. 10 , at 38 degrees 28 ' high , it is just south , as by practice may plainly appear . but if the suns altitude be above 45 , then the degrees will go beyond the end of the rule : to supply this defect , do thus : substract 45 out of the number you would have , and double the remainder , then lay the rule down with some piece of the same thickness , in a streight line with the moveable leg ; then take the distance from the tangent of the remainder doubled ( counted from 60 to the end of the rule , in the line next the edge ) to the center , lay that distance in the same streight line from the tangent doubled , and that shall be the tangent of the angle above 45 , whereunto you must lay the thread for the finding the azimuth , when the sun is above 45 degrees high . chap. vi. to finde the hour of the night by the moon . first by the help of an almanack , get the true time of the new moon , then compute her true place at that time , which is always the place of the sun ( very nigh ) at the hour and minute of conjunction ; then compute how many days old the moon is , then by the line of numbers say : if 29 dayes 13 hours , ( or on the line 29. 540 ) require 860 degrees , or 12 signs , what shall ●ny less number of days and part of a day require ? the answer will be : the moons true place at that age . having ●ound her true place , then take her al●itude , and lay the thred on the moons place found , and work as you did for the sun , and note what hour you finde ; then consider if it be new moon , the hour you finde is thētrue hour , likewise in the full ; but if it be before or after , you must substract by the line of numbers thus : if 29 days 540 parts require 24 hours , what shall any number of days and parts require ? the answer is : what you must take away from the moons hour found , to make the true hour of the night which was required . but for more plainness sake , i will reduce these operations to so many propositions , before i come to an example . prop. 1. to finde the moons age. first , it is most readily and exactly done by an ephemerides , such a one as you finde in mr. lilly's alman●ck , or ( as to her age onely ) in any book or sheet-almanack ; but you may do it indifferently by the epact thus , ( by the rules of the 152 page in the appendix to the carpenters rule . ) adde the epact , the moneth , and the day of the mone●h together , and the sum , if under 30 , is the moons age ; but if above , consider if the moneth have 30 or 31 days , then substract 29 or 30 out , and the remainder is the moons age in days . example . august 2. 1660. epact 28. month 6. day 2. added makes 36. now august or sixt moneth , hath 31 days , therefore 30 being taken away , 6 days remains for the moons age required . prop. 2. to finde the moons place . by the ephemerides aforesaid in mr. lilly's almanack , you have it ser down every day in the year ; but to finde it by the rule , do thus : count six days back from august 2. viz. to iuly 27. there lay the thread , and in the line of the suns place , you have the moons place required , being then near alike ; then in regard the moon goes faster than the sun , that is to say , in 29 days 13 hours , 12 signs , or 360 degrees ; in 3 days 1 sign , 6 degr . 34 min. 20 sec. in one day o signs , 12 degr . 11 min. 27 sec. in one hour , 30 min. 29 sec. ( or half a minute : ) adde the signs , and degrees , and minutes the moon hath gone in so many days and hours , if you know them together , and the sun shall be the moons true place , being added to what she had on the day of her change ; but far more readily , and as exactly by the line of numbers ( or rule of three ) say , if 29. 540. require 360. what 6 facit 73 1 / 4 , that is , 2 signs , 13 degrees , 15 minutes , to be added to 14 degrees in ♌ , and it makes 27 1 / 4 in ♎ , the moons place for that day . or thus , multiply the moons age by 4 , divide the product by 10 , the quotient sheweth the signs , and the remainder multiplied by 3 , sheweth the degrees which you must adde to the suns place on the day required , and it shall be the moons true place required for that day of her age. example . iuly 27. the sun and moon is in leo 14 degr . 0. august 2. being 6 days , adde the moons motion 2. 13d 9 ' . makes being counted , virgo ♍ , libra ♎ . 27 deg . 9 the place required ; which on the rule you may count without all this work or trouble : but for plainness sake i am constrained so to do . or thus . prop. 3. to finde the moons hour . to do this , you must do the work of the second chapter , and second and third proposition ; where note that the moons place found , is to be used as the day of the moneth , or suns declination . example . the moon being 27 degrees in libra , and 20 degrees high , i finde the our to be 31 ' past 9 , if on the east-side ; or 29 ' past 2 , if on the west-side ●f the meridian . prop. 4. to finde the true hour of the night . having found the moons hour , as before , consider the moons age , then say by the line of numbers , or rule of three , if 29. 540 part require 24 , what shall 6 days require facit 4 hours and 52 minutes ? which taken from 9. 31. rest ● . 29. the true time required . example . moons hour — 9 31 time to be subst . ● 52 true time remain 3 39. the time required . this work is done more readily by the two lines fitted for that purpose ; for look for 6 the moons age in one , and you shall finde ● hours 52 minutes , the time to be substracted in the other . chap. vii . to finde the hour of the night by the fixed stars . for the doing of this , i have made choice of twelve principal fixed stars , all within the two tropicks ; many more might be added , but these will very well serve the turn : the names of them , and their right ascension in hours and minutes , is set on the rule , and the star is placed in his true declination on , or among the moneths ; and for to know the stars next to a tutor , a celestial globe , or a nocturnal , of all the chief stars from the pole to the equinoctial , and to be had at the sun-dial in the mi●nories , is the best ; the uses whereof do follow . first know the star you observe , then observe his altitude , and laying the thread on the star by the second chapter , second and third proposition , get the stars hour , then out of the righ● ascension of the star , take the righ● ascension of the sun , ( found by the fifth proposition of the third chapter ) for that day , and note the difference for this difference added to the star hours found , shall shew the hour o● the night . example . on the first of november i observe the altitude of the bulls-eye , and find it to be 30. then by second and third of the second chapter , i finde the hour to be 7. 54 past the meridian the suns right ascension that day finde to be 15 hours 8 minutes , the stars right ascension is 4 hours 16 minutes ; which taken , the greatest from the least , by adding of 24 hours , re● 13. 08. then 7. 54. the stars hou● added , makes 21. 02. from which taking 12 hours , rest 9. 02. the hour o● the night required . for more plainness , note the work of two or thre● examples . stars right ascension being set on the rule — 4 — 16 suns right asc. nov. 1. — 15 — 08 — substraction being made , by adding 24 , remain — 13 — 08 to which you must adde the stars hour found — 7 — 54 then the remainder , taking away 12 hours , is — 9 — 02 the true hour . again may 15 by arcturus , at 50 0 ' high westwards . the right asc. of arcturus , is 14h● — 0 ' suns right asc. may 15. is — 4 — 10 — the right asc. of the sun taken from the r. asc. of the star , rest ● — 50 the stars hour at 50 degr . high , found to be — 02 — 13 — which being added to the difference before , makes — 1● — 03 or — 11 — 03. again january 5. by the great-dog at 15 degrees high east . the right asc. of great-dog , is 06-29 the right asc. of the sun , is 19 — 50 — substr . made , by adding 24 h. is 10-39 stars hour at 15 degr . high , is 9 — 32 ante m. or p. septen . which added to the difference found , and 12 substracted , remains 8. 11 ' . for the true hour of the night required to be found , and so of any other star se● down in the rule , as by the trial and pr●ctice , will prove easie and ready to the ingenuous practitioner . but by the line of 24 , or twice 12 hours , and the help of a pair of compasses , you may perform it withou● writing it down , thus : take the righ● ascension of the sun , out of that lin● of hours between your compasses ( being always counted under 12 ) an● set the same from the right ascensio● of the star , toward a lesser number , o● the beginning of the hours , and th● point shall stay at the remainder that is to be added to the stars hour found , then open the compasses from thence to the beginning of the hours , and adde that to the stars hour found , and it shall reach to the hour of the night required . example . feb. 6. 1660. by arcturus 20 degrees high : take 10 hours 1 minute between your compasses , and set it from 14 hours , or 2 hours beyond 12 , and it shall stay at 3. 59 : then take 3. 59 , and adde it to 6. 24 , the stars hour at 20 degrees high , and it shall be 10. 23 , the hour of the night required . chap. viii . to finde the amplitude or azimuth of the fixed stars ; also their rising , setting , and southing . 1. first for the amplitude , take the stars declination from the particular scale of altitudes , and lay it from 90 in the azimuth-line , and it shall shew his amplitude from east or west toward south or north , according to the declination , and time of day , morning or evening . the same work is for the sun . example . the bulls-eye hath 25. 54 degrees of amplitude , so hath the sun at 15 degrees 48 minutes of declination . 2. to finde the azimuth , work as you did for the sun at the same declination the star hath , by chapter 5. and you shall have your desire . example . december 24. at 6 degrees high , by the bulls-eye i finde the azimuth to be 107. 53. from the south . 3 to finde the stars rising and setting , lay the thread on the star among the moneths , and in the line of hours it shews the stars rising and setting , as you counted for the sun ; but yet note this is not the time in common hours , but is thus found : adde the complement of the suns ascension , and the stars right ascension , and the stars hour last found together , and the sun , if less than 12 ; or the remain 12 being substracted , shall be the time of his rising in common hours ; but for his setting , adde the stars setting last found to the other numbers , and the sum or difference shall be the setting . example . for the bulls-eye on the 23 of december , it riseth at 2 in the afternoon , and sets at 4. 46 in the morning . 4. to finde the time of the southing of any star on the rule , or any other whose right ascension and declination is known , substract the suns right ascension from the stars , increased by 24 , when you cannot do without , and the remainder , if less than 12 , is the time required , in the afternoon or night before 12 ; but if there remain more than 12 , substract 12 , and the residue is the time from mid-night to mid-day following . example . lyons-heart on the tenth of march , the suns ascension is 0 2 ' . lyons-heart whole right asc . is 9 50 ' time of southing is 9 48 ' at night . 5. to finde how long any star will be above the horizon . lay the thread to the star , and in the hour-line it sheweth the ascensional difference , counting from 90 ; then note if the star have north declination , adde that to 6 hours , and the sum is half the time ; if south , substract it from 6 , and the residue is half the time ; and the complement of each to 24 being doubled , is the whole nocturnal arch under the horizon . example . for the bulls-eye , his ascensional difference will be found to be one hour , 23 minutes , which added to 6 hours and doubled , makes 14. 46 , the diurnal ark of the star , and the residue from 24 is 9. 14. for the nocturnal ark , or the time of its being under the horizon . chap. ix . to perform the fore-going work in any latitude , as rising , amplitude , ascensional difference , latitude , hour , and azimuth , wherein i shall give onely the rule , and leave out the examples for brevity sake . 1. for the rising , and setting , and ascensional difference , being all one , do thus : take the suns declination out of the general scale of altitudes , then set one foot of the compasses in the colatitude on the same scale , and with the other lay the thred to the nighest distance ; then the thred so laid , take the nighest distance from the latitude to the thread , with that distance set one foot in the suns declination , counted from 90 toward the center , and the thread laid to the nearest distance , shall in the degrees shew the ascensional difference required , counting from 90 at the head toward the end of the rule ; and if you reduce those degrees and minutes to time , you have the rising and setting before and after 6 , according to the declination and time of the year . 2. to finde the suns amplitude . take the suns declination , and setting one foot in the colatitude , with the other lay the thread to the nearest distance , and on the degrees it sheweth the suns amplitude at rising or setting , counting as be●ore from 90 to the left end of the rule . 3. having amplitude and declination , to finde the latitude . take the declination from the general scale , and set one foot in the amplitude , the thread laid to the nearest distance in the line of degrees , it sheweth the complement of the latitude required , or the converse . 4. having latitude , suns declination and altitude , to find the height at 6 , and then at any other time of the day and year . count the declination in the degrees from 90 toward the end , thereto lay the thread , the least distance from which to the latitude in the general scale , shall be the suns height at 6 in the summer , or his depression in the winter . the compasses standing at this distance , take measure on the general scale of altitudes , from the beginning at the pin towards 90 , keeping one point there , open the other to the suns altitude , thus have you substracted the height at 6 , out of the suns altitude ; but in winter you must adde the depression at 6 , which is all one at the same declination with his height at 6 in summer , and that is done thus : put one point of the compasses so set in the general scale to the suns altitude , then turn the other outwards toward 90 , there keep it , then open the compasses to the beginning of the scale , then have you added it to the suns altitude ; having this distance , set one foot in the colatitude on the general scale , lay the thread to the nearest distance ; the thread so laid , take the nearest distance from 90 to the thred , then set one foot in the declination , counted from 90 , and on the degrees it sheweth the hour from 6 , reckoning from the head , or from 12 , counting from the end of the rule . i shall make all more plain , by making three propositions of it , thus : prop. 1. to finde the hour in the aequinoctial . take the altitude from the beginning of the general scale of altitudes , and set one foot in the colatitude , the thread laid to the nearest distance ( with the other foot ) in the degrees , shall shew the hour from 6 , counting from 90 , and allowing for every 15d 1 hour , and 4 min : for every degree . prop. 2. to finde it at just 6. is before exprest by the converse of the first part of the fourth , which i shall again repeat . prop. 3. to finde it at any time do thus . count the suns declination in the degrees , thereunto lay the thred , the least distance , to which from latitude in the general scale , shall be the suns altitude at 6 ; which distance in summer you must substract from , but in winter you must add to the suns present altitude ; having that distance , set one foot in the coaltitude , with the other lay the thread to the neerest distance , take again the neerest distance from 90 to the thread , then set one foot in the suns diclination counted from 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the hour required . example . at 10 declination north , and 30 high , latitude 51. 32 , the hour is found to be 8. 25 , counting 90 for 6 , and so forward . again at 20 degrees of declination south , and 10 degrees of altitude , i finde the hour in the same latitude to be 17 minutes past 9. having latitude , delination , and altitude , to finde the suns azimuth . take the sine of the declination , put one foot in the latitude , the thread laid to the neerest distance : in the degrees , it sheweth the suns height at due east or west , which you must in summer substract from the suns altitude , as before on the general scale of altitudes , with which distance put one foot in the colatitude , and lay the thread to the neerest distance , then take the neerest distance from the sine of the latitude , fit that again in the colatitude , and the thread laid to the heerest distance , in the degrees shall shew the suns azimuth required . 6. but in winter you must do thus : by the second proposition of the ninth chapter , finde the suns amplitude for that day , then take the altitude from the general scale of altitudes , and putting one point in colatitude , lay the thread to the neerest distance , then the neerest distance from the latitude must be added to the suns amplitude ; this distance so added must be set from the coaltitude , and the thread laid to the highest distance , and in the line of degrees , it gives the azimuth from south , counting from the end of the rule , or from the east or west , counting from the head or 90 degrees . example . at 15 degrees of declination and 10 altitude , latitude 51. 32. the azimuth is 49. 20. from the south , or 40 degrees and 40 ' from east or west . chap. x. to finde all the necessary quesita for any erect declining sun-dial both , particularly and general , by the lines on the dial side , also by numbers , sines , and tangents artificial , being logarithms on a rule . 1. first a particular for the substile . count the plains declination on ●he azimuth scale , from 90 toward the end , and thereunto lay the thread , in the line of degrees it shews the distance of the substile from 12. example . at 10 degrees declination , i find 7. 51. for the substile . 2. for the height of the stile above the substile . take the plaines declination from 90 in the azimuth line , but counted from the south end , between your compasses : and measure it in the particular scale of altitudes , and it shall give the height of the stile required . example . at 30 declination is 32. 35. 3. for the inclination of meridians : count the substile on the particuler scale of altitudes , and take that distance between your compasses , measure this distance on the azimuth line from 90 toward the end , and counting that way it sheweth the inclination of meridians required . example . at 15 the substile , the inclination of meridians will be found to be 24. 36. 4. to finde the angle of 6 from 12. take the plaines declination from the particular scale of altitudes , and lay it from 90 on the azimuth scale , and to the compasses point lay the thread : then on the line of degrees you have the complement of 6 from 12 , counting from 60 toward the end . note this rule ( as this line is drawn ) doth not give this angle exactly , neither will it be worth the while to delineate another line for this purpose . but if it be required it may be done , but i rather prefer this help , the greatest error is about the space of 45 minutes of the first degree in the particular scale of altitudes ; so that if you conceive those 45 minutes to be divided as the particular scale of altitudes is , like a natural sine , and if your declination be 30 , then take half the space of the 45 minutes less , and that shall be the true distance to lay on the azimuth line from 90 whereunto to lay the thread . example . a plaine declining 30 degrees , the angle will be found to be 32. 21. whose complement 57. 49. is the angle required . 5. to perform the same generally by the general scale of altitudes ; and first for the stile . lay the thred to the complement of the latitude , counted in the degrees from the head toward the end , then the nightest distance from the complement of the plaines declination to the thread , taken and measured on the general scale , from the center , shall be the stiles height required . 6. to finde the inclination of meridians take the plaines declination , from the general scale , and fit it in the complement of the stiles elevation , and lay the thread to the neerest distance , and on the degrees it sheweth the inclination of meridians required . 7. for the substile , count in the inclination of meridians on the degrees from 90 , and thereto lay the thread , then take the least distance from the latitudes complement to the thread , set one foot of that distance in 90 , and lay the thread to the neerest distance , and in the degrees it shall shew the substile from 12 required . 8. for the angle of 6 from 12. take the side of the square , or the measure of the parallel from 12 , and fit it in the cosine of the latitude , and lay the thread to the nighest distance , then take out the nearest distance from the sine of the latitude to the thread , then fit that over in the sine of 90 , and to the nearest distance lay the thread , then take the nearest distance from the sine of the plains declination to the thread , and it shall reach on the parallel line , or side of the square , from the horizon to 6 a clock line required . four canons to work the same by the artificial sines & tangents . inclination of meridians . as the sine of the latitude , to the sine of 90 : so the tangent of the declination , to the tangent of inclination of meridians . stiles elevation . as the sine of 90 , to the cosine of the declination : so the cosine of the latitude , to the sine of the stiles elevation , substile from 12. as the sine of 90 , to the sine of the declination : so the cotangent of latitude , to tangent of the substile from 12 ▪ for 6 and 12. as the co●tangent of the latitude , to the sine of 90 : so is the sine of declination , to the cotangent of 6 from 12. for the hours . as the sine of 90 , to the sine of the stiles height : so the tangent of the hour from the proper meridian , to the tangent of the hour from the substile . the way to work these canons on the sines and tangents , is generally thus : as first , for the inclination of meridians , set one point in the sine of the latitude , open the other to the sine of 90 , that extent applied the same way , from the tangent of the plains declination , will reach to the tangent of the inclination of meridians required . chap. xi . to draw a horizontal dyal to any latitude . first draw a streight line for 12 , as the line a b , then make a point in that line for a center , as at c , then through the center c , raise a perpendicular to a b , for the two six a clock hour-lines , as the line d e ; then draw two occult lines parallel to a b , as large as the plain will give leave , as d e , and e g then fit c d in the sine of the latitude , in the general scale , and lay the thread to the nighest distance , then take the nearest distance from 90 to the thread , and set it from d and e in the two occult lines , to f and g , and draw the line f and g parallel to the two sixes , ( or make use of the sines on the other side , thus : fit a d , or c d in the sine of the latitude , and take out the sine of 90 , and lay it as before from d and e , ) then fit d f , or e g in the tangent of 45 degrees , on the other side of the rule , and lay off 15 , 30 , and 45 , for every whole hour , or every 3 degrees and 45 minutes for every quarter , from d and e , toward f and g , for 7 , 8 9 , and for 3 , 4 , and 5 a clock hour points . lastly , set c d , or b e in the tangent of 45 , and lay the same points of 15 , 30 , 45 , both wayes , from b or 12 , for 10 , 11 , and 1 , 2 , and to all those points draw lines for the true hour-lines required , for laying down the stiles height ; if you take the latitudes complement , out of the tangent-line as the sector stood , to prick the noon hours , and set it on the line d f , or e g , from d or e downwards from d to h , it will shew you where to draw c h for the stile , then to those lines set figures , and plant the dial horizontal , and the stile perpendicular , and right north and south , and it shall shew when the sun shineth , the true hour of the day . note well the figure following . chap. xii . to draw a vertical , direct , south , or north dyal . first draw a perpendicular line for 12 a clock , then in that line at the upper end , in the south plain : and at the lower end in a north plain , appoint a place for the center , through which point cross it at right angles , a horizontall diall a south diall for 6 and 6 , as you did in the horizontal plain , as the lines a b , and c d , on each side 12 make two parallels , as in the horizontal , then take a d the parallel , and fit it in the sine of the latitudes complement , and take out the sine of 90 and 90 , and lay it in the parallels from d and c , to e and f , and draw the line e f , then make d e , and b e tangents of 45 , and lay down the hours as you did in the horizontal , and you shall have points whereby to draw the hour lines . for the north you must turn the hours both ways for 4 , 5 , 8 and 7 in the morning and 4 , 5 , 7 and 8 , at night the height of the stile must be the tangent of the complement of the of the latitude when the sector is set to lay off the hours from d , as here it is laid down from c to g , and draw the line a g for the stile . for illustration sake note the figure . chap. xii . to draw an erect east or west dial. first by the fifth proposition of the second chapter draw a horizontal line , as the line a b at the upper part of the plaine . then at one third part of the line a b , from a the right end if it be an east plaine , or from b the left end , if it be a west plain , appoint the center c , from which point c draw the semicircle a e d , and fit that radius in the sine of 30 degrees , ( which in the chords is 60 degrees ) then take out the sine of half the , latitude , and lay it from a to e , and draw the line c e for six in the morning on the east , ( or the contrary way for the west . ) then lay the sine of half the complement of the latitude , from d to f , and draw the line c f , for the contingent or equinoctial line , to which line you must draw another line parallel , as far an east diall . a west diall . assunder as the plaine will give leave , then take the neerest distance from a to the six a clock line , or more or less as you best fancy , and fit it in the tangent of 45 degrees , and prick down all the houres and quarters , on both the equinoctial lines , both ways from six , and they shall be points , whereby to draw the hoor lines by , but for the two houres of 10 and 11 there is a lesser tangent beginning at 45 , and proceeding to 75 , which use thus : fit the space from six to three , in the little tangent of 45 , and then and then lay of 60 in the little tangents from 6. to 10 and the tangent of 75 from 6 to 11 , and the respective quarters also if you please , so have you all the hour●s in the east , or west diall , the distance from six to nine or from six to three , in the west , is the height of the stile , in the east and west diall , and must stand in the six a clock line , and parallel to the plaine . chap. xiii . to finde the declination of any plain . for the finding of the declination of a plain , the most usual and easie way , is by a magnetical needle fitted according to mr. failes way , in the index of a declinatory ; or in a square box with the 90 degrees of a quadrant on the two sides , or by a needle fitted on the index of a quadrant , after all which ways , you may have them at the sign of the sphere and sun-dial in the minories , made by iohn brown. but the work may be very readily and exactly performed by the rule , either by the sun or needle in this manner following , of which two ways that by the sun is always the best , and most exact and artificial , and the other not to be used ( if i may advise ) but when the other failes , by the suns not shining , or as a proof or confirmation of the other . and first by the needle because the easiest . for this purpose you must have a needle well touched with the loadstone of about three or four inches long , and fitted into a box somewhat broader then one of the legs of the sector , with a lid to open and shut ; and on the inside of the lid may be drawn a south erect dial , and a wire to set the lid upright , and a thread to be the gnomon or stile to that dial : it will not be a miss also to extend the lines on the horizontal part for the same thread is a stile for that also . also on the bottom let there be a rabbit , or grove , made to fit the leg of the rule or sector ; so as being pressed into it , it may not fall off from the rule , if your hand should shake , or you cease to hold it there . this being so fitted , the uses follow in their order . put your box and needle on that leg of the rule , that will be most fit for your purpose , and also the north end of the needle toward the wall , if it be a south wall ; and the contrary , if a north , as the playing of the needle will direct you , better then the way how in a thousand words , then open or close the rule , till the needle play right over the north and south-line , in the bottom of the box : then the complement of the angle that the sector standeth at ( which may always be under 90 degrees ) is the declination of the plain . but if it happen to stand at any angle above 90 , then the quantity thereof above 90 , is the declination of the wall . to finde the quantity of the angle the sector stands at may be done two ways : first by protraction , by laying down the rule so set on a board , and draw two lines by the legs of the sector , and finde the angle by a line of chords . secondly , more speedily and artificially , thus : by the lines of sines being drawn to 2. 4 , or 6 degrees asunder . the sector so set , take the parallel sine of 30 and 30 , and measure it on the lateral sines from the center , and it shall reach to the sine of half the angle the line of sines stands at , being more by 2 , 4 , or 6 degrees then the sector stands at , because it is drawn one , two , or three degrees from the inside . or else take the latteral sine of 30 from the center , and keeping one foot fixed in 30 , turn the other till it cross the line of sines on that line next the inside , and counting from 90 , it shall touch at the angle the line of sines standsat , being two degrees more then the sector stands at ; the lines being drawn so , will be ( as i conceive ) most convenien . take an example . i come to a south-east-wall , and putting my box and needle on my rule , with the cross or north-end of the needle toward the wall , and the rule being applyed ●lat to the wall , on the edge thereof , on the evenest place thereof , and held level , so as the needle may play well with the head of the rule toward your right hand , as you shall finde it to be in an east-wall most convenient ; then i open or close the rule , till the needle play right over the north and south line in the bottom of the box ; then having got the angle , ( take off the box , or if you put it on the other side that labor may be saved . ) i take the parallel sine of 30 , and measuring it from the center it reaches , suppose to the sine of 20 , then is the line of sines at an angle of 40 degrees , but the sector at two degrees less , viz. 38 degrees , whose complement 52 is the declination ; then to consider which way , minde thus : first it is south , because the sun being in the south , shines on the wall . secondly consider , the sun being in the east , it shines also on the wal , therefore it is east plain : thus have you got the denomination which way , and also the quantity how much that ways . or if you take the latteral sine of 30 from the center , and turn the point of the compasses from 30 towards 90 , on the other leg you shall finde it to reach to the sine of 50 degrees , whose complement , counting from 90 , is 40 , or rather 38 , for the reason before-said , or else adde 2 to 50 , and you have the angle required , without complementing of it , being the true declination sought for . thus by the needle you may get the declination of any wall , which in cloudy weather may stand you in good stead , or to examine an observation by the sun , as to the mis-counting or mistaking therein ; but for exactness the sun is alwayes the best , because the needle , though never so good ▪ may be drawn aside by iron in the wall , and also by some kinde of bricks , therefore not to be too much trusted unto . to finde a declination by the sun. first open the rule to an angle of 60 degrees , as you do to finde the hour of the day , and put a pin in the hole , and hang the thread and plummet on the pin ; also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use . then apply the head leg to the wall , if the sun be coming on the plain , and hold the rule horizontal or level , then hold up the long thread till the shadow falls right over the pin , or the center hole , at the same instant the shadow shall shew on the degrees , how much the sun wants of coming to be just against the plain , which i call the meridian or pole of the plain , which number you must write down thus , as suppose it fell on 40 , write down 40-00 want : then as soon as may be take the suns true altitude , and write that down also , with which you must finde the suns azimuth ▪ then substract the greater out of the less , and the remainder is the declination required . but for a general rule , take this : if the sun do want of coming to the meridian of the place , and also want of coming to the meridian of the plain , then you must alway substract the greater number out of the less ▪ whether it be forenoon or after-noon , so likewise when the sun is past the meridian , and past the plain also . but if the signes be unlike , that is to say , one past the plain , or meridian , and the other want either of the plain , or meridian , then you must add them together , and the sum is the declination from the south . which rule for better tenaciousness sake take in this homely rime . signs both alike substraction doth require , but unlike signs addition doth desire . the further illustration by two or three examples . suppose on the first of may , in the forenoon , i come and apply the rule , being opened to his angle of 60 degrees to the wall , ( viz. the head leg , or the leg where the center is ) and holding up my thickest thread and plummet , so as the shadow of it crosseth the center , and at the same instant also on 60 degrees , then i say the sun wants 60 deg . of coming to the meridian of the plain ; at the same instant , or as soon as possible i can , i take the suns altitude , as before is shewed , and set that down , which suppose it to be 20 degrees , then by the rules before , get the suns azimuth for that day , and altitude ; which in our example will be found to be 94 degrees from the ( south or ) meridian , then in regard the signs are both alike , i. e. want , if you substract one out of the other , there remains 34 the declination required ; but for the right denomination which way , either north or south , toward either east or west , observe this plain rule : first , if the sun come to the meridian or pole of the plain , before it come to the meridian or pole of the place , then it is always an east-plain ; but if the contrary , it is a west-plain , that is to say , if the sun come to the meridian or pole of the place , before it comes to the meridian or pole of the plain , then it is a west-plain . also if the sum or remainder , after addition or substraction , be under 90 , it is a south-plain ; but if it be above 90 , it is a north-plain . also note , that when the sum or remainder is above 90 , then the complement to 180 , is always the declination from the north toward either east or west ; so that according to these rules in our example it is 34 degrees south-east . again in the morning , iune 13. i apply my rule to the wall , and i finde the sun is past the pole or meridian of the plain 10 degrees , and the altitude at the same time 15 degrees , the azimuth at that altitude , and day in this latitude , will be found to be 109 degrees want of south or pole of the place ; therefore unlike signs , and to be added , and they make up 119 degrees , whose complement to 180 is 61 ; for 61 and 119 added , make up 180 , therefore this plain declineth 61 degrees from the north toward the east . again the same day in the afternoon , i finde the azimuth past the south or meridian of the place 30 degrees , and at the same time the sun wants in coming to the meridian or pole of the plain 10 degrees , here by addition i finde the declination to be 40 degrees south-west . note what i have said in these three examples , is general at all times ; but if it be a fair day , and time and opportunity serve , to come either just at 12 a clock , when the sun is the meridian or pole of the place ; or just when the sun is in the meridian or pole of the plain , then your work is onely thus : first if you come to observe at 12 , then applying your rule to the wall , and holding up the thread and plummet , how much so ever the sun wants or is past the pole of the plain , that is the declination , if it be past it is east-wards ; if it wants , it is south-west-wards ; if neither , a just south plain , and then the poles , or meridians of place and plain , are the same . but secondly , if you come when the sun is just in the pole of the plain , then whatsoever you finde the suns azimuth to be , that is the declination ; if it wants of south , it declineth east-wards ; if it be past , it declineth west-wards . thus i have copiously , ( and yet very briefly ) shewed you the most artificial way of getting the declination of any wall , howsoever situated . note if the sun be above 15 degrees wanting of the meridian of the plain , your rule will prove defective in taking the plains meridian when the center leg is next the wall , then you must turn the other leg to the wall , and then you finde a supply for all angles to 45 degrees past the plain . but for the supply of the rest which is 45 degrees , do thus : open the rule till the great line of tangents & the outside of the leg make a right angle , for which on the head you may make a mark for the ready setting , then making the inside of the leg at the end of 45 , as a center , the tangents on the other leg supply very largely the defect of the othersides . or if you set on the box and needle on the rule , and open or shut the rule till the shadow of the thread shew just 12 , then the angle the sector stands at , is the complement of what the sum wants , or is past the meridian or pole of the plain . chap. xiv . to draw a vertical declining plain to any declination . first draw a perpendicular-line for 12 , as a b then design a point in that line for the center , as c , at the upper end , if it be a south plain ; or contrary , if it be a north plain , then on that center describe an arch of a circle on that side of 12 , which is contrary to the plains declination , as d e , and in that arch lay off from 12 the substile , and on that the stiles height , and the hour of 6 , being found by the tenth chapter , and draw those lines from c the center , then draw two parallels to 12 , as in the direct south : then fit the distance of the parallels in the secant of the declination , and take the secant of the latitude , and set it from the center c on the line of 12 , to f , and on the parallel from 6 to g , and draw a line by those two points f and g , to cut the other parallel in h : then have you found 6 , 3 , and 9 , then fit 6 g in the tangent of 45 , and prick off 15 , 30 , and the respective quarters both ways from 6 , for the morning , and afternoon hours , then fit f g in 45 , and lay off the same points from f both ways for 10 , 11 , 1 , and 2 , and the quarters also , if you please , and those shall be points to draw the hour-lines by . the stile must be set perpendicular over the substile , to the angle found by the rules in the tenth chapter , and then the dial shall shew the true hour of the day , being drawn fit to his proper declination . another way to perform this geometrically for all erect dyals with centers . when you have drawn a line of 12 , and appointed a center , make a geometrical square on that side 12 , as the stile must stand on , as a b c d , the perpendicular side of which square may also be the parallel as before . again , fit the side of the square in the cosine of the latitude , and take out the sine of the latitude , and fit that over in the sine of 90 , then take out the 〈◊〉 sine of the declination , and lay it from d to g for the hour of 6 , and draw the line a g for the 6 a clock hour line . then again , fit the side of the square ( or the distance of the parallel in the other way , when you want a secant , or your secant too little ) in the sine of 90 , and take out the cosine of the declination , fit that in the cosine of the latitude , then take out the sine of 90 , and lay it from the center on the line of 12 , and 〈◊〉 6 in the side of the square , and by those two points draw the contingent line ▪ and then fit those points or distances in the tangent of 45 , and lay down the hours as in the former part of this chapter ; but if you want the hours before 9 in west-decliners , or the hours after 3 in east-decliners , and the 6 fall too high above the horizontal line on that side , you may supply this defect thus : take the measure with your compasses from 3 to 4 and 5 , upwards in the west-decliners , or from 9 to 7 and 8 in east-decliners , and lay it upwards from 9 or 3 on the deficient side , in the parallel as it should have been from 6 downward in south or upward in north plains , and you shall see all your defective points to be compleatly supplied , whereunto draw the hour-lines accordingly . chap. xv to draw the hour-lines on an upright declining dial , declining above 60 degrees ▪ in all erect decliners , the way of finding the stile , substile 6 and 12 , and inclination of meridians , is the same ; but when you come to protract , or lay down the hour-lines , you shall finde them come so close together , as they will be useless , unless the stile be augmented . the usual way for doing of which , is to draw the dial on a large floor , and then cut off so much and at such a distance , as best serveth your turn , but this being not always to be affected , for want of conveniency , and large instruments , it may very artificially be done by a natural tangent of 75 or 80 degrees , fitted on the legs of a sector in this manner . the example i shall make use of , let be a plain declination 75 degrees , south , west . first on the north edge of the plain draw a perpendicular-line a b , representing 12 a clock line ; then on the center draw an occult arch of a circle as large as you can , as b d , therein lay off an angle of 37. 30 for the substile , ( though indeed this line will not prove the very substile , yet it is a parallel to it : ) then cross it with two perpendiculers for the two contingent lines , at the most convenient places , one in the upper part , and the other in the lower part of the plain , as the two lines c e , and f g do shew , then by help of the inclination of meridians , make the table for the hours in this manner ; for this declination , it is 78 : 09 , now every hour containing 15 degrees , and every quarter three degrees 45 ' i find that the substile will fall on neer a quarter of an hour past five at night , therefore if you take 75 , the measure of five hours out of 78. 9 , there remaines 03. 09. for the first quarter from the substile next 12. again , if you take 78. 09 , the inclination of meridians out of 78. 45 , the measure of 5 hours one quarter , and there remaines 00. 36. for the first quarter on the other side of the substile , then by continual adding of 03. 45 , to 03. 09 , and to 00. 36 , i make up the table as heer you see or else you may against 12 , set down 78. 09 , and then take out 3. 45 , as often as you can , till you come to the substile , and then what the remainder wants of . 3. 45 , must be set on the other side of the substile , and 3. 45 added to that , till you come to as late as the sun will shew on the plain , as here you see to 1 / 4 past 8. 45 36 . sub . 48 09 2 41 51 8 03 095 51 54 . 38 06 . 06 54. 55 39 . 34 21 . 10 39. 59 24 . 30 36 . 14 24. 63 09 1 26 51 7 18 094 66 54 . 23 06 . 21 54. 70 39 . 19 21 . 25 39. 74 24 . 15 36 . 29 24. 78 09 12 11 51 6 33 093     08 06 . 36 54.     04 21 . 40 39.     00 36 . 44 24.     the stile or gnomon is to be erected right over the substile , to the angle of 9. 16 , and augmented as much as from h to ● , and from i to ● , at the nighest distance from the point h and i , drawnin the substile line on the plain , this way is as easy , speedy , artificial and true , as any extant , ( if your scales be true , ) and improveth the plain at the first , to the most possible best advantage that can be . note also , if in striving to put as many hours as you can on the plain , the sum of the two natural numbers added together , comes to above 40000 : then you must reject something of one of them , for they will not be comely nor convenient , nor far enough asunder . i might have enlarged to declining , reclining plaines , out my intention is not to make a business of it , but onely to give a taste of the usefulness , and convenience of a joynt-rule , as now it is improved . an advertisement relating to dialling . these directions are sufficient for any horizontal , direct , and declining vertical dials , but for all other , as east or west inclining and reclining , direct , or reclining and inclining , and declining polars ; and all declining and inclining : and reclinimg plains , the perfect knowledge of their affections and scituations is very hard to conceive , but much more hard to remember by roat , or knowingly . a representation of which instrument take in this place , let the circle e s w n , represent the horizon s ae p n the meridian , ae w ae e the equinoctial , p c p the pole , being the points of an axes made of th●ead , and passing through the center c w c e being shaddowed , represents the plain , set to a declination , and to a reclination 45 degrees , f the foot that doth support it ; moreover the plain hat● an arch fitted to it , to get the stiles height , and to set it to any angle ; also the horizon and plain is made to turn round , which could not well be expressed in this figure , or representation of the instrument : a more liveley and easie help cannot likely be invented , and for cheapness the like may be in a half circle , or made of past-board . the use and application of which figure , i shall not now speak of , partly because of the facillity in the using thereof , and partly because of the difficulty in description thereof ; and lastly , wheresoever you shall buy the same , at either of the three places , the use will be taught you gratis . also note , that half of the instrument may be made either in b●ass , or pastboard , and be made to fold down in a book , and perform the uses thereof indifferently well , for many purposes , as to the affections of dials . then having discovered the affections , you may by the canno●s in the seventh chapter of mr. windgates r●le of proportion , finde all the requisites , and then to speed the laying down the hours , you may do it by help of the tangent of 45 , as you did in erect decliners ; for after you have drawn the hours of 6 , and three , or nine , and made a parallelogram by two lines parallel to 12 , and one parallel to six , then making the distance from six to nine , or three in the line parallel to 12 , a tangent of 45 , and the distance from 12 to 9 , or 3 in the parallel to 6 , likewise a tangent of 45 , in the sector , and laying off 15 , 30. 45 , or the respective quarters from 6 , and 12 , in those two lines , and they shall be the true points to draw the hour lines , by laying a ruler to them , and the center of the dial ; but for those that have no centers , the rule of augmenting the stile in far decliners , serveth for these also . chap. xvi to finde a perpendicular altitude at one or two stations , and observations by by the degrees on the rule . open the rule to his angle of 60 degrees , then looking up to the top or point of altitude you would observe the height of , as you do when you take the suns altitude , and note the degrees and parts the thread cuts , and write it down in chaulk or ink , that you forget it not . then measure from the place you stand to the foot or base of the object , ( being right under the top of the object , whose height you would measure ) in feet , yards , or any other parts . thirdly , consider this being a right angled plain triangle , if you have the angle at the top , the angle at the base is always the complement thereof . these things being premised , the proportion holds , as the fine of the angle opposite to the measured side , ( or base ) being the complement of the angle found , is to the base or measured side : so is the angle found , to the height required . always remember to adde the height of your eye from the ground , ( at the time of taking the angle ) to the altitude found . for the operation of this , extend the compasses from the sine of the complement of the angle found to the number of the measured side , on the line of numbers that distance applied the same way , from the sine of the angle found , shall reach on the line of numbers to altitude required . example at one station . i open my rule , and hang on the thread and plummet on the center , and observing the angle at c , i finde it to be 41. 45 , and the angle at b the complement of it 48. 15 , and the measure from c to a 271 feet : then the work being so prepared , is thus : as the sine of 48. 15 , is to 271 the measure of the side opposite to it : so is the sine of the angle 41. 45 , to 242 the measure of the side a b , opposite to the angle at c , the height required . again , at the station d , 160 foot from a , i observe and finde the angle d to be 56. 30 , the angle at a is the complement thereof , viz. 33. 30. this being prepared , i extend my compasses from the sine of 33. 30 to 160 , on the line of numbers the same extent will reach from the sine of 56. 30 , to 242 on the line of numbers , lacking a small fraction , with which i shall not trouble you . an example at two stations . as the sine of the difference , which is the angle c b d 14. 45 , is to the side measured , viz. d c 111 feet on the numbers : so is the sine of the angle at c 41. 45 , to the measure of the side b c , the hypothenusa , or measure from your eye to the top of the object , viz. 290 feet . again for the second operation . as the sine of 90 the right angle at a , is to 290 the hypothenusa b d : so is the sine of 56. 30 , the angle at the first station d , to 242 , the altitude b a , the thing required : : so also is the sine of the angle at b 33. 70 , the complement of 56 , 30. to 160 the distance from d to a. to perform the same by the line of sines , drawn from the center on the flat-side , and the line of lines , or equal parts or inches in ten parts . to work these or any other questions by the line of natural sines and tangents , on the flat-side , drawn from the center , it is but changing the terms , thus : as the measured distance taken out of the line of lines , or any scale of equal parts , is to the sine of the angle , opposite to that measured side , fitted across from one leg to the other , the sector so standing , take out the parallel sine of the angle opposite to the enquired side , and that measure shall reach on the line of equal parts , to the measure of the altitude requi●●● ▪ example as before . take out of the lines or inches 2. 71 , and fit it in the sine of 48. 15 , across from one legge to the other , which i call a parral sine , ( but when you measure from the center onwards the end , i call it a latteral sine , ) then take out the parallel sine of 41. 45 , and measure it on the line of inches , or equal parts , and it shall reach to 2 inches 42 parts , or 242 , the altitude required . after the same manner may questions be wrought on the line of lines , sines , or tangents alone , or any one with the other , by changing the logarithmetical canon from the first to the second , or third , and the second or third to the first to second ; as the case shall require , from a greater to a less , and the contrary ; for the fourth is always the same ; of which in the use of the sector , by edmund gunter , you may finde many examples , to which i refer you . also without the lines of sines either natural or artificial , you may find altitudes , by putting the line of quadrat , or shadows , on the rule as in a quadrant , then the directions in the use of the quadrat , page ▪ 146 of the carpenters rule , will serve your turn , which runs thus : as 100 ( or 50 according as it is divided ) to the parts cut by the thread , so is the distance measured , to the height required ; which work is performed by the line of numbers onely . or again , as the parts cut , to 100 or 50 : so is the height to the distance required . but when the thread falls on the contrary shadow , that is , maketh an angle above 45 , then the work is just the contrary to the former . what is spoken here of taking of altitudes , may be applied to the taking of distances ; for if the sector be fitted with a staff , and a ball-socket , you may turn it either horizontal , or perpendicular , and so take any angle with it , very conveniently and readily by the same rules and directions as were given for the finding of altitudes . chap. xvii . the use of certain lines for the mensuration of superficial and solid bodies , usually inserted on ioynt-rules for the use of work-men , of several sorts and kindes . first the most general and received line for mensuration of magnitudes , is a foot divided into 12 inches , and those inches into 8 , 10 , 12 , or more parts ; but this being not so apt for application to the numbers , i shall not insist of it here , but rather refer you to the carpenters rule ; yet nevertheless those inches , laid by a line of foot measure , doth by occular inspection onely , serve to reduce foot measure to inches , and inches likewise to foot measure , and some other conclusions also . 1. as first , the price of any commodity at five score to the hundred , either tale or weight , being given , to finde the price of one in number , or one pound in weight . as suppose at two pence half-peny a pound , ( or one ) i demand to what cometh the hundred weight , ( or five-score , ) counting so many pound to the hundred weight ? if you look for two inches and a half , representing two pence two farthings , right against it on the foot measure , you have 21 very near ; for if you conceive the space between 20 and 21 , to be parted into 12 parts , this will be found to contain ten of them , for the odde ten pence . but for the more certain computation of the odde pence , look how many farthings there is in the price of one pound , twice so many shillings , and once so many pence is the remainder , which if it be above 12 , the 12 or 12s , being substracted , the remainder is the precise number of pence , above the shillings there expressed ; and on the contrary , at any price the c hundred , or 5 score , to finde the price of one , or 1l . as suppose at 40s . the c. or 5 score , look for 40 in the foot measure , and right against it in the inches , you have 4 inches , 3 quarters , and 1 / 4 of a quarter , which in this way of account is 4 pence 3 farthings , and about a quarter of one farthing . thus by the lines , as they are divided , it proceeds to 12 pence a pound ; but if you conceive the inches to be doubled , and the foot measure also , you shall have it to 24 d. or 48 d. the pound , or one in tale , of any commodity . as at 18 d. a piece , or pound , the price comes to 7l . 10s . the c. for then every ten strokes is 20s in the foot measure , and every inch is 2 pence , and every eighth one farthing . 2. secondly , for the buying of timber at 50 foot to a load , at any price the load , how much a foot . here in resolving this , the inches are to be doubled , and the foot measure taken as it is : as at 40 shillings the load , 40 in the foot measure stands right against 4 inches 3 quarters and better , which being doubled , is 9d . 2 far . 1 / 2 far . near , for the price of one foot ; and on the contrary , at 5d . a foot , is 41s . 8 d. a load , &c. 3. for the great hundred of 112l . to the hundred , let the space of 12 inches be divided into 112 parts , then the like rule holds for that also . for the inches being divided into quarters , every quarter is a farthing , and every eighth half a farthing , and every division of the 112 is a shilling , and every alteration of a farthing in the price of a pound , makes a groat in the hundred , as thus : at 3 pence a pound is 28s . the c. at 3d. 1 q. a pound , 30s . and 4d . the c. at 3d. 1 / 2 the pound , 32s . 8 d. the c. at 3 d. 3 farthings , 35s . the c. thus you see that every fraction at a farthing advance , is 4 pence in the hundred ; but for any other account , as 3 pence farthing half farthing , then count the fraction , as 1 , 12th part of a shilling , and nearer you cannot come by a bare occular inspection ; but the price of the hundred being given , the price of the pound you have as near by this occular inspection , as any usual coin is reducable , viz. to the 32 part of a peny , or nigher if you please . again note here also , you may double , or quadruple the price : as to 24d . or 48d . the pound , or any price between . as for example : at 13d . a pound , is 6l . 1s . 4d . the c. at 32d . or 2s . 8d . the pound , is 14l . 19s . the c. and the like by dupling and quadrupling the inches , and the 112 parts , that layeth by it . 4. these lines of equal parts serve as scales , for the protracting of any draught of house or field , or the like ; also for addition or substraction of any small number . 5. note that the line of foot measure , may be applied for the reducing of any odde fraction to a decimal fraction , as you may fee it in page 64. of mr. windgats arithmetick made easie . 2. the use of the lines of decimal timber and board measure . the lines of decimal timber and board measure , are fitted to agree with the tenths , or foot measure , as those lines in the first chapter of the carpenters rule , are fitted to the inches , and the use of them is thus : and first for the decimal board measure . suppose a board is 1 foot 50 broad , i look for 150 on that line , and from that place to the end of the rule forwards , toward 100 , so much in length must you have to make a foot of superficial or board measure . 2. or else thus : if you apply the end of the rule next 100 , to one edge of the breadth of a board , or glass , then right against the other edge of the board , on that line of decimal board measure , you shall finde the 10ths and 100s , ( or feet 10ths & 100 ) parts of a foot , that you must have in length to make a foot superficial at that breadth . example . i come to a board and applying the upper end next 100 , even to one edge of the board , the other reacheth to 0. 8 tenths , then i say that 8 tenths of a foot length at that breadth , makes a foot . 3. the use of decimal timber measure . the use of this is much like the board measure , onely here you must have a respect to the squareness of the piece , and not to the breadth onely ; for after you know how much the piece of timber or stone is square , in feet and 100 parts , then look that number on the line of decimal timber measure , and from thence to the end of the rule , is the length that goes to make a foot of timber . example . at 14 , or 1. 40. parts of a foot square , look the same on the rule , and from thence to the end where 40 is , is the length of a foot of timber , at that squareness , being about 51 parts of a foot divided into a 100 parts . 5. the use of the line of decimal yard measure , also running yard measure , according to the inches or decimal parts of a foot . the decimal yard measure , is nothing else but a yard or 3 foot divided into a 100 parts , and used in the same manner as the foot measure is , for if you take the length , and the breadth in that measure , and multiply it together , you shall have the content , in yards and 100 parts of a yard . example . suppose a peece of plastering is 4 yards 78 parts one way ; and 7. 35 parts another way : being multiplied together makes 35 yards , and 9954. of 10000 , which is very neer 36 yards . 5. but the decimal running yard measure is fitted to the foot measure , and the use is thus : suppose a room is to be measured that is 7 foot 8 tenths high , and i would know how much makes a yard , at that breadth or height ; look foor 7f . 8 10ths on the line of decimal running yard measure , and the space on the rule , from thence to the end next 100 is the true length , that goeth to make up a yard of superficial measure , at that breadth or height . but if the peece be between 4 foot 5 10 broad , and 2 foot , then the table at the end of the line , will supply the defect : or you may change the terms , and call the length the breadth , and the contrary . but if it be under 2 foot broad , then if you do as you did with the board measure , you shall have your desire . example . at 1 foot 3 10th broad , 6 foot 9 10ths make a yard . 6. but if the running yard measure be made to agree with the inches , then measure the height of the room in feet and inches , and if you take a pair of compasses , and measure from that place , to the end of the rule , then turn the compasses set at that distance as many times as you can about the room , so many yards is there in the room . 7. the use of the line of decimal round measure , commonly called girt-measure , which is when the circumference of a round cillender , or piller given in inches or ten parts of a foot . first ( for girt-measure according to inches , being the most usual measure ) now much the pillar is about , then look for the same number on the line of girt-measure , and from thence to the end of the rule , is the length that goeth to make a foot of timber . but if it be under 30 inches about , then you must have above two foot in length , and then a table at the end of the line , or a repetition in another line , will supply the defect . but if the line of girt-measure be divided according to foot measure , then use it as before , seeking the decimal part on the line , and from thence to the end is a foot . 8. the use of a line of solid measure , by having the diameter of a round piece given in inches , or foot measure . take the diameter with a rule , or a pair of callipers , and learn the measure either in inches , or foot measure according as your line of diameter is divided . then look for the same number on the line of diameter , and from thence to the end of the rule forward , is the length that makes a foot of timber at that diameter ( or measure cross the end of the round piece of timber or stone . ) the tables of all the under measure for all these lines follow . decimal superficiall under m.   10th . f. 1000   10 f. 1000 p.   1 100. 00     3. 848   2 50. 000     3. 706   3 33. 300     3. 570   4 25. 000     3. 450   5 20. 000 3 3 3. 332   6 16. 600     3. 217   7 14. 300     3. 115   8 12. 500     3. 025   9 11. 120     2. 940 1 1 10. 000   5 2. 850   1 9. 100   6 2. 780   2 8. 340   7 2. 700   3 7. 720   8 2. 628   4 7. 150   9 2. 560   5 6. 670   4 2. 500   6 6. ●60     ● . 440   7 5. 888     2. 382   8 5. 5●5     2. 336   9 5. 260     2. 273 2 2 5. 000     2. 213     4. 760     2. 173     4. 5●6     2. 127     4. 350     ● . 083     4. 170     2. 042   5 4 ▪ 000   5 2. 000 decimall superficiall . m.   f. 1000. p. 01 f. 1000. p.   1. 962   1. 320   1. 923   1. 304   1. 816   1. 286   1. 850   1. 268   1. 820 8 1. 250   1. 785   1. 237   1. 756   1. 220   1. 726   1. 207   1. 697   1. 192 6 1. 669   1. 178   1. 640   1. 164   1. 615   1. 151   1. 589   1. 138   1. 563   1. 125   1. 538 9 1. 112   1. 516   1. 100   1. 493   1. 087   1. 472   1. 076   1. 450   1. 063 7 1. 430   1. 052   1. 409   1. 041   1. 391   1. 030   1. 373   1. 020   1. 353   1. 011   1. 337 10 1. 000 decimal solid under measure .   f. 1000. p. 10 f. 1000. p. 1 10000. 000   14. 805 2 2500. 000   13. 735 3 1100. 000   12. 780 4 630. 000   11. 916 5 400. 000 3 11. 125 6 277. 900   10. 415 7 200. 430   9. 760 8 150. 660   9. 125 9 120. 350   8. 625 1 100. 0000   8. 150   82. 800   7. 700   96. 500   7. 310   59. 390   6. 900   51. 100   6. 565   44. 500 4 6. 250   39. 150   5. 945   34. 650   5. 664   30. 850   5. 404   27. 750   5. 465 2 25. 000   4. 938   22. 700   4. 720   20. 675   4. 530   18. 920   4. 342   17. 400   4. 162   16. 000 5 4. 000 decimall solid under measure .   f. 1000. p. 01. f. 100. p.   3. 825   1. 738   3. 7●0   1. 694   3. 524   1. 651   3. 430   1. 608   3. 310 8 1. 568   3. 188   1. 528   3. 078   1. 493   2. 968   1. 458   2. 873   1. 420 6 2. 780   1. 390   2. 688   1. 356   2. 602   1. 323   2. 521   1. 297   2. 442   1. 266   2. 366 9 1. 236   2. 294   1. 208   2. 227   1. 185   2. 160   1. 160   2. 100   1. 131   2. 043   1. 109 7         1. 985   1. 084   1. 93●   1 , 061   1. 878   1. 041   1. 830   1. 021   1. 781 10. 1. 000 vnder yard-measure for feet and inches from one inch'to four feet six inches f. f. 1000. f. f. 1000. in.   in.   1 108.000   3. 850 2 54. 000   3. 720 3 36. 000   3. 600 4 27. 000 6 3. 482 5 21. 600   3. 373 6 18. 000   3. 271 7 15. 420   3. 175 8 13. 520   3. 085 9 12. 000 3 3. 000 10 10. 300   2. 922 11 9. 820   2. 842 1. 9. 000   2. 769   8. 320   2. 710   7. 740   2. 633   7. 201 6 2. 572   6. 760   2. 512   6. 350   2. 455 6 6. 000   2. 400   5. 680   2. 345   5. 400   2. 298   5. 140 4 2. 250   4. 906 1 2. 203   4. 695 2 2. 160 2 5. 500 3 2. 119   4. 320 4 2. 073   4. 160 5 2. 037   4. 000 9 2. 000 vnder yard measure according to decimal or foot measure f. 10. f. 1000. p.   f. 1000. p. 1 90. 000 4 3. 7●0 2 45. 000 5 3. 600 3 30. 000 6 3. 461 4 22. 500 7 3. 332 5 18. 000 1 3. 211 6 15. 000 9 3. 104 7 12. 880 3 3. 000 8 11. 200 1 2. 903 9 10. 000 2 2. 812 1 9. 000 3 2. 728 1 8. 190 4 2. 648 2 7. 510 5 2. 572 3 6. 930 6 2. 502 4 6. 430 7 2. 435 5 6. 000 8 2. 370 6 5. 625 9 2. 310 7 5. 290 4 2. 250 8 5. 000 1 2. 195 9 4. 735 2 2. 142 1 4. 500 3 2. 093 1 4. 285 4 2. 046 2 4. 092 5 2. 000 3 3912     vnder girt-measure . inc. about f. in . 100.   f. in . 100 1 1809.6.81 24 3.1.87 2 452. 4. 74 25 2. 10. 74 3 201. 0. 77 26 2. 8. 12 4 113. 1. 18 27 2. 5. 87 5 72. 4. 60 28 2. 3. 70 6 50. 3. 19 29 2. 1. 83 7 39. 3. 22 30 2. 0. 13 8 28. 4. 00 31 1. 10. 60 9 22. 4. 09 32 1. 9. 21 10 18. 1. 15 33 1. 7 : 94 11 14. 11. 46 34 1. 6. 78 12 12. 6. 80 35 1. 5. 72 13 10. 8. 09 36 1,4,75 14 9. 2. 79 37 1. 3 , 86 15 8. 0. 51 38 1 , 3 , 04 16 7. 0. 82 39 1 , 2 , 28 17 6. 3. 14 40 1 , 1 , 57 18 5. 7. 30 41 1 , 0 , 92 19 5. 0. 15 42 1 , 0 , 31 20 4. 6. 28 43 0 , 11 , 74 21 4. 1. 24 44 0 , 11 , 22 22 3. 8. 87 45 0 , 10 , 72 23 3. 5. 04     vnder measure for the diameter in inches and quarters . in. over f. in 10   f. in 100 , 29 3 0 , 0 , 00 , 4 , 8 , 31 , 73● , 1 , 00 , 4 , 4 , 06 , 326 , 0 , 10 , 4 , 0 , 27 1 183 , ● , 80 7 3 , 8 , 89 , 118 , 2 , 16 , 3 , 5 , 85 , 81 , ● , 00 , 3 , 3 , 10 , 59 , 10 , 24 , 3 , 0 , 61 2 45 , 10 , 90 8 2 , 10 , 36 , 36 , 2 , 41 , 2 , 8 , 31 , 29 , 3 , 93 , 2 , 6 , 44 , 24 , 3 , 86 , 2 , 4 , 73 3 20 , 4 , 40 9 2 , 3 , 15 , 17 , 4 , 25 , 1 , 1 , 70 , 15 , 0 , 00 , 1 , 0 , 37 , 13 , 1 , 86 , 1 , 11 , 13 4 11 , 5 , 14 10 1 , 9 , 99 , 10 , 1 , 78 , 1 , 8 , 93 , 9 , 0 , 62 , 1 , 7 , 95 , 8 , 1 , 80 , 1 , 7 , 04 5 7 , 3 , 98 11 1 , 6 , 18 , 6 , 7 , 80 , 1 , 5 , 38 , 6 , 0 , 72 , 1 , 4 , 63 , 5 , 6 , 53 , 1 , 3 , 93 6 5 , 1 , 10 12 1 , 3 , 27     13 1 , 1 , 02 a table of the number of bricks in a rodd of walling at any feet high , from 1 to 20 for 1 and 1 1 / 2 feet high . at 1 brick thick . at 1 brick & 1 / 2 thick 1 176 264 2 352 528 3 528 792 4 704 1056 5 880 1320 6 1136 1704 7 1232 1848 8 1408 2112 9 1584 2376 10 1760 2640 11 1936 2904 12 2112 3168 13 2288 3432 14 2464 3696 15 2640 3960 16 2816 4224 16 1 / 2 2904 4356 17 -2992 4488 18 -3168 4752 19 -3344 5010 20 -3520 5280 if you would have this table for 1 / 2 a brick , take the half of the table for one brick . if for two bricks then double it . if for two and a 1 / 2 then ad both these together ; if for three , double that for one brick and 1 / 2. if you have any number of feet of brick work , at half a brick , one brick , or two bricks , or more , and you would reduce it to one brick and half , then say by the line of numbers as 1. 2. 4. 5 or 6 is to three , so is the number of feet at 1 / 2 1. 2. 2 1 / 2 or three bricks to the number of feet at one and 1 / 2. the use of four scales , called circumfence , diameter , square equal , square inscribed . suppose you have a circle whose diameter is 10 inches , or 10 feet : and to this circle you would finde the circumference , or the side of a square equal , or inscribed , or having any one of the three , to finde the other three , do thus : take the measure of the circumference , diameter , or either of the squares , which is first given , and open the compasses to the number of the given measure , in its respective scale : the compasses so set , if you apply it in the scale whose number you would know , you shall have your desire . example . suppose a circle whose diameter is 10 inches , and to it i would know the circumference , take 10 out of the diameter scale , and in the circumference scale it shall reach to 31 42 , and on the line of square equal 8 86 , and on square inscribed 7. 07. for illustration sake , note the figure . the use of the line to divide a circle into any number of parts . take the semidiameter , or radius of the circle between your compasses , and fit it over in six and six of the line of circles , then what number of parts you would have , take off from that point by the figure in the line of circles , and it shall divide the circle into so many parts . as suppose i would have the former circle divided into nine parts , take the measure from the center to the circle as exactly as you can , fit that over in 6 and 6 , then take out 9 and 9 , and that shall divide it into so many parts ; but if you would divide a wheel into any odde parts , as 55. 63. or 49 parts , you shall finde it an almost impossible thing , to take a part so exact that in turning about so many times , shall not miss at last : to help which the parts the rule giveth shall fit you exact enough for all the odde parts , then the even will easie be had by dividing , therefore usually the rule is divided but to 30 or 40 parts . so that for this use as the finding the side of an 8 or 10 square piece , as the mast of a ship , or a newel , or a post , this will very readily , and exactly help you . chap xviii . the use of mr. whites rule , for the measuring of timber and board , either by inches or foot measure . 1. and first for superficial or board measure , by the inches , the breadth and length being given in inches , and feet and parts , slide or set 12 on any one side , to the breadth in inches or parts on the other side , then just against the length found on the first side , where 12 was on the second side you shall have the content in feet , and 10ths , or 100 parts required . which by the rules of reduction by the foot measure , you may reduce to inches and 8 parts . example . at three inches broad , and 20 foot long , you shall finde it to be 5 foot just : but at 7 inches broad , and the same length it will come to 11 foot 7 10th fere , or 8 inches . 2. the breadth being given in inches , to finde how many inches in length goes to make a foot of board or flat measure . set 12 on the first side , to the breadth in inches on the second side , then look for 12 on the second side , and right against it on the first side , is the number of inches , that goes to make a foot superficial , at that breadth . example . at three inches broad you shall finde 48 inches to make a foot . 3. to work multiplication on the s●iding , or whites rule . set one on the first side , to the multiplicator on the second side , then seek the multiplicand on the first , and right against it on the second , you shall finde the product . example . if 9 be multiplied by 16 , you shall finde it to be 144. 4. to work division on the same rule . set the divisor on the first side to one on the second , then the dividend on the first , shall on the second shew the quotient required . example . if 144 be to be divided by 16 , you shall finde the quotient to be 9. 3. to work the rule of 3 direct . set the first term of the question , sought out on the first line , to the second term on the other , ( or second line : ) then the third term sought on the first line , right against it on the second , you shall finde the fourth proportional term required . example . if 15 yard 1 / 2 , cost 37s . 6d . what cost 17 3 / 4 ? facit 42s . 10d . 3 q. for if you set 15 1 / 2 right against 37 1 / 2 , then look for 17 3 / 4 on the first line , ( where 15 1 / 2 was found , and right against it on the second line , is neer 42 the fractions are all decimal , and you must reduce them to proper fractions accordingly to work the rule of 3 reverse . 4. set the first term sought out on the first line , to the second being of the same denomination or kind to the second line , or side . then seek the third term on the second side , and on the first you shall have the answer required . example . 5. if 300 masons build an edifice in 28 days , how many men must i have to perform the same in six days , the answer will be found to be 1400. 6. to work the double rule of 3 direct . this is done by two workings : as thus for example . if 112 l. or 1 c. weight , cost 12 pence the carraiage for 20 miles , what shall 6 c , cost , 100 miles ? say first by the third rule last mentioned , as 1 c. weight to 12 , so is 6 c. weight to 72. pence , secondly say if 6 c. cost 72 pence or rather 6s , for 20 miles ? what shall 100 miles require ? the answer is 30 s. for if you set 20 against 6 then right against 100 is 30 , the answer required . the use of mr. whites rule in measuring timber round , or square , the square or girt being given in inches , and the length in feet and inches . 1. the inches that a piece of timber is square , being given : to finde how much in length makes a foot of timber , look the number of inches square on that side of the timber line , which is numbred with single figures from 1 to 12 , and set it just against 100 on the other or second side , then right against 12 at the lower , ( or some times the upper ) end , on the first line , in the second you have the number of feet and inches required . example . at 4 1 / 2 inches square , you must have 7 foot 1 inch 1 / 3 to make a foot of timber . but if it be above 12 inches square , then use the sixth problem of the 5th chapter of the carpenters rule , with the double figured side and compasses . 2. but if it be a round smooth stick , of above 12 inches about , and to it you would know how much in length makes a true foot , then do thus . set the one at the beginning of the double figured side , next your left hand , to the feet and inches about , counted in the other side , numbred with single figures from 1 to 12 , then right against three foot six 1 / 2 inches , in the single figures side next the right hand , you have in the first side the number of feet , and inches required . example . a piece of 12 inches about , requires 11 f 7 in : fere to make a foot . again a piece of 15 inches about , must have 8 foot 1 / 2 an inch in length , to make a foot of timber . 3. but if you would have it to be equal to the square , made by the 4th part of a line girt about the piece , then instead of three foot 6 1 / 2 inches make use of four foot , and you shall have your desire . 4. the side of a square being given in inches , and the length in feet , to find the content of a piece of timber . if it be under 12 inches square then work thus : set 12 at the beginning or end of the right hand side , to the length counted on the other side , then right against the inches square on the right side is the content on the left side example . at 30 foot long , & 9 inches square , you shall find 16 foot 11 inches for the working this question , 12 at the end must be used . but if it be above 12 inches square , then ser one at the beginning , or 10 at the end of the right hand side , to the length counted on the other side , then the number of inches or rather feet and inches , counted on the first side , shall shew on the second the feet and parts required . example . at 1 f. 6 inch . square , and 30 foot long , you shall finde 67 feet and about a 1 / 2. 5. to measure a round piece by having the length , and the number of inches about , being a smooth piece , and to measure true , and just measure , then proceed thus : set 3 f. 6 1 / 2 inches on the right side , to the length on the other side , then the feet and inches about , on the first side , shall shew on the second or left , the content required . as at 20 inches about , and 20 foot long , the content will be found to be about 4 foot 5 inches . but if you give the usual allowance , that is made by dupling the string 4 times , that girts the piece : then you must set 4 foot on the right side , to the length on the other , then at 1 foot 8 inches about , the last example you shall finde but three foot 6 inches . 6. ● astly , if the rule be made fit for foot measure , onely then the point of 12 is altogether neglected , and one onely made use of as a standing number : and the point at three foot 6 1 / 2 will be at three foot 54 parts ; and the four will be the same , and the same directions in every respect , serve the turn . and because i call it mr. whites rule , being the contriver thereof , according to feet and inches , i have therefore fitted these directions accordingly , and there are sufficient to the ingenious practitioner . chap. xix . certain propositions to finde the hour , and the azimuth , by the lines on the sector . prop. 1. having the latitude , and complements of the declination , and suns altitude , and the hour from noon , to finde the suns azimuth , 〈◊〉 that time . take the right sine of the complement of the suns altitude , and mak 〈◊〉 it a parallel sine in the sine of th 〈◊〉 hour from noon , ( counting 15 degree 〈◊〉 for an hour , and 1 degree for for minutes ) counted from the center . the sector so set , take the right sine of the complement of the declination , and carry it parallel till the compasses stay in like sines , and the sine wherein they stay shall be the sine of the azimuth required . or else thus : take the right sine of the declination , make it a parallel in the cosine of the suns altitude , then take the parallel sine of the hour from noon , and it shall be the latteral or right sine of the azimuth from the south required . if it be between six in the morning , and 6 at night ; or from the north , if it be before or after six : and so likewise is the azimuth . prop. 2. having the azimuth from south or north , the complement of the suns altitude , and declination , to finde the hour . take the latteral , or right sine of the complement of the suns altitude , make it a ga●●llel in the cosine of the declination : the sector so sett , ake out the parallel sine of the azimuth , and measure it from the center , and it shall reach to the right sine of the hour from noon required . or else as before . as the right sine of the complement of the suns declination : is to the parallel sine of the azimuth , so is the right sine of coaltitude , to the parallel sine of the hour from noon , counting as before . prop. 3. having the complements of the latitude , suns altitude , and declination . to finde the suns azimuth from the north part of the horizon . 1. first of the complement of the latitude , and suns present altitude finde the difference . 2. and secondly count it on the line of sines from 90 toward the center . 3. take the distance from thence , to the sine of the suns declination ; but note when the latitude and declination differ , as in winter you must counte the declination beyond the center , and you must call it the suns distance from the pole . 4. fourthly , make that distance a parallel sine in the complement of the latitude . 5. fifthly , then take out the parallel sine of 90. 6. and sixthly , make it a parallel sine in the coaltitude . 7. seventhly , then the sector so set , take out parallel , radius , or sine of 90. and eighthly , measure it on the line of sines from 90 towards , ( and if need be beyond ) the center : and it shall reach to the versed sine of the suns azimuth from the north , or if you count the other way from the south , note that in working of these , if the line of sines be too big , then you have two or three smaller sines on the rule , where on to begin and end the work . example . latitude 51. 32 , declination 18 ▪ 30 , altitude 48. 12 , you shall finde the azimuth to be 130 from the north , or 50 from the south . prop. 4. having the complements of the latitude and declination , or suns distance from the pole , and the sun altitude given , to finde the hour from east or west , or else from noon . 1. first of the complement of the latitude , and suns distance from the pole , finde the difference . 2. count this from the sine of 9● toward the center . 3. take the distance from thence to the sine of the suns altitude . 4. make that distance a parallel sine of the complement of the latitude . 5. take out the parallel sine of 90 degrees , and 6. make that a parallel sine in the codeclination , then 7. take out the parallel sine o● 90 again , and 8. measure it from the sine of 90 toward the center , and it shall shew the versed sine of the hour from the north , or the sine of the hour from east or west ; or if you reckon from 90 , the hour from noon required . example . latitude 51. 32 , declination north 20. 14 , altitude 50. 55 , you shall finde the hour from the north to be 10 houres , or 10 a clock in the forenoon , or 4 hours past 6 , or two short of noon , according to each proper reckoning . prop. 5. having the latitude , the complement of the suns declination , the suns present altitude , and meridian altitude for that day , to finde the hour . make the lateral secant of the latitude , a parallel sine in the codeclination , then take the distance from the suns meridian altitude , to his present altitude , and lay it from the cer on both sides of the line of sines , and take the parallel distance between those two points , and measure it from 90 on a line of sines , of the same radius the secants be , ( as the small adjoyning sine is , ) and it shall shew the versed sine of the hour from noon , or the right sine before or after 6 , towards noon or night . prop. 6. having the latitude , declination , and suns altitude , to finde the suns azimuth . take the latteral secant of the latitude , and make it a parallel in the complement of the altitude : then take the distance , between the sun● of the complement● of of museum the latitude and altitude , ( if under 90 , ) and the sons declination , and lay it from the center on the line of sines , that parallel distance taken , and measured on the sines ( of the same radius the secant was ) from 90 , shall shew the versed sine of the azimuth from noon . but if the sum of the colatitude , and coaltitude exceed 90 , then take the excess above 90 , out of the natural sine from the center toward 90 , and add that to the sine of the suns declination towards 90 , and then the parallel distance between those two points , shall be the azimuth required , from noon , but when the latitude and declination are unlike , as with us ( in the northern parts ) in winter , then you must take the declination out of the excess , or the lesser out of the greater , and lay the rest from the center , and the parallel distance , shall be versed sine of the azimuth from noon . example . at 18 15 altitude , latitude 51 32 , declination 13 15 south . the sum of the colatitude , and the coaltitude is 109. 37 , then count the center for 90 , the right sine of 10 for a 100 and 19 for 109 , and 37 minutes forwarder , there set the point of the compasses , then take from thence to the right sine of the declination , and lay this distance from the center on the line of sines , and the parallel space between , is the versed sine of the azimuth required . prop. 7. having the length of the shaddow of any object standing perpendicular , and the length of that object , to finde the altitude . take the tangent of 45 , and make it a parallel in the length of the shaddow in the line of lines , then the parallel distance between the length of the object that casts the shaddow , taken from the line of lines , and measured on the line of tangents from the center , shall reach to the suns altitude required . example . if the object be 40 parts long , and the shaddow 80 parts , the altitude will be found to be 26. 35. but if you have the altitude , and shaddow , and would know the height of the object , then work thus : take the length of the shaddow out of the line of lines , or any other equal parts , and make it a parallel tangent of 45 , then take out the parallel tangent of the suns altitude , and measure it on the line of lines , ( or the same equal parts ) and it shall shew the length of the object that caused the shaddow : the same rule doth serve in taking of altitudes by the rule , as in the 18 chapter , accounting the measure from the station to the object , the length of the shaddow , and the suns altitude , the angle at the base . prop. 8. to finde the suns rising and setting in any latitude . take the latteral cotangent of the latitude , make it a parallel in the sines of 90 and 90 , then take the latteral tangent of the suns declination , and carry it parallel in the sines till it stay in like sines , that sine shall be the asentional difference between six , and the time of rising before or after 6 , counting 15 degrees to an hour . prop. 9. to finde the amplitude in any latitude . take the latteral sine of 90 , and make it a parallel in the cosine of the latitude , then the parallel sine of the declination , taken and measured in the line of sines from the center , shall give the amplitude required . prop. 10. to finde the suns height at six in any latitude . take the lateral or right sine of the declination , and make it a parallel in the sine of 90 , then take out the parallel sine of the latitude , and measure it in the line of sines from the center , and it shall reach to the altitude required . note in working of any of these propositions , if the sines drawn from the center , prove too large for your compasses , or to make a parallel sine or tangent to a small number of degrees , then you may use the smaller sine or tangent adjoyning , that is set on the rule , and it will answer your desire . and note also in these propositions , the word right , or latteral sine or tangent , is to be taken right on from the center or beginning of the lines of sines , or tangents ; and the word parallel always across from one leg to the other . prop. 11. to finde the suns height at any time , in any latitude . as the right sine of 90 , is to the parallel cotangent of the latitude : so is the latteral or right sine of the hour from 6 , to the parallel tangent of a fourth ark ; which you must substract from the suns distance from the pole , and note the difference . then , as the right of the latitude , to the parallel cosine of the fourth ark : so is the parallel cosine of the remainder , to the latteral sine of the altitude required . prop. 12. to finde when the sun shall come to due east , or west . take the tangent of the latitude from the smaller tangents , make it a parallel in the sine of 90 , then take the latteral tangent of the declination from the smaller tangents , and carry it parallel in the sines , till it stay in like sines , and that sine shall be the sine of the hour required from 6. prop. 13. to finde the suns altitude at east or west ( or vertical circle . ) as the latteral sine of declination , is to the parallel sine of the latitude : so is the parallel sine of 90 , to the latteral sine of the altitude required . prop. 14. to finde the stiles height in upright declining dials . as the right sine of the complement of the latitude , to the parallel sine of 90 : so the parallel sine complement of the plains declination , to the right sine of the stiles elevation prop. 15. to finde the substiles distance from the meridian . as the lateral tangent of the colatitude , to the parallel sine of 90 : so the parallel sine of the declination , to the latteral tangent of the substile from the meridian . prop. 16. to finde the inclination of meridians . as the latteral tangent of the declination , to the parallel sine of 90 : so is the parallel sine of the latitude , to the latteral cotangent in the inclination of meridians . prop. 17. to finde the hours distance from the substile in all plains . as the latteral tangent of the hour from the proper meridian , to the parallel sine of 90 : so is the parallel sine of the stiles elevation , to the latteral tangent of the hour from the substile . prop. 18. to finde the angle of 6 from 12 , in erect decliners . as the latteral tangent of the complement of the latitude , to the parallel sine of the declination of the plain : so is the parallel sine of 90 : to the latteral tangent of the angle between 12 and 6. thus you see the natural sines and tangents on the sector , may be used to operate any of the canons that is performed by logarithms , or the artificial sines and tangents , by changing the terms from the first to the third , and the second to the first , and the third to the second , and the fourth must always be the fourth , in both workings being the term required . chap. xx. a brief description , and a short-touch of the use of the serpentine-line , or numbers , sines , tangents , and versed sine contrived in five ( or rather 15 ) turn . 1. first next the center is two circles divided one into 60 , the other into 100 parts , for the reducing of minutes to 100 parts , and the contrary . 2. you have in seven turnes two in pricks , and five in divisions , the first radius of the sines ( or tangents being neer the matter , alike to the first three degrees , ) ending at five degrees and 44 minutes . 3. thirdly , you have in 5 turns the lines of numbers , sines , tangents , in three margents in divisions , and the line of versed sines in pricks , under the line of tangents , according to mr. gunters cross staff : the sines and tangents beginning at 5 degrees , and 44 minutes where the other ended , and proceeding to 90 in the sines , and 45 in the tangents . and the line of numbers beginning at 10 , and proceeding to 100 , being one entire radius , and graduated into as many divisions as the largeness of the instrument will admit , being from 10 to 50 into 50 parts , and from 50 to 100 into 20 parts in one unit of increase , but the tangents are divided into single minutes from the beginning to the end , both in the first , second , and third radiusses , and the sines into minutes ; also from 30 minutes to 40 degrees , and from 40 to 60 , into every two minutes , and from 60 to 80 in every 5th minute , and from 80 to 85 every 1oth , and the rest as many as can be well discovered . the versed sines are set after the manner of mr. gunters cross-staff , and divided into every 10th minutes beginning at 0 , and proceeding to 156 going backwards under the line of tangents . 4. fourthly , beyond the tangent of 45 in one single line , for one turn is the secants to 51 degrees , being nothing else but the sines reitterated beyond 90. 5. fifthly , you have the line of tangents beyond 45 , in 5 turnes to 85 degrees , whereby all trouble of backward working is avoided . 6. sixthly , you have in one circle the 180 degrees of a semicircle , and also a line of natural sines , for finding of differences in sines , for finding hour and azimuth . 7. seventhly , next the verge or outermost edge is a line of equal parts to get the logarithm of any number , or the logarithm sine and tangent of any ark or angle to four figures besides the carracteristick . 8. eightly and lastly , in the space place between the ending of the middle five turnes , and one half of the circle are three prickt lines fitted for reduction . the uppermost being for shillings , pence , and farthings . the next for pounds , and ounces , and quarters of small averdupoies weight . the last for pounds , shillings , and pence , and to be used thus : if you would reduce 16 s. 3 d. 2 q. to a decimal fraction , lay the hair or edge of one of the legs of the index on 16. 3 1 / 2 in the line of l. s. d. and the hair shall cut on the equal parts 81 16 ; and the contrary , if you have a decimal fraction , and would reduce it to a proper fraction , the like may you do for shillings , and pence , and pounds , and ounces . the uses of the lines follow . as to the use of these lines , i shall in this place say but little , and that for two reasons . first , because this instrument is so contrived , that the use is sooner learned then any other , i speak as to the manner , and way of using it , because by means of first , second , and third radiusses , in sines and tangents , the work is always right on , one way or other , according to the canon whatsoever it be , in any book that treats of the logarithms , as gunter , wells , oughtred , norwood , or others , as in oughtred from page 62 to 107. secondly , and more especially , because the more accurate , and large handling thèreof is more then promised , if not already performed by more abler pens , and a large manuscript thereof by my sires meanes , provided many years ago , though to this day not extant in print ; so for his sake i claiming my intrest therein , make bold to present you with these few lines , in order to the use of them : and first no●e . 1. which soever of the two legs is set to the first term in the question , that i call the first leg always , and the other being set to the second term , i call the second leg . 2. secondly , if one be the first or second term , then for the better setting the index exactly , you may set it to 100 , for the error is like to be the least neerest the circumference . 3. thirdly , be sure you keep a true account of the number of turnes between the first and second term . 4. fourthly , observe which way you move , from the first to the second term . to keep the like from the third to the fourth , exept in the back rule of three , and in such cases as the canon requires the contrary . 5. fifthly in multiplication , one is always the first term , and the multiplycator or multiplycand the second , and the product always the fourth . also note that in multiplycation the product of two numbers multiplyed , shall be in as many places as both the multiplycator , and multiplycand , except the least of them , be less then the two first figures of the product ; moreover , for your more certain assinging of the two last figures of four or six , which is as many as you can see on this instrument , multiply the two last in your minde , and the product shall be the figure , as in page 28 of the carpenters rule . 6. in division , the multiplicator is always the first term , and one the second , the dividend the third , and the quotient the fourth ; also the quotient shall have as many figures as the dividend hath more then the divisor , except the first figures of the divisor be greater then the dividends , then it shall have one less . also note , the fraction after division , is a decimal fraction , and to be reduced as before . 7. note carefully whether the fourth proportional ought to be a greater or a less , and resolve accordingly , and note if one cometh between the third and fourth term , then must the fourth be raised a radius or a figure more , and be careful to set the hairs exactly over the part representing the number or minutes of any degree . 8. always in direct proportion , and astronomical calculation , set the first leg to the first term , and the second leg to the second term , and note how many circles is between , then set the first leg to the third term , and right under the second leg , the same way , and so many turnes between the third and fourth , is always the fourth term required . example . as 1 , to 47 , so is 240 , to 11280. as the sine of 90 , to 51 degrees 30 minutes , so is the sine of 80 to 50. 26. and so of all other questions according to their respective canons by the logarithms in other books as in mr. oughtreds circles of porportions , from page 62 to 107 , and others . here followeth the working of certain propositions by the serpentine-line . those that i shall insert , are onely to shew the manner of working , and knowing of that once , all the canons for all manner of questions , either in arithmetick , geometry , navigation , or astronomy , by any other author , as mr. gunter , mr. oughtred , mr. windgate , mr. norwood , or others , may be speedily resolved , and as exactly as by the tables , if the instrument be well and truly made . and first ●or the hour , according to gunter . prop. 1. having the latitude , declination , and the suns altitude , to finde the hour . add the complement of the altitude , the complement of the suns present altitude , and the distance of the sun , from the elevated pole together , and nore their sum , and half sum , and find the difference between their half sum and the complement of the suns present altitude , then work thus : for 36 42 degrees high , at 23. 32 declination , lat . 51. 32. lay the first leg , viz. that next your right hand being here most convenient , on the sine of 90 , keeping that fixed there , lay the other leg to the cosine of the latitude , viz. 38. 28. and note the turns between , which here is none between , but it is found in the next over it , then set the first leg to the sine of 66 , 28 , the suns distance from the pole , and in the circle just over it you shall have the sine of 34. 47. for the fourth ark or sine . then in the second operation . set the first or left leg to the sine of 34. 47. of the fourth sine last found , then keeping that fixed there , set the other leg to the sine of the half sum , viz. 79. 37. then remove the first leg to the sine of the difference between the half sum , and the coaltitude , viz. 25. 49. and then in the next circle , the other leg shall shew the sine of 48. 34. whose half distance toward 90 , being found by the scale of logarithms on the outermost circle , will discover the sine of an ark , whose complement being doubled , and turned into time ( by counting 15 degrees to an hour ) will give the hour required ; but by help of the versed sines all this trouble is saved ; for when the index or second leg cuts the sine of 48. 34. at the same instant it cuts the versed sine of 60 , the hour from noon required , being 8 in the morning , or 4 in the afternoon , at 23. 32. of declination , in the latitude of 51. 32. prop. 2. to work the same another way , according to mr. collins . the latitude and declination given , to finde the suns height at 6 a clock , dec. 23. 31. lat. 51. 32. lay the left leg on the sine of 90 , and the other to the sine of 23. 32 , and you shall finde one turn between upwards , then the first leg laid on the sine of the latitude 51 , 32 , the other leg shall shew the sine of 18. 3. ( in the second circle above 51 , 32. ) for the suns height at 6 required , and this is fixed for one day . then in summer time , or north declinations , by help of the line of natural sines , in the second line , finde the difference between the suns present altitude , and the latitude at 6 , but in winter or southern ( signs or ) declinations , add the two altitudes together , in this manner . lay one leg of the index to the natural sine of the altitude at 6 , and the other to the altitude proposed , the two legs so set , bring one of them ( viz the right ) to the beginning of the line of natural sines , and the other shall stay at the difference required , but in winter set one leg to the beginning of the sines , and open the other to the height at 6 , or rather depression under the horizon at 6 , ( which is all one at like declinations , north and south ) then set the first leg to the present altitude of the sun , and the other shall shew the sine of the sum of both added together ; which sum or difference is thus to be used : lay the left leg to the cosine of the declination , and the other to the secant of the latitude , counted beyond 90 , as far as the secant of 9 , 40 ; or rather lay the left leg on the co-sine of the latitude , and the other to the secant of the suns declination , then the first leg laid on the sine of the sum in winter , or difference in summer , shall cause the other leg to fall on the sine of the hour from 6 , toward noon in winter and summer , except the altitude in summer be less then the altitude at 6 , then it is the hour from 6 , toward mid-night . prop. 3. having the latitude , suns altitude and declination , to finde the suns azimuth from east or west . lat. 51. 32. declin . 23. 30. alt. 49 56. first you must get the suns altitude , or depression in the vertical circle by this cannon . lay the first leg to the sine of the latitude , and the second to the sine of 90 , and you shall finde them both to be on the same line , then the first leg laid on the sine of the declination , shall cause the second ( being carried with the first , without moving the angle first set ) to fall on the sine of 30. 39. the suns altitude in the vertical circle ; with which you must do , as you did before with the altitude at 6 , and the present altitude , to finde the sum and difference by help of the line of natural tangents , then this proportion holds . lay the first leg to the cosine of the altitude , ( by counting the altitude from 20 ) and the second leg to the tangent of the latitude , and observe which way , and the turns between ; then the first leg removed and laid to the sine of the sum ( before found ) in winter , or the difference in summer , shall cause the second leg to fall on the sine of the azimuth of the sun , from east or west toward noon , if winter ; and also in summer , when the suns altitude is more than his altitude at the vertical circle ; but if less from the east or west , toward north or mid-night meridian : thus in our example , it will be found to be the sine of 30 degrees , or 60 from the south , the sine of the difference being found to be 14. 49. prop. 4. to finde the azimuth , according to mr. gunter , by having the latitude , suns altitude , and declination given . first by the suns declination get his distance from the pole , which in summer or north declinations , is always the complement of the declination , ( likewise in south latitudes , and south declinations ) but when the latitude and declination is unlike , then you must adde 90 to the declination , and the sum is the distance from the elevated pole. having found the distance from the pole , adde that and the complement of the latitude , and suns altitude together , finde the difference between their half sum , and the suns distance from the pole , then the proportion will be thus , as in this example : 13 declination , 41. 53. alt. lat. 51. 32. lay one leg on the sine of 90 , and the other to the sine of 38 , 28 , being so set , remove the first leg to the sine complement of the altitude 48. 07. and the second legg shall fall on a fourth sine , which will be found to be 27. 36. then set the first or left leg to 27. 36. the fourth sine , and the second to 81. 47. the sine of the half sum , then removing the first leg to the sine of the difference , shall cause the second to shew two circles lower , the versed sine of 130 , the azimuth required , being counted from the north part of the horizon , whose complement to 180 from the south is 50 degrees . two other canons to finde the hour of the day , and azimuth of the sun , by one operation , by help of the natural sines : and first for the hour . having the latitude , the suns meridian , and present altitude , and declination , to finde the hour from noon . first lay one leg to the meridian altitude , in the line of natural sines , and the other to the sine of the altitude in the same line , then bring the right leg to the beginning of the line of sines , and the other 〈…〉 difference , which difference 〈…〉 keep . then lay one leg on the 〈◊〉 the declination , and the other 〈◊〉 secant of the latitude , and note the turnes between , or rather lay the first leg to the cosine of the latitude , and the other to the secant of the declination , then the legs being so set , bring the first or left leg to the sine of the difference first found , and the other leg shall shew the versed sine of the hour from noon , if the versed sines had been set thus , i. e. the versed sine of 90 , against the sine of 90 , as in some instruments it is : but to remedy this defect , do thus : keep the right leg there , and open the other to the versed sine of 0. or sine of 90 , and note the turnes between , then lay the leg that was on the sine of 90 , ( or versed sine of 0 , ) to the versed sine of 90 , and the other leg shall shew the versed sine of the hour from noon , counting from 90. example . at 45 13 degrees altitude , declination north 23 32. latitude 51 32 , the meridian altitude is 62 , ( being found by adding colatitude , and declination together , and in southern declinations by substraction . ) then the natural sine of 45. 42 , taken from 62 , shall be the sine of 9. 38. then as the cosine of the latitude 38. 28 , is to the secant of declination 23. 32 , so is the sine of 9. 38. to the sine of 17. 05 ) or versed sine of 45 ▪ if they were placed and numbred , as in some instruments they be : but to help i● in this , say : as the versed sine of 0 , is to the verses of 114. 26 , so is the versed sine of 90 , to the versed sine of 135 , whose complement to 180 is the angle or hour from noon required . secondly , for the suns azimuth . having the latitude , declination , and suns altitude , to finde the azimuth from south or north. first add the complements of the coaltitude and cola●itude together , then if the sum be under 90 , take the distance between the cosine of it , and the sine of the declination , in the line of natural sines , and measure it in the line of sines from the beginning , and it shall give the sine of the difference ; but if the sum exceed 90 , then when the latitude and declination is alike , add the excess to the declination ; but if contrary substract one out of other , and measuring the sum or remainder from the beginning of the sines , you have the difference which you must keep . then lay the first or left leg to the cosine of the altitude , and the second to the secant of the latitude , or else lay the first to the cosine of the latitude , and the other to the secant of the altitude , and more the turnes between , then lay the first leg to the sine of the difference before found , and the other shall shew the versed sine of the azimuth from noon required , if the versed sines be set as before is expressed , that is to say 90 of right sines , and versed sines together , and numbred forwards as the sines be : but in the use of this instrument , the remedy aforesaid supplyeth the defect . example . at 10. 19. altitude , 23. 32. declination , 51. 32. latitude , to finde the azimuth . the sum of the coaltitude and co-latitude is 118.09 , the excess above 90 , with right sine of declination added is 60. 30 , found by natural sines , then say , as the cosine of latitude , to the secant of altitude , so is the sine of the difference 60. 30 , to the versed sine of the azimuth , but here to the versed sine of an ark beyond radius unknown , then as the versed sine of 0 , to that ark , so is the versed sine of 90 , to the ver sine of 60 , the azimuth from noon , whose complement to 180 , is 125 the azimuth from sonth required . having been so large in these , i shall in the rest contract my self as to the repetition , and onely give the canon for the propositions following , the way of working being the same in all other , as in these before rehearsed , and note also what is to be done by the serpentine-line , is to be done by the three same lines of numbers , sines and tangents on the edge of the sector , by altering the term leg , to to the point of the compasses . the canons follow . prop 4. having latitude , declination , and hour given , to finde the suns altitude at that hour or quarter . and first for the hour of 6. as the sine of 90 , to the sine of the latitude 51. 30 : so is the sine of the declination 23. 30 , to the sine of the altitude at 6. 18. 13. secondly , for all hours in the equinoctial . as the radius or sine of 90 , to the cosine of the latitude 51. 32 : so is the sine of the suns distance from 6 ( in hours and minutes , being turned into degrees and minutes 30 , for 8. or 4 , ) to the sine of the altitude of the sun at the time required 18. 07 , but for all other times say , as the sine of 90 , to the cotangent of the latitude 38. 28. so is the sine of the suns distance from 6 , 30 , 0 , to the tangent of the 4 arke 21 40. which fourth arke must be taken out of the suns distance from the pole 66 , 21 , ( in cancer ) leaveth a residue 44 48 , which is called a fift ark . but for the hours before and after 6 , you must add the fourth arke to the suns distance from the pole , and the sum is the fifth ark . then say , as the cosine of the fourth ark 78 , 20 , is to the sine of the latitude 51 , 32 : so is the cosine of the residue 45 , 12 , to the sine of the suns altitude at 8 , 36 , 42 , at that declination . prop. 6. having the latitude , declination , and azimuth , to finde the suns altitude at that azimuth . and first to finde the suns altitude at any azimuth in the aequator . then , as the sine of 90 , to the cosine of the suns azimuth from the south 50 , 0 , so is the cotangent of the latitude 38 , 28 , to the tangent of 27 , 03 ▪ the suns altitude , at that azimuth required . secondly , to find it at all other times , do thus : as the sine of the latitude 51 , 32 , to the sine of the suns declination , 23 , 32 : so is the cosine of the suns altitude in the aequator , at the same azimuth from the vertical , viz. 30 , to the sine of a 4th . ark 28 , 16. which fourth ark must be added to the suns altitude at the aequator in all azimuths under 90 , from the meridian , where the latitude and declination are alike . but in azimuths more then 90 from the meridian , take the altitude in the aequator out of the fourth ark , and the sum or remainder shall be the altitude required , viz. 42 , 56. but when the latitude and declination are unlike , as with us in winter time , then take the fourth ark out of the altitude at the aequator , and you shall have the altitude belonging to that azimuth required . prop. 7. having the hour from noon , and the altitude to find the suns azimuth at that time . as the cosine of the altitude , to the sine of the hour , so is the cosine of the suns declination , to the sine of the azimuth required . prop. 8. having the suns azimuth , altitude , and declination , to find tho hour of the day . as the cosine of the declination , to the sine of the suns azimuth : so is the cosine of the altitude , to the sine of the hour . prop. 9 , having the latitude and declination , to find when the sun shall be due east or west . as the tangent of the latitude , to the tangent of the suns declination , so is the sine of 90 , to the cosine of the hour from noon . prop. 10. having the latitude and suns declination , to find the amplitude . as the cosine of the latitude , to the sine of the declination : so is the sine of 90 , to the sine of the amplitude from the east or west , toward north or south , according to the time of the day and year . prop. 11. the latitude and declination given , to find the time of the suns rising before or after 6. as the cotangent of the latitude , to the sine of 90 : so is the tangent of the suns declination , to the sine of the suns ascentional difference between the hour of 6 and the suns rising . prop. 12. having the suns place , to find his declination , and the contrary . as the sine of 90 , to the suns distance from the next equinoctial point , so is the sine of the suns greatest declination , to the sine of his present declination required . prop. 13. the greatest and present declination given , to find the suns right ascension . as the tangent of the greatest declination , to the sine of 90 : so the tangent of the present declination , to the right ascension required . onely you must regard to give it a right account by considering the time of the year , and how many 90s . past . pror . 14. to find an altitude by the length , and shadow of any perpendicular object . lay the hair on one legg to the length of the shadow found on the line of numbers , and the hair of the other leg to the length of the object that caused the shadow found on the same line of the numbers ; then observe the lines between , and which way when the legs are so set , bring the first of them to the tangent of 45 , and the other leg shall ●hew on the line of tangents , so many turns between , and the same way the tangent of the altitude required . thus may you apply all manner of quest . to the serpentine-line & work them by the same canons , that you use for the logarithms in all or most authors . prop. 15. to square , and cube a number , and to findethe square root , or cube roat of a number . the squaring of a number , is nothing else but the multiplying of the number by it self , as to square 12 is to multiply 12 by 12 , and then the cubing of 12 , is to multiply the square 144 by 12 , & that makes 1728 , and the way to work it , is thus : set the first leg to 1 , and the other to 12 , then set the first to 12 , and then the second shall reach to 144 , then set the first to 144 , and the second shall reach to 1728 , the cube of 12 required : but note , the number of figures in a cube , that hath but one figure is certainly found by the line , by the rule aforegoing : but if there be more figures then one , so many times 3 must be added to the cube , and so many times two to the square . to find the square root of a number , do thus : put a prick under the first , the third , the 5th , the 7th , & the number of pricks doth shew the number of figures in the root ; and note if the figures be even , count the 100 to be the unit , if odde as 3 , 5 , 7 , 9 , &c. the 10 at the beginning must be th● unit , as for 144 , the root consists of two figures , because there is two pricks under the number , and if you lay the index to 144 in the numbers , it cu●s on the line of logarithms 15870 , the half of which is 7915 whereunto if you lay the index , it shall shew the 12 the root required ; but if you would have the root of 14+44 , then divide the space between that number , and 100 you shall finde it come to 8 , 4140 that is four turnes , and 4140 for which four turnes , you must count 80000 , the half of which 8,4140 , is 4,2070 , whereunto if you lay the index , and count from 1444 ●r 100 , at the end you shall have it cut at 38 lack four of a 100. to extract the cubique root of a number , set the number down , and put a point under the 1 , the 4th , the 7th , and 10th , and look how many pricks , so many figures must be in the root , but to finde the unity you must consider , if the prick falls on the last figure , then the 10 is the unit at the beginning of the line , as it doth in 1728 , for the index laid on 1728 , in the log●rithms , sheweth 2,3760 , whose third part 0,7920 counted from 10 , falls on 12 the root , but in 17280 , then you must conceive five whole turnes , or 1000 to be added , to give the number that is to be divided by three , which number on the outermost circle in this place , is 12 , +3750. by conceiving 10000 to be added , whose third part counted from 10 , viz. two turnes or 4.125 , shall fall in the numbers to be near 26. but if the prick falls of the last but 2 , as in 172800 then 100 at the end of the line , must be the unit , and you must count thus : count all the turnes from 172830 to the end of the line , and you shall finde them to amount to 7,6250 , whose third part 2 , 5413 counted backward from 100 , will fall on 55,70 the cubique root required . prop. 16. to work questions of interest or progression , you must use the help of equal parts , as in the extraction of roots , as in this question , if 100 l. yield 106 in one year , what shall 253 yield in 7 year ? set the first leg to 10 at the beginning , in this case representing a 100 , and the other to 106 , and you shall finde the legs to open to 253 of the small divisions , on the logarithms , multiply 253 by 7 , it comes to 1771 , now if you lay the hair upon 253 , and from the place where the index cuts the logarithms count onwards 1771 , it shall stay on 380 l. 8 s. or rather thus : set one leg to the beginning of the logarithms , and the other to 1771 either forward or backward , and then set the same first leg to the sum 253 , and the second shall fall on 380. 8 s. according to estimation ; the contrary work is to finde what a sum of money due at a time to ●ome , is worth in ready money : this being premised here , is enough for the ingenious to apply it to any question of this nature , by the rules in other authors . however you may shortly expect a more ample treatise , in the mean time take this for a taste and farewell . the use of the almanack . having the year , to finde the day of the week the first of march is on in that year , and dominical letter also . first if it be a leap-year , then look for it in the row of leap-year , and in the column of week-days , right over it is the day required , and in the row of dominical letters is the sunday letters also : but note the dominical letter changeth the first of ianuary , but the week day the first of march , so also doth the epact . example . in the year 1660 , right over 60 which stands for 1660 , there is g for the dominical or sunday letter , beginning at ianuary , and t for thursday the day of the week the first of march is on , and 28 underneath for the epact that year , but in the year 1661. being the next after 1660 the leap-year , count onwards toward your right hand , and when you come to the last column , begin again at the right hand , and so count forwards till you come to the next leap-year , according to this account for 61 , t is the dominical letter , and friday is the first of march. but to finde the epact , count how many years it is since the last leap-year , which can be but three , for every 4th is a leap-year , and adde so many times 11 to the epact in the leap-year last past , and the sum , if under 30 , is the epact ; if above 30 , then the remainder 30 or 60 , being substracted is the epact for that year . example for 1661.28 the epact for 1660 , and 11 being added makes 39 from which take 30 , and there remaineth 9 , for the epact for the year 1661 the thing required . note that in orderly counting the years , when you come to the leap-year , you must neglect or slip one , the reason is , because every leap-year hath two dominical letters , and there also doth the week day change in the first of march , so that for the day of the month , in finding that the trouble of remembring the leap-year is avoided . to find the day of the month. having found the day of the week , the first of march is on the respective year ; then look for the month in the column , and row of months : then all the daies right under the month are the same day of the week the first of march was on , then in regard the days go round , that is change orderly every seven days , you may find any other successive day sought for . example . about the middle of march 1661 on a friday , what day of the month is it ? first the week day for 1661 is friday , as the letter f on the next collumn beyond 60 she●et● ; then i look for 1 among the months , and all the days right under , viz. 1 , 8 , 15 , 22 , 29. in march , and november 61 , are friday , therefore my day being friday , and about the middle of the month , i conclude it is the 15th day required . again in may 1661. on a saturday about the end of may , what day of the month ? may is the third month , by the last rule i find that the 24 and 31 are fridays , therefore this must needs be the 25 day , for the first of iune is the next saturday . finis . errata . page 23. l. 4. adde 1660 p. 24. l. 6. for 5 hours r. 4. l. 9. for 3. 29. r. 4. 39. 1. 12 for 5. 52. r. 4 , 52. l. 13. for 3. 39. r. 4. 39. l. 17 for 5 hours 52. r. 4. 52. p. 27. l. ult . dele or 11. 03. p 31. l 4. for sun r. sum . p. 50. l. 8. for b r. a. p. 50 d chap. xii . p. 51. r. 16. for 6. 10. 1. 6 to 10. p. 71. l. 6. for 7 / 4 r. 1 / 4. l. penult . for 2 afternoon , r. 1. p. 74 l. ult . for 1. r. 1 , 2. p. 83. l. 18. for bc r. bd. p. 69 l. 17. add measure , p. 129 l. 24. for right of , r. right sine of . p. 114 l. 9 for 18 3. r 18 13. p. 147 1. 2 for 20 , r. 90 p. 163. l. 16. for of , r. on . horlogiographia optica. dialling universall and particular: speculative and practicall. in a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. with diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. illustrated by diverse opticall conceits, taken out of augilonius, kercherius, clavius, and others. lastly, topothesia, or, a feigned description of the court of art. full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. together with many usefull instruments and dials in brasse, made by walter hayes, at the crosse daggers in more fields. / written by silvanus morgan. morgan, sylvanus, 1620-1693. this text is an enriched version of the tcp digital transcription a89305 of text r202919 in the english short title catalog (thomason e652_16). textual changes and metadata enrichments aim at making the text more computationally tractable, easier to read, and suitable for network-based collaborative curation by amateur and professional end users from many walks of life. the text has been tokenized and linguistically annotated with morphadorner. the annotation includes standard spellings that support the display of a text in a standardized format that preserves archaic forms ('loveth', 'seekest'). textual changes aim at restoring the text the author or stationer meant to publish. this text has not been fully proofread approx. 273 kb of xml-encoded text transcribed from 81 1-bit group-iv tiff page images. earlyprint project evanston,il, notre dame, in, st. louis, mo 2017 a89305 wing m2741 thomason e652_16 estc r202919 99863048 99863048 115230 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a89305) transcribed from: (early english books online ; image set 115230) images scanned from microfilm: (thomason tracts ; 100:e652[16]) horlogiographia optica. dialling universall and particular: speculative and practicall. in a threefold præcognita, viz. geometricall, philosophicall, and astronomicall: and a threefold practise, viz. arithmeticall, geometricall, and instrumentall. with diverse propositions of the use and benefit of shadows, serving to prick down the signes, declination, and azimuths, on sun-dials, and diverse other benefits. illustrated by diverse opticall conceits, taken out of augilonius, kercherius, clavius, and others. lastly, topothesia, or, a feigned description of the court of art. full of benefit for the making of dials, use of the globes, difference of meridians, and most propositions of astronomie. together with many usefull instruments and dials in brasse, made by walter hayes, at the crosse daggers in more fields. / written by silvanus morgan. morgan, sylvanus, 1620-1693. goddard, john, fl. 1645-1671, engraver. [16], 144 p. : ill. (woodcuts, metal cuts) printed by r. & w. leybourn, for andrew kemb, and robert boydell, and are to be sold at st. margarets hill in southwark, and at the bulwark neer the tower, london : 1652. with an additional title page, engraved and initialed by john goddard. the first leaf contains verses "on the frontispiece". annotation on thomason copy: the "2" in the imprint date is crossed out and date altered to 1652; "febr. 4th". reproduction of the original in the british library. eng dialing -early works to 1800. globes -early works to 1800. sundials -england -early works to 1800. a89305 r202919 (thomason e652_16). civilwar no horlogiographia optica.: dialling universall and particular: speculative and practicall. in a threefold præcognita, viz. geometricall, phil morgan, sylvanus 1652 34401 50 0 14 0 0 0 19 c the rate of 19 defects per 10,000 words puts this text in the c category of texts with between 10 and 35 defects per 10,000 words. 2007-08 tcp assigned for keying and markup 2007-08 apex covantage keyed and coded from proquest page images 2008-05 john pas sampled and proofread 2008-05 john pas text and markup reviewed and edited 2008-09 pfs batch review (qc) and xml conversion horologiographia optica dialing universall and perticuler . speculatiue and practicall together with the discription of the courte of arts by a new method by sylvanus morgan jj. sculp horologiographia optica . dialling vniversall and particular : speculative and practicall . in a threefold praecognita , viz. geometricall , philosophicall , and astronomicall : and a threefold practise , viz. arithmeticall , geometricall , and instrumentall . with diverse propositions of the use and benefit of shadows , serving to prick down the signes , declination , and azimuths , on sun-dials , and diverse other benefits . illustrated by diverse opticall conceits , taken out of augilonius , kercherius , clavius , and others . lastly , topothesia , or , a feigned description of the covrt of art . full of benefit for the making of dials , use of the globes , difference of meridians , and most propositions of astronomie . together with many usefull instruments and dials in brasse , made by walter hayes , at the crosse daggers inmore fields written by silvanus morgan . london , printed by r & w. leybourn , for andrew kemb , and robert boydell ▪ ● and are to be sold at st margarets hill in southwark , and at the bulwark neer the tower . 1657. to william bateman , esqrs. to anthony bateman , esqrs. to thomas bateman , esqrs. sons to the late honourable thomas bateman , esq chamberlain of london , deceased . gentlemen , your late father being a patron of this honourable city , doth not a little invite me to you , though young , yet to patronise no less then the aspiring of coelum , which , as the poets feign , was the ancientest of the gods , and where you may see sol only of the titans , favouring jupiters ▪ signe , and by their power and operation hath established arts or learning , the fable rather according to that establishment which god hath given them , they are , i say , sought out of those that take their pleasure therein : pardon my boldness , i beseech you , if like prometheus i have made a man of clay ; and now come to light my bundle of twigs at the chariot of the sun , desiring that you would infuse vigor in that which cannot at all move of it self , & if your benevolence shall but shine upon it , the angles of incidence & reflection shall be all one : your love invites me to be so bold as to think you worthy of my labour , wherein , if faults shall arise in the cuspis of the ascendent , they shall also have their fall upon my selfe . and if any shall be offended at this worke , my device shall be a dyall with this mottoe , aspicio ut aspiciar , only to all favourers of art i am direct erect plaine , as i am , gentlemen , to you , and desire to be yours in the best of my services , s. m. to the reader . reader , i here present thee with some coelestiall operations drawn from the macrocosmall world , if i should tel you of plurality , it may seem absurd , but i 'le distinguish the word . mundus the world is somtimes taken archtypically , and so is god , only in whose divine minde is an example of all things . mundus the world is somtimes taken angelicall , and this is the hierarchicall government of angels in ceruphins , cherubins , and thrones . mundus the world is somtimes taken elementary , and this is the philosophers common place : the salamander in fire , the birds in air , the fish in water , and men and beasts on earth . somtimes macrocosmally , considering the whol universe , as well aetheriall as subterene , yea , and every orb , and this is imaginarily set down in the praecognita astronomicall . somtimes microcosmally , as in the little world man , and this is described in the last chapter of the praecognita philosophicall . somtimes typically , and that either geographicall or gnomonicall , or mentally in the minde of the workman . geographically in maps or globes , or sphears in plano . gnomonicall in this present art of dialling , of which it may be said that umbra horas phoebi designat climate nostro nodus , quod signum sol tenet arte docet . and by which they must necessarily trace out our times by the orbiculation of the rady of the circle of the body of the sunne . again , the world is mentally considered in the minde of an artist , as in painting , graving , carving , &c. but having thus defined the word , you may think from hence that i am with democrates platonissans , acquainting thee with infinity of worlds , and in his words , stanza 20. — — and to speake out though i detest the sect of epicurus for their manner vile . yet what is true i may not well neglect of truths incorruptible , ne can the stile of vicious pen her sacred worth defile . if we no more of truth should deign to speak then what unworthy mouthes did never soyle , no truths at all mongst men would finde a place but make them spéedy wings , & back to heaven apace . howsoever thou hast here a field large enough to walke in , which if thou affect the light , thou mayst trace out the truth , and i presume i have done that for thee who art a learner , the most plain wayes that were ever published , and have studyed not to make it the art of shadows , so much as the shadow of that art whose gnomons may be said to touch the poles , and whose planes may be severall planispheres , a scale to the geometrician , a pole to the navigator , a chart to the geographer , a zodiaque to the astronomer , a table of houses to the astrologian , the meridian and needle to the surveyor , a dyall to us all , to put us in minde of that pretious time which saith to us fugio , fuge , and which time shall be swallowed up of eternity , when there shall be but one day without tropicall distinctions , where thou shalt not need helps from any other , nor from me who am thine , s. m. in solarium . hic tibi cum numero spectantur nodus & umbra , quae tria quid doceant , commemorare libet umbra notat dextrè quota cursitet hora dici , hincque monet vitam sic properare tuam ast in quo signo magni lux publica mundi versetur mira nodulus arte docet si vis scire , dies quot quilibet occupet horas , id numerus media sede locatus habet . on my friend mr. silvanus morgan , his book of dialling . the use of dials all men understand ; to make them few : & i am one of those . i am not of the mathematick band : nor know i more of vers , then vers from prose . but though nor diallist i am , nor poet : i honour those in either doe excell ; our author 's skill'd in both alike , i know it , shadows , and substance , here run parallel . consider then the pains the author took , and thank him , as thou benefit'st by 's book . edward barwick . on the author and his book . dares zoil or momus for to carp at thee , and let such ideots as some authors be boldly to prosecute or take in hand such noble subjects they not understand , only for ostentation , pride , or fame , or else because they 'd get themselves a name , like that lewd fellow , who with hatefull ire , flinch'd not , but set diana's court on fire : his name will last and be in memory from age to age ▪ although for infamie . what more abiding tombe can man invent then books , which ( if they 'r good ) are permanent and monuments of fame , the which shall last till the late evening of the world be past : but if erroneous , sooth'd with vertues face , their authors cridit's nothing but disgrace . if i should praise thy book it might be thought , friends will commend , although the work be nought , but i 'le forbeare , lest that my verses doe belie that praise that 's only due to you . good wiue requires no bush , and books will speak their authors credit , whether strong or weak . w. leybourn . errata . reader , i having writ this some years since , while i was a childe in art , and by this appear to be little more , for want of a review hath these faults , which i desire thee to mend with thy pen , and if there be any errour in art , as in chap. 17 , which is only true at the time of the equinoctiall , take that for an oversight , and where thou findest equilibra read equilibrio , and in the dedication ( in some copies ) read robert bateman for thomas , and side for signe , and know that optima prima cadunt , pessimas aeve manent . pag. line correct . ● 10 equall lines 18 16 galaxia 21 1 galaxia 21 8 mars ▪ 24 12 scheame 35 1 hath 38 8 of the tropicks & polar circles 40 22 ab is 44 31 artificiall 46 ult heri 49 4 forenoon 63 29 ab 65 11 6 80 16 bd 92 17 arch cd 9 ult in some copies omit center 126 4 happen 126 6 tovvard b ▪ 127 26 before 126 prop. 10 ▪ for sine read tang . elev .   figure of the dodicahedron false cut pag. 4 lf omitted at end of axis 25 for a read d 26 in the east and west diall a omitted on the top of the middle line , c on the left hand , b on the right 55 small arch at b omitted in the first polar plane 58 for e read p on the side of the shadowed line toward the left hand i omitted next to m , and l in the center omitted 81 k omitted in figure 85 on the line fc for 01 read 6 , for 2 read 12 , line mo for 15 read 11 96 a small arch omitted at e & f , g & h omitted at the ende of the line where 9 is 116 i & l omitted on the little epicicle . 122 the argvment of the praecognita geometricall , and of the work in generall . what shall i doe ? i stand in doubt to shew thee to the light ; for momus still will have a flout , and like a satyre bite : his serpentarian tongue will sting , his tongue can be no slander , he 's one to wards all that hath a fling his fingers ends hath scan'd her . but seeing then his tongue can't hurt , fear not my little book , his slanders all last but a spurt , and give him leave to look and scan thee thorough , and if then this momus needs must bite at shadows which dependant is only upon the light . withdraw thy light and be obscure . and if he yet can see faults in the best that ever writ , he must finde fault with me . how ere proceed in private and deline the time of th' day as oft as sun shall shine : and first define a praecognitiall part of magnitude , as usefull to this art . the praecognita geometrical . the arts , saith arnobius , are not together with our mindes , sent out of the heavenly places , but all are found out on earth , and are in processe of time , soft and fair , forged by a continuall meditation ; our poor and needy life perceiving some casual things to happen prosperously , while it doth imitate ▪ attempt and try , while it doth slip , reform and change , hath out of these same assiduous apprehensions made up small sciences of art , the which afterwards , by study , are brought to some perfection . by which we see , that arts are found out by daily practice , yet the practice of art is not manifest but by speculative illustration , because by speculation : scimus ut sciamus , we know that we may the better know : and for this cause i first chose a speculative part , that you might the better know the practice ; and therefore have first chose this speculative part of practicall geometry , which is a science declaring the nature , quantity , and quality of magnitude , which proceeds from the least imaginable thing . to begin then , a point is an indivisible , yet is the first of all dimension ; it is the philosophers atome , such a nothing , as that it is the very energie of all things , in god it carryeth its extreams from eternity to eternity : in the world it is the same which moses calls the beginning , and is his genesis : 't is the clotho that gives clio the matter to work upon , and spins it forth from terminus à quo , to terminus ad quem : in the alphabet 't is the alpha , and is in the cuspe of the ascendant in every science , and the house of life in every operation . again , a point is either centricall or excentricall , both which are considered geometrically or optically , that is , a point , or a seeming point : a point geometrically considered is indivisible , and being centrall is of magnitude without consideration of form , or of rotundity , with reference to figure as a circle , or a globe , &c. or of ponderosity , with reference to weight , and such a point is in those balances which hang in equilibra , yet have one beam longer than the other . if it be a seeming point , it is increased or diminished optically , that is , according to the distance of the object and subject . 't is the birth of any thing , and indeed is to be considered as our principall significator , which being increased doth produce quantity which is the required to magnitude ; for magnitude is no other then a continuation of quantity , which is either from a line to a plain superficies , or from a plain superficies to a solid body : every of which are considered according to the quantity or form . the quantity of a line is length , without breadth or thicknesse , the forme either right or curved . the quantity of a superficies consisteth in length and breadth , without thicknesse , the form is divers , either regular or irregular ; regular are triangles , squares , circles , pentagons , hexagons , &c. an equilaterall triangle consisteth of three right lines & as many angles , his inscribed side in a circle contains 120 degrees . a square of four equall right lines , and as many right angles , and his inscribed side is 90 degrees . a pentagon consisteth of five equall lines and angles , and his inscribed side is 72 degrees of a circle . a hexagon is of six equall lines and angles , and his side within a circle is 60 degrees , which is equall to the radius or semidiameter . an angle is the meeting of two lines not in a streight concurring , but which being extended will crosse each other ; but if they will never crosse , then they are parallel . the quantity of an angle is the measure of the part of a circle divided into 360 degrees between the open ends , and the angle it self is the center of the circle . the quantity of a solid consists of length , breadth , and thickness , the form is various , regular or irregular : the five regular or platonick bodies are , the tetrahedron , hexahedron , octohedron , dodecahedron , icosahedron . tetrahedron is a solid body consisting of four equall equilaterall triangles . a hexahedron is a solid body consisting of six equal squares , and is right angled every way . an octahedron is a solid body consisting of eight equal equilaterall triangles . a dodecahedron is a solid body consisting of 12 equall pentagons . an icosahedron is a solid body consisting of 20 equal equilaterall triangles : all which are here described in plano , by which they are made in pasteboard : or if you would cut them in solid it is performed by mr. wells in his art of shadows , where also he hath fitted planes for the same bodies . a parallel line is a line equidistant in all places from another line , which two lines can never meet . a perpendicular is a line rightly elevated to another at right angles , and is thus erected . suppose ab be a line , and in the point a you would erect a perpendicular : set one foot of your compasses in a , extend the other upwards , anywhere , as at c , then keeping the foot fixed in c , remove that foot as was in a towards b , till it fall again in the line ab , then if you lay a ruler by the feet of your compasses , keep the foot fixed in c , and turn the other foot toward d by the side of the ruler , and where that falls make a marke , from whence draw the line da , which is perpendicular to ab . and so much shall suffice for the praecognita geometricall , the philosophicall followeth . the end of the praecognita geometricall . the argvment of the praecognita philosophicall . not to maintain with nice philosophie , what unto reason seems to be obscure , or shew you things hid in obscurity , whose grounds are nothing sure . 't is not the drift of this my book , the world in two to part , nor shew you things whereon to looke but what hath ground by art . if art confirm what here you read , sure you 'l confirmed be , if reason wonte demonstrate it , learn somwhere else for me . there 's shew'd to you what shadow is , and the earths proper place , how it the middle doth possesse , and how heavens run their race . resolving many a proposition , which are of use , and needfull to be known . the praecognita philosophical . chap i. of light and shadows . he that seeketh shadow in its predicaments , seeketh a reality in an imitation , he is rightly answered , umbram per se in nullo praedicamento esse , the reason is thus rendred as hath been , it is not a reality , but a confused imitation of a body , arising from the objecting of light , so then there can be no other definition then this , shadow is but the imitation of substance , not incident to parts caused by the interposition of a substance , for , umbra non potest agere sine lumine . and and it is twofold , caused by a twofold motion of light , that is , either from a direct beam of light , which is primary , or from a secondary , which is reflective : hence it is , that sun dials are made where the direct beams can never fall , as on the seeling of a chamber or the like . but in vain man seeketh after a shadow , what then , shall we proceed no farther ? surely not so , for qui semper est in suo officio , is semper orat , for there are no good and lawful actions but doe condescend to the glory of god , and especially good and lawfull arts . and that shadow may appear to be but dependant on light , it is thus proved , quod est & existit in se , id non existit in alio : that which is , and subsisteth in it selfe , that subsisteth not in another : but shadow subsisteth not in it selfe , for take away the cause , that is light , and you take away the effect , that is shadow . hence we also observe the sun to be the fountain of light , whose daily and occurrent motions doth cause an admirable lustre to the glory of god ; seeing that by him we measure out our times , seasons , and years . is it not his annuall revolution , or his proper motion that limits our year ? is it not his tropicall distinctions that limits our seasons ? is it not his diurnall motion that limits out our dayes and houres ? and man truly , that arch type of perfection , hath limited these motions even in the small type of a dyall plane , as shall be made manifest in things of the second notion , that is , demonstration , by which all things shall be made plain . chap ii. of the world , proving that the earth possesseth its own proper place . we have now with the philosopher , found out that common place , or place of being , that is , the world , will you know his reason ? 't is rendred , quia omnia reliqua mundi corpora in se includit . i 'le tell you of no plurality , not of planetary inhabitants , such as the lunaries ▪ lest you grabble in darkness , in expecting a shadow from the light without interposition , for can the light really without a substance be its own gnomon ? surely no , neither can we imagine our earth to be a changing cynthia , or a moon to give light to the lunary inhabitants : for if our earth be a light ( as some would have it ) how comes it to passe that it is a gnomon also to cast a shadow on the body of the moon far lesse then it selfe , and so by consequence a greater light cannot seem to be darkned on a lesser or duller light , and if not darkned , no shadow can appear ? but from this common place the world with all its parts , shall we descend to a second grade of distinction , and come now to another , which is a proprius locus , and divide it into proper places , considering it as it is divided into coelum , solum , salum , heaven , earth , sea , we need not so far a distinction , but to prove that the earth is in its own proper place , i thus reason : proprius locus est qui proxime nullo alio interveniente continet locatum : but it is certain that nothing can come so between the earth as to dispossesse it of its place , therefore it possesseth its proper place , furthermore , ad quod aliquid movetur , id est ejus locus , to what any thing moves that is its place : but the earth moves not to any other place , as being stable in its own proper place . and this proper place is the terminus ad quem , to which ( as the place of their rest ) all heavie things tend , in quo motus terminantur , in which their motion is ended . chap iii. shewing how the earth is to be understood to be the center . a center is either to be understood geometrically or optically , either as it is a point , or seeming a point . if it be a point , it is conceived to be either a center of magnitude , or a center of ponderosity , or a center of rotundity : if it be a seeming point , that is increased or diminished according to the ocular aspect , as being somtime neerer , and somtime farther from the thing in the visuall line , the thing is made more or lesse apparent . a center of magnitude is an equal distribution from that point , an equality of distribution of the parts , giving to each end alike , and to each a like vicinity to that point or center . a center of ponderosity is such a point in which an unequall thing hangs in equi libra , in an equall distribution of the weight , though one end be longer or bigger than the other of the quantity of the ponderosity . a center of rotundity is such a center as is the center of a globe or circle , being equally distant from all places . now the earth is to be understood to be such a center as the center of a globe or sphear , being equally distant from the concave superficies of the firmament , neither is it to be understood to be a center as a point indivisible , but either comparatively or optically : comparatively in respect of the superior orbs ; optically by reason of the far distance of the one from the earth ; as that the fixed stars being far distant seeme , by the weaknesse of the sense , to be conceived as a center indivisible , when by the force and vigour of reason and demonstration , they are found to exceed this globe of earth much in magnitude ; so that what our sense cannot apprehend , must be comprehended by reason : as in the circles of the coelestiall orbs , because they cannot be perceived by sense , yet must necessarily be imagined to be so . whence it is observable , that all sun dials , though they stand on the surface of the earth , doe as truly shew the houre as if they stood in the center . chap iv. declaring what reason might move the philosophers and others to think the earth to be the center , and that the world moves on an axis , circa quem convertitur . ocular observations are affirmative demonstrations , so that what is made plain by sense is apparent to reason : hence it so happeneth , that we imagine the earth to move as it were on an axis , because , both by ocular and instrumentall observation , in respect that by the eye it is observed that one place of the skie is semper apparens , neither making cosmicall , haeliacall or achronicall rising or setting , but still remaining as a point , as it were , immoveable , about which the whole heavens are turned . these yet are necessary to be imagined for the better demonstration of the ground of art ; for all men know the heavens to be supported only by the providence of god . thus much for the reason shewing why the world may be imagined to be turned on an axis , the demonstration proving that the earth is the center , is thus , not in maintaining unlikely arguments , but verity of observation ; for all gnomons casting shadow on the face of the earth , cast the like length or equality of shadow , they making one & the same angle with the earth , the sun being at one and the same angle of height to al the gnomons . as in example , let the earth be represented by the small circle within the great circle , marked abcd , and let a gnomon stand at e of the lesser circle , whose horizon is the line ac , and let an other gnomon of the same length be set at i , whose horizon is represented by the line bd , now if the sun be at equall angles of height above these two horizons , namely , at 60 degrees from c to g , and 60 from b to f , the gnomons shall give a like equality of shadows , as in example is manifest . now from the former appears that the earth is of no other form then round , else could it not give equality of shadows , neither could it be the center to all the other inferior orbs : for if you grant not the earth to be the middle , this must necessarily follow , that there is not equality of shadow . for example , let the great circle represent the heavens , and the lesse the earth out of the center of the greater , now the sunne being above the horizon ac 60 d. and a gnomon at e casts his shadow from e to f , and if the same gnomon of the same length doth stand till the sun come to the opposite side of the horizon ac , and the sun being 60 degrees above that horizon , casts the shadow from e to h , which are unequall in length ; the reason of which inequality proves that then it did not stand in the center , and the equality of the other proves that it is in the center . hence is also most forceably proved that the earth is compleatly round in the respect of the heavens , as is shewed by the equality of shadows , for if it were not round , one and the same gnomon could not give one and the same shadow , the earth being not compleatly round , as in the ensuing discourse and demonstration is more plainly handled and made manifest . and that the earth is round may appeare , first , by the eclipses , when the shadow of the earth appeareth on the body of the moon , darkning it in whole or in part , and such is the body such is the shadow . again , it appears to be round by the orderly appearing of the stars , for as men travell farther north or south they discover new stars which they saw not before , and lose the sight of them they did see . as also by the rising or setting of the sun or stars , which appear not at the same time to all countries , but by difference of meridians , and by the different observations of eclipses , appearing sooner to the easterly nations then those that are farther west : neither doe the tops of the highest hils , nor the sinking of the lowest valleys , though they may seeme to make the earth un-even , yet compared with the whole greatnesse , doe not at all hinder the roundnesse of it , and is no bigger then a point or pins head in comparison of the highest heavens . thus having run over the systeme of the greater world , now let us say somthing of the compendium thereof , that is man . chap v. of man , or the little world . man is the perfection of the creation , the glory of the creator , the compendium of the world , the lord of the creatures . he is truly a cosmus of beauty , whose eye is the sunne of his body , by which he beholds the never resting motions of the heavens , contemplatively to behold the place of motion ; the place of his eternall rest . lord , what is man that thou shouldest be so mindefull of him , or the son of man that thou so regardest him ? thou hast made a world of wonder in his face . thou hast made him to be a rationall creature , endowed ▪ him with reason , so that his intellect becomes his primum mobile , to set his action at work , nevertheles , man neither moves nor reigns in himselfe , and therefore not for himselfe , but is born not to himselfe , but for his countrey ; therefore he ought to employ himselfe in such arts as may be , and prove to be profitable for his countrey . man is the atlas that supports the earth , a perfect world , though in a second birth : i know not which the compleat world to call , the senslesse world , or man the rationall : one claims compleat in bignesse and in birth , saith she 's compleat , for man was last brought forth . man speaks again , and stands in his defence because he 's rationall , hath compleat sense . nature now seeing them to disagree , sought for a means that they united be : concluded man , that he should guide the sphears , limit their motion in dayes , and moneths , and years : he thinking now his office not in vain , limits the sun unto a diall plain : girdles the world in circles , zones , and climes , to shew his art unto the after times . nature that made him thus compleat in all , to please him more , him microcosmus call , a little world , only in this respect of quantity , and not for his defect : pray , gentle reader , view but well their feature , which being done , pray tell me who 's the greater ? for he hath given me certain knowledge of the things that are , namely to know how the world was made , and the operations of the elements , the beginning , ending , and midst of times , the alteration of the turning of the sun , and the change of seasons , the circuite of years and position of stars , wisd. 7. 17. the ende of the praecognita philosophicall . the argvment of the praecognita astronomicall . you 'r come to see a sight , the world 's the stage , perhaps you 'l sayt's but a star-gazing age , what come you out to see ? one use an instrument ? can speculation yeild you such content ? that you can rest in learning but the name of pegasus , or of swift charleses wane ? and would you learn to know how he doth move about his axis , set at work by jove ? if you would learn the practice , read and then i need not thus intreat you by my pen to tread in arts fair steps , or to attain the way , go on , make haste , relinquent do not stay : or will you scale olympick hils so high ? be sure you take fast hold , astronomie : then in that fair spread canopie no way from thee is hid , no not galezia . they that descend the waters deepe doe see gods wonders in the deepe , and what they be : they that contemplate on the starry skie do see the works that he hath fram'd so high . learn first division of the world , and how 't is seated , i doe come to shew you now . the praecognita astronomical . chap i. of the division of the world , by accidentall scituation of the circles . cosmus , the world , is divided by microcosmus the little world , into substantiall and imaginary parts : now the substantiall are those materiall parts or substance of which the world is compacted and made a body , by the inter-folding of one sphear within another , as is the sphear of saturn , jupiter , mars , sol , &c. and these of themselves have a gentle and proper motion , but by violence of the first mover , have a racked motion contrary to their own proper motion : whence it appears , that the motion of the heavens are two , one proper to the sphears as they are different in themselves , the other common to all . by phebus motion plainly doth appear , how many dayes doe constitute one yeare . will you know how many days doe constitute a year , he telleth you who saith , ter centum ter viginti cum quinque diebus sex horas , neque plus integer annus habet . three hundred sixty five dayes , as appear , with six houres added , make a compleat year . the just period of the suns proper revolution . perpetuus solis distinguit tempora motus . the imaginary part traced out by mans imagination , are circles , such is the horizon , the equator , the meridian , these circles have of themselves no proper motion , but by alteration of place have an accidentall division , dividing the world into a right sphear , cutting the parallels of the sun equally or oblique , making unequall dayes and nights : whence two observations arise : first , where the parallels of the sun are cut equally , there is also the dayes and nights equall . secondly , where they are cut oblique , there also the dayes and nights are unequall . the variety of the heavens are diversly divided into sphears , or severall orbs , and as the poets have found out a galazia , the milkie way of juno her brests , or the way by which the gods goe to their palaces , so they will assigne to each sphear his severall god . goddesse of heralts . caliope in the highest sphears doth dwell , astrologie . amongst the stars urania doth excell , philosophie . polimnia , the sphear of saturn guides , gladnesse , sterpsicore with jupiter abides . historie , and clio raigneth in mans fixed sphear . tragedic . melpomine guides him that gvids the year solace . yea , and erata doth fair venus sway . loud instruments . mercury his orbe euturpe doth obey . ditty . and horned cynthia is become the court of thalia to sing and laugh at sport . where they take their places as they come in order . the sphear is said to be right where the poles have no elevation , but lie in the horizon , so that to them the equinoctiall is in the zenith , that is , the point just over their heads . the sphear is oblique in regard of its accidentall division , accidentally divided in regard of its orbicular form ; orbicular in regard of its accidentall , equall variation orbicular , it appears before in the praecognita philosophicall , his equall variation is seen by the equall proportion of the earth answering to a coelestiall degree , for circles are in proportion one to another , and parallel one to another are cut equally , so is the earth to the heavens ; having considered them as before , we will now consider another sort of sphear , which is called parallel . this parallel sphear is so that the parallels of the sun are parallel to the horizon , having the poles in their zenith , being the extream intemperate , colde , and frozen zone : ovid in his banishment complaines thus thereof . hard is the fright in scythia i sustain , over my head heavens axis doth remain . chap ii. of the circles of the horizon , the equator , and the meridian . the greatest circle of a sphear is that which divides it in two equall parts , and that because it crosseth diametrically , and the diameter is the longest line as can be struck in a circle , and therefore the greatest , which great circles are represented in the following figure , representing the circles of a sphear in an oblique latitude , according to the latitude or elevation of the pole here at london , which is 51 deg. 32 min. being north latitude , because the north pole is elevated . the horizon is a great circle dividing the part of heauen seen , from where we imagine an antipodes , the inhabitants being to us an antipheristasin , our direct opposites , so that while the sun continues visible to us , it is above our horizon , and so continues day with us , while it is night with our opposites ; and when the sun goes down with us it appears to them , making day with them while it remaineth night with us , and according to the demonstration , is expressed by the greot circle marked nsew , signifying the east , west , north , and south parts of the horizon . so now if you imagine a circle to be drawn from the suns leaving our sight , through those azimuth points of heaven , then that circle there imagined is the horizon , and is accidentally divided as a man changes his place , and divides the world in a right or oblique sphear . the meridian is a great circle scituated at right angles to the horizon , equally passing between the east and west points , and consequently running due north and south , and passeth through the poles of the world , being stedfastly fixed , it is represented by the great circle marked ndsc , and is accidentally divided , if we travell east or west , but in travailing north or south altereth not , & when the sun touches this circle , it is then mid-day or noon : now if you imagine a circle to passe from the north to the south parts of the horizon , through your zenith , that circle so imagined is your meridian , from which meridian we account the distance of houres . the aequinoctiall likewise divides the world in two equall parts , crossing at right angles between the two poles , and is therefore distant from each pole 90 degrees , and is elevated from the horizon on the contrary side of the poles elevation , so much as the pole wants of 90 deg. elevation , demonstrated in the scene by the circle passing from a to b , and is accidentally elevated with the poles as we change our horizon , and when the sun touches this circle , the dayes and nights are then equall , and to those that live under this citcle the dayes and nights hang in equilibra continually , and the sun doth move every houre 15 degrees of this circle , making the houre lines equall , passing 15 degrees in one houre , 30 degrees in two houres , 45 degrees in three houres , 60 degrees for four , and so increasing 15 degrees as you increase in houres . this i note to the intent you may know my meaning at such time as i shall have occasion ro mention the aequinoctiall distances . the axis of the world is that which the stile in every diall represents , being a line imaginary , supposed to passe through the center of the world , from the south to the north part of the meridian , whose outmost ends are the poles of the world , this becomes the diameter , about which the world is imagined to be turned in a right sphear having no elevation , in an oblique to be elevated above the horizon and the angle at the center , numbred on the arch of the meridian between the apparent pole and the horizon , is the elevation thereof , represented by the streight line passing from e to f , the arch en being accounted the elevation thereof , which according to our demonstration is the latitude of london . the stars that doe attend the artick or north pole , are the greater and lesser beare , the last star in the lesser bears tale is called the pole star , by reason of its neerness to it : this is the guide of mariners , as appeareth by ovid in his exile , thus you great and lesser bear whose stars doe guide sydonian and graecian ships that glide even you whose poles doe view this lesser ball , under the western sea neere set at all . the stars that attend the southern pole is the cross , as is seen in the globes . lord be my pole , make me thy style , lord then thy name shall be my terminus ad quem . video coelos opera manuum tuarum , lunam & stellas que tu fundasti . chap iii. of the severall sorts of planes , and how they are known . dyals are the dayes limiters , and the bounders of time , whereof there are three sorts : horizontall , erect , inclining : horizontall are alwayes parallel to the horizon : erect , some are erect direct , others erect declining : inclining also are direct or declining : for more explanation the figure following shall give you better satisfaction , where the horizon marked with diverse points of the compasse shall explain the demonstration : now if you imagine circles to passe through the zenith a , crossing the horizon in his opposite points , as from sw through the verticall point a , passing to the opposite point of south-west to north-east , those , or the like circles , are called azimuthes , parallel to which azimuthes all erect sciothericals doe stand . those planes that lie parallel to the horizontall circle are called horizontall planes , and his style makes an angle with the pole equall to the elevation thereof ; then the elevation of the pole is the elevation of the style . erect verticals are such which make right angles with the horizon , and lie parallel to the verticall point , and these , as i told you before , were either direct or declining . direct are those that stand in a direct azimuth , beholding one of the four cardinall quarters of the world , as either direct east , west , north , or south , marked with these letters news , or declining from them to some other indirect azimuth or side-lying points . erect north and south are such as behold those quarters , and cuts the meridian at right angles , so that the planes crosse the meridian due east and west , and the poles are their styles , equally elevated according to the aequinoctiall altitude , being the complement of the poles elevation . for in all north faces , planes , or dials , the style beholds the north pole , and in all south faces , the style beholds the south pole : therefore , where the north pole is elevated , there the north pole must be pointed out by the style , and where the south pole is elevated vice versa . the second sort of verticals are declining , which ate such that make an acute angle with the quarter from which they decline ; for an acute angle is lesse then a right angle , and a right angle is 90 degrees : these declining planes lying in some accidentall azimuthe . for supposing a diall to turn from the south or north towards the east or west , the meridian line of the south declines eastward , happening in these azimuthes or between them . south declining east south declining west s by e 11 15 or to these points of the west decliners , or between them . s by w 11 15 s s e 22 30 s s w 22 30 s e by s 33 45 s w by s 33 45 south-east 45 00 south west 45 00 s. e by e 56 15 s w by w 56 15 e s e 67 30 w s w 67 30 e by s 78 45 w by s 78 45 east 90 00 west . 90 00 again , north decliners , declining toward the east and west , doe happen in these azimuthes or between them . north declining east north declining west n by e 11 15 or to these points of the west decliners , or between them . n by w 11 15 n n e 22 30 n n w 22 30 n e by n 33 45 n w by n 33 45 north-east 45 00 north west 45 00 n e by e 56 15 n w by w 56 15 e n e 67 30 w n w 67 30 e by n 78 45 w by n 78 45 east . 90 00 west . 90 00 by which it appeareth that every point of the compasse is distant from the meridian 11 degrees 15 minutes . the third sort of planes are inclining , or rather reclining , whose upper face beholds the zenith , and in that respect is called reclining , but if a diall be made on the nether side , and thereby respect the horizon , it is then called an incliner , so that the one is the opposite to the other . these planes are likewise accidentally divided , for they are either direct recliners , reclining from the direct points of east , west , north ; and south , and in this sort happens the direct polar and aequinoctiall planes , as infinite more according to the inclination or reclination of the plane , or they are as erect planes doe become declining recliners , which looke oblique to the cardinall parts of the world , and obtusely to the parts they respect . suppose a plane to fall backward from the zenith , and by consequence it falls towards the horizon ; then that represents a reclining plane , such you shall you suppose the aequinoctiall circle in the figure to represent , reclining from the north southwards 51 degrees from the zenith , or suppose the axis to represent a plane lying parallel to it , which falls from the zenith northward reclining 38 degrees , one being aequinoctiall , the other a polar plane . but for the inclining decliners you shall know them thus , forasmuch as the horizon is the limiter of our sight , and being cut at right angles representeth the east , west , north , and south points , it may happen so that a plane may lie between two of these quarters in an accidentall azimuth , and so not beholding one of the cardinall quarters is said to decline : again , the said plain may happen not to stand verticall , which is either inclining or reclining , and so are said to be inclining decliners : first , because they make no right angle with the cardinal quarters : secondly , because they are not verticall or upright . there are other polar planes , which lie parallel to the poles under the meridian , which may justly be called meridian plains , and these are erect direct east and west dials , where the poles of the plane remain , which planes if they recline , are called position planes , cutting the horizon in the north and south points , for circles of position are nothing but circles crossing the horizon in those points . chap iv. shewing the finding out of a meridian line after many wayes , and the declination of a plane . a meridian line is nothing else but a line whose outmost ends point due north and south , and consequently lying under the meridian circle , and the sun comming to the meridian doth then cast the shadow of all things northward in our latitude ; so that a line drawn through the shadow of any thing perpendicularly eraised , the sun being in the meridian , that line so drawn is a meridian line , the use whereof is to place planes in a due scituation to their points respective , as in the definition of this circle i shewed there was accidentall meridians as many as can be imagined between place and place , which difference of meridians is the longitude , or rather difference of longitude , which is the space of two meridians , which shews why noon is sooner to some then others . the meridian may be found divers wayes , as most commonly by the mariners compasse , but by reason the needle hath a point attractive subject to errour , and so overthroweth the labour , i cease to speake any further . it may be found in the night , for when the starre called aliot , seems to be over the pole-starre , they are then true north , the manner of finding it , mr. foster ▪ hath plainly laid down in his book of dyalling , performed by a quadrant , which is the fourth part of a circle , being parted into 90 degrees . it may also be fouhd as master blundevile in his booke for the sea teacheth , being indeed a thing very necessary for the sea , which way is thus : strike a circle on a plain superficies , and raise a wire , or such like , in the center to cast a shadow , then observe in the forenoon when the shadow is so that it just touches the circumference or edge of the circle , and there make a mark ; doe so again in the afternoon , and at the edge where the shadow goes out make another mark , between which two marks draw a line ; which part in halfe , then from that middle point to the center draw a line which is a true meridian . or thus , draw a great many circles concentricall one within another , then observe by the circles about noone when the sun casts the shortest shadow , and that then shall represent a true meridian , the reason why you must observe the length of the shadow by circles & not by lines is , because if the sun have not attained to the true meridian it wil cast its shadow from a line , and so my eye may deceive me , when as by circles the sun casting shadow round about , still meetes with one circumference or other , and so we may observe diligently . secondly , it is proved that the shadow in the meridian is the shortest , because the sun is neerest the verticall point . thirdly , it is proved that it is a true meridian for this cause , the sun , as all other luminous bodies , casts his shadow diametrically , and so being in the south part casts his shadow northward , and is therefore a true meridian . but now to finde the declination of a wall , if it be an erect wall draw a perpendicular line , but if it be a declining reclining plane , draw first an horizontall line , and then draw a perpendicular to that , and in the perpendicular line strike a style or small wyre to make right angles with the plane , then note when the shadow of the style falleth in one line with the perpendicular , and at that instant take the altitude of the sun , and so get the azimuthe reckoned from the south , for that is the true declination of the wall from the south . the distance of the azimuthes from the south , or other points , are mentioned in degrees and minutes in the third chapter , in the definition of the severall sorts of planes : or by holding the streight side of any thing against the wall , as is the long square abcd , whose edge ab suppose to be held to a wall , and suppose again that you hold a thrid and plummet in your hand at e , the sun shining , and it cast shadow the line ef , and at the same instant take the altitude of the sun , thereby getting the azimuthe as is taught following , then from the point f , as the center of the horizon . , and from the line fe , reckon the distance of the south , which suppose i finde the azimuthe to be 60 degrees from the east or west , by the propositions that are delivered in the end of this booke , and because there is a quadrant of a circle between the south , and the east or west points , i substract the distance of the azimuthe from 90 degrees , and it shall leave 30 , which is the declination of the wall , equall to the angle efg : but to finde the inclination or reclination , i shall shew when i come to the use of the universall quadrant , or having first found the meridian line , you may prick down the azimuthe . chap v. shewing what houre-lines may be drawn upon any plane . light being the cause primary of shadows , shadows being but the imitation of the secondary cause , that is substance , doth delineate unto us the passing away of time , by receiving light on the substance casting shadow . the sun , though he never moves from the line ecliptique wherein he hath his annuall or yearly motion , yet have a declination from the aequinoctiall north or south , making his diurnall or daily motion , altering the dayes and nights according to all the diversities thereof : for the sun being in the aequinoctiall hath no declination , but in his diurnall motion still declyning from the aequinoctiall makes his progresse towards the north or south , describeth many parallel circles , being parallel to the aequinoctiall , whose farthest distance from either side is 23 deg. 30 minutes , so that so many degrees that the sun is distant from the aequinoctiall , so much is its declination . now if you imagine the circle before described to represent the meridian circle which crossed diametrically , which diameter shall represent the aequinoctiall , then laying down the greatest declination , on either side of it , drawing two lines at that distance , on either side of the aequinoctiall , parallel to it , represent the tropicks , the upper representing the tropick of cancer , marked with ge , the other the tropick of capricorn , marked with hi : and if from each severall degree you draw parallels too , they doe represent the parallels of the sun , which shall shew the diurnall motion of the sun : now if you crosse these parallels with a line from e to h , that then represents the ecliptique ; now if you crosse the aequino-ctiall at right angles with another line , that line represents the axis of the world : then if you lay down from the poles the elevation thereof , to wit , the north and south poles , according to the elevation of the north pole downward , where the number of degrees end make a mark ; then account the same elevation from the south pole upward , and there also make a mark , from which two marks draw a right line , which shall represent your horizon , and cuts the parallels of the sun according to the time of his abiding above the horizon . first , an east and west diall lies parallel to the meridian , therefore the sun in the meridian cannot shine on them ; neverthelesse , though an east and west diall cannot have the houre of 12 on it , yet an east or west position may , because it crosseth the horizon in the north and south . secondly , a direct north diall can have but morning and evening houres on it , and then of no use but when the sun hath north declination , for then his amplitude or distance from the east and west is northward , and so at morning or night shines on the face thereof . thirdly , a north reclining may shew all the houres all the year , if it recline from the north southward , the quantity of the complement of the least meridian altitude , but if but the complement of the elevation of the aequinoctiall , and so become a polar plane , it can then but shew while the sun is in the north signes , for the dyall lying parallel to the aequinoctiall while the sun is in south declination cannot shine on the plane because it lies under . all upright planes declining from the south may have the houre line of 12 , so also may all north decliners , but not in the temperate zone , which is contained between the degrees . south incliners also may have the line of 12 , whose upper face is not below the least meridian altitude , as also if greater then the greatest meridian altitude , then doth the upper face want it . fifthly , all north recliners reclining more then the greatest meridian altitudes complement , may have all the houres but will shew but one part of the yeare . sixthly , all south declinets or recliners may have the line of 12 on them . and now having proceeded thus far in some theoricall demonstration or grounds of dials for the geometricall projection , we will in the next chapter lay down the theoricall demonstration for the arithmeticall calculation , and so proceed to our practicall way of operation as ensueth . chap vi . being the definition of the severall lines of sines , tangents , and secants , to be understood before we can come to arithmeticall calculation . a tangent is a right line without the peripherie to the extremity of the secant to the radius being perpendicular eraised , such is represented by the line bc. a secant is a right line drawn from the center through the circumference to the tangent , such is represented by the line ab , the semidiameter of the same circle is called the radius . you may furthermore for very convenient uses have those lines placed on a ruler , for if from one degree of one quadrant of a semicircle you draw lines to the same degree of the other quadrant , cutting the line ga , that line so cut shall be a line of sines , and if from the centre you draw lines to the tangent line through every degree of the quadrant , that line so cut is a tangent line , whose use is most exquisite and infinite for the solution of many excellent propositions . chap vii . being the fundamentall diagram for the geometricall projection of dials . the style being the representation of the axis of the world , doth become the gnomon or substance casting shadow on all planes lying parallel to some circle or other , as to circles of azimuthes in all verticall dials . so that the figure following is a representation of divers semidiameters , doth plainly shew the theoricall ground of the practick part hereof . where the line in the demonstration , noted the semidiameter of the horizon , signifies the horizon , for so supposing it to represent an horizontall diall , the style or axis must be elevated above it , according to the poles elevation above the horizon , and then the semidiameter or axis of the world represents the style or axis casting shadow being the line ac . the geometricall projection of dials . where note by the way , that if you set one foot of the compasses in b , and with the semidiameter of the equator , fix the other foot in the line bc , keeping that last foot fast , and at that center draw a quadrant divided into six parts , & a ruler from the center of the equator through each division , shall divide the line ab as a contingent line , and if from c to these marks on the line ab you draw lines , it shall be the houre lines of a verticall diall . but supposing a diall to stand verticall , or upright to the horizon ab , as the line bc , then that is represented by the semidiameter of the verticall , and his style again represented by the semidiameter or axis ac , being distant from the verticall equall to the complement of the poles elevation , and again , the aequinoctiall crossing the axis at right angles , the semidiameter thereof is represented by the line bd , the reason why the angle at a hath to his opposite angle at c , the complement of the angle at a , is grounded on this , the three angles of any right lined triangle are equall to two right angles , and a right angle consists of 90 degrees : now the angle at b is 90 degrees , being one right angle , and the angle at a being an angle of 51 degrees , which wants of 90 39 degrees , which is the angle at c , all which being added together doe make 180 degrees , being two right angles : here you see that having two angles , the third is the complement of 180 degrees . chap viii . of the proportion of shadows to their bodies . seeing the zenith makes right angles with the horizon , and a right angle consisteth of 90 degrees , the middle point betwixt both is 45 degrees , the sun being at that height , the shadow of all things perpendicularly raised , are equal to their bodies , so also is the radius of a circle equall to the tangent of 45 degrees : and if the sunne be lower then 45 degrees it must necessary follow the shadow must exceed the substance , because the sun is nigh the horizon , and this is called the adverse or contrary shadow . contrarily , if the sun exceed this middle point , the substance then exceeds the shadow , because the sun is neerer the verticall point . mr. diggs in his pantometria laying down the manifold uses of his quadrant geometricall , doth there shew , that having received the sun beams through the pinacides or sights , that when the suns altitude cuts the parts of right shadow , then the shadow exceeds the substance erected casting shadow as 12 exceeds the parts cut : but in contrary shadow contrary effects . chap ix . to finde the declination of the sun . to give you orontius his words , it is convenient to take the beginning from the greatest obliquation of the sun , because on that almost the whole harmony of all astronomicall matters seeme to depend , as shall be manifest from the discourse of the succeeding canons . wherefore prepare of commodious and elect substance , a quadrant of a circle parted into 90 equall parts , on whose right angled radius let be placed two pinacides or sights to receive the beams of the sun . then erect it toward the south in the time of the solsticials , either in cancer the highest annuall almicanther , or in capricorn the lowest annuall ▪ meridian altitude , also observe the equilibra , or equality of day and night in the time of the aequinoctials , from the meridian altitude thereof substract the least meridian altitude , which is , when the sun enters in the first minute of capricorn , the remainer is the declination , or substract the aequinoctiall altitude from the greatest meridian altitude , the remainer is the declination of the greatest obliquity of the sun in the zodiaque . the height of the sun is observed by the quadrant when the beames are received through the sights by a plummet proceeding from the center , noting the degree of altitude by the thrid falling thereon . you may also take notice that for the continuall variation of the suns greatest declination it ought to be observed by faithful instruments : for as orontius notes that claudius , ptolomie found it to be 23 degrees 51 minutes and 20 seconds , but in the time of albatigine the same number of degrees yet but 35 minutes , alcmeon found it of little lesse , to wit 33 minutes , purbachi and some of his disciples doe affirme the same to be 23 degrees only 28 minutes , yet johanes regiomontan . in the tables of directions , hath alotted the minutes to be 30 , but since dominick maria an italian , and johannes varner of norimburg testifie to have found it to be 29 minutes , to which observation our works doe exactly agree . albeit all did observe the same well neere by like instruments , neverthelesse , not justly by exact construction , or by insufficient dexterity of observation some small difference might happen , but not so much as from ptolomie to our time . having this greatest declination , to finde the present declination is thus , by calculation : as the radius , is to the sine of the greatest declination ; so is the sine of the suns distance from the next aequinoctiall point , that is aries or libra , to the declination required : wherefore in the naturall sines , as in the rule of proportion , multiply the second by the third , divide by the first , the quotient is the sine of the declination . or by the naturall sines , adde the second and third , and substract the first , the remainer is the sine of the present declination . degre . ♈ ♎ ♉ ♏ ♊ ♐ degre . d m d m d m 0 0 0 11 29 20 10 30 1 0 24 11 50 20 23 29 2 0 47 12 11 20 35 28 3 1 11 12 31 20 47 27 4 1 35 12 52 20 58 26 5 1 59 13 12 21 9 25 6 2 23 13 32 21 20 24 7 2 47 13 52 21 30 23 8 3 10 14 11 21 40 22 9 3 34 14 30 21 49 21 10 3 58 14 50 21 58 20 11 4 21 15 8 22 7 19 12 4 45 15 27 22 15 18 13 5 8 15 45 22 23 17 14 5 31 16 3 22 30 16 15 5 55 16 21 22 37 15 16 6 18 16 38 22 43 14 17 6 41 16 56 22 50 13 18 7 4 17 12 22 55 12 19 7 27 17 29 23 0 11 20 7 49 17 45 23 5 10 21 8 12 18 1 23 9 9 22 8 34 18 17 23 13 8 23 8 57 18 32 23 17 7 24 9 19 18 47 23 20 6 25 9 41 19 2 23 22 5 26 10 3 19 16 23 24 4 27 10 25 19 30 23 26 3 28 10 46 19 44 23 27 2 29 11 8 19 57 23 27 1 30 11 29 20 10 23 28 0 de ♓ ♍ ♒ ♌ ♑ ♋ de but i have here added a table of declination of the part of the ecliptique from the aequinoctiall , the use whereof you may discern is very plain , for if you finde the signe on the top , and the degrees downward , the common angle shall be the declination of the sun that day . as if the sun being in the 10 degree of taurus or scorpio , the declination shall bee 14 degrees 50 minutes , and if you finde the signe in the bottome , you shall seeke the degrees on the right hand upward , so the 20 degreee of leo or aquarius hath the same declination with the former . the ende of the praecognita astronomicall . the argvment of practicall sciothericy optical . reader , read this , for i dare this defend , thy posting life on dials doth depend , consider thou how quick the houre 's gone , alive to day , to morrow life is done : then use thy time , and alwayes beare in minde , times hary forehead , yet he 's ball'd behinde , here 's that that will deline to thee and shew how quick time runs , how fast thy life doth goe : yet ( festina lente ) learn the praecognit part , and so attain to practice of this art , whereby you shall be able for to trace out such a path , where sol shall run his race , and make the greater cosmus to appear , delineating day and time of year . horologium vitae . latus ad occasum , nunquam rediturus ad ortum vivo hodie , moriar cras , here natus eram . horologiographia optica . chap i. shewing the making of an horizontall plane to an oblique spheare . from the theoricall demonstration before , take the semidiameter of the horizon with your compasses , then draw the line ab , representing the meridian or line of 12 , and setting one foot in a , describe the quadrant cab , and ca must be at right angles to ab , to which quadrant draw the tangent line fa , which is the line of contingence , then take from the theorical demonstration the semidiameter of the aequator , and placing that on the line ab desctibe a quadrant touching the line of contingence also within the other , represented by the quadrant h e i which divide into 6 parts , and a ruler laid to the center e , make marks where the ruler toucheth the line of contingence , which must be continued beyond f , that so the houre lines may meet with the line bf , where it crosseth that line make marks : then removing the ruler to the center a of the horizontall semicircle , draw lines through each mark of the line of contingence which shall be the houres , number the morning houres from the meridian towards your left hand , and evening or afternoon houres towards the right . the style must be an angle equall to the elevation of the pole , the 12 houre must lie under the meridian circle . the arithmeticall calculation . as the radius , is to the tangent of the aequinoctiall distance of the houre from the meridian ; so is the signe of the elevation of the pole , to the tangent of the houres distance from the meridian . the definition of the aequinoctiall distance is in the definition of the aequinoctiall circle , chap. 1. praecognita astronomicall . the figure of an horizontall diall , for the latitude of london 51d . 30m . south the houres of the afternoon must be the same distance from the meridian , 1 and 11 , 2 and 10 , 3 and 9 , and so of the rest , this is very plain , neither wants any expositor , only you may on the horizontal plane , prick down beyond the houre of 6 a clock , the morning houres of 4 and 5 , and the evening houres of 7 and 8 , by reason that the sun wil shine on the horizontall plane as soone as it is above the horizon . the figure of a south verticall plane , for the latitude of london , which is parallel to the prime verticall . the semidiameter of the verticall is but the tangent of the elevation of the pole to the radius of the horizon . and the semidiameter of the horizon , the tangent of the elevation of the equator to the radius of the verticall . chap ii. shewing the making of a direct verticall diall for an oblique sphear , that is , a direct north or south diall plane . every plane hath a verticall point , and for the making of a verticall diall for the latitude of london , out of the theoricall demonstration chap. 7. praecog. . astron ▪ take the semidiameter of the verticall , and with that , as with the semidiameter of the horizon , describe a quadrant , & draw the tangent line fg , and with the semidiameter of the aequator finish all as in the horizontall : the style must proceed from the center a , and be elevated from the meridian line af , so much as is the complement of the elevation of the pole , and must point toward the invisible pole , viz. the south pole , and hath but 12 houres on it . the arithmeticall calculation . as the radius , is to the tangent of the aequinoctiall distance of the houre from the meridian ; so is the co-sine , that is , the complement sine of the elevation , to the tangent of the houre distance from the meridian required . chap iii. shewing the making of a direct north verticall diall for an oblique sphear , as also a more easie way of drawing the south or horizontall planes . the north diall is but the back side of the south diall ; and differeth little from it , but in naming of the houres , for accounting the sixth houre from the meridian in the direct south verticall , to be the same in the direct north verticall , and accounting the first houres on the east side of the south , on the west side of the north plane , and so vice versa , the first houres on the west side of the south , on the east side of the north plane , as by the figure appeareth . and because the north pole is elevated , the style must point up toward it the visible pole . it must have but the first and last houres of the south plane , because the sun never shines but at evening or morning on a north wall in an oblique sphear , and but in somer , because then the sun hath north declination , but in a right sphear , it may shew all the houres as a south diall , but for a season of the yeare . but if you will make the verticall plane or horizontall in a long angled parallelogram , you shall take the secant of the elevation of the pole , which is the same with ac in the fundamentall diagram , and make that your meridian line , and shall take the sine of the elevation of the pole above the meridian , which in a direct south or north is equall to the elevation of the aequinoctiall , and in the fundamentall diagram is the line de , and prick it down from a and c at right angles with the line ac , and so inclose the long square badbcd , it shall be the boūds of a direct north or south diall ; lastly , if from the fundamental diagram you prick down the several tangents of 15 , 30 45 , from band d on the lines bb and dd , & the same distances from c toward b and d , & lastly if from the center a , you draw lines to every one of those marks , they shall be the houre-lines of an erect direct south diall . to make an horizontall diall by the same projection you shall take the secant of 38 deg. 30 min. the elevation of the equator , which in the fundamentall scheme is the line af , for the meridian , and the sine of the elevation of the pole , which in the fundamentall diagram is the same with da , and prick that down from the meridian at right angles both wayes , as in the former planes , and so proceed as before from the six of clock houre and the meridian , with the severall tangents of 15 , 30 , 45 , you shall have constituted a horizontall plane . i have caused the pricked line that goes crosse , and the other pricked lines which are above the houre line of six , to be drawn only to save the making of a figure for the north direct diall , which is presented to you if you turn the book upside down , by this figure , contained between the figures of 4 , 5 , 6 , the morning houres , and 6 , 7 , 8 , the evening . and because the north pole is elevated above this plane 38 deg. 30 min. the axis must be from the center according to that elevation , pointing upward as the south doth downward , so as a is the zenith of the south , c must be in the north . the arithmeticall calculation is the same with the former , also a north plane may shew all the houres of the south by consideration of reflection : for by opticall demonstration it is proved , that the angles of incidence is all one to that of reflection : if any be ignorant thereof , i purposely remit to teach it , to whet the ingenious reader in labouring to finde it . the figure of a direct east and west diall for the latitude of london , ▪ 51 deg. 30 min. east diall . west diall . chap iv. shewing the making of the prime verticall planes , that is , a direct east or west diall . for the effecting of this diall , first draw the line ad , on one end thereof draw the circle in the figure representing the equator ; then draw two touch lines to the equator , parallel to the line ad , these are they on which the houres are marked : divide the equator in the lower semicircle in 12 equall parts , then apply a ruler to the center , through each part , and where it touches the lines of contingence make marks ; from each touch point draw lines to the opposite touch point , which are the parallels of the houres , and at the end of those lines mark the easterly houres from 6 to 11 , and of the west from 1 to 6. these planes , as i told you , want the meridian houre , because it is parallel to the meridian . now for the placing of the east diall , number the elevation of the axis , to wit , the arch dc , from the line of the equator , to wit , the line ad : and in the west diall number the elevation to b ; fasten a plummet and thrid in the center a , and hold it so that the plummet may fall on the line ac for the east diall , and ab for the west diall , and then the line ad is parallel to the equator , and the dial in its right position . and thus the west as well as east , for according to the saying , contrariorum eadem est doctrina , contraries have one manner of doctrine . here you may perceive the use of tangent line , for it is evident that every houres distance is ●●t the tangent of the aequinoctiall distance . the arithmeticall calculation . 1 having drawn a line for the houre of 6 , whether east or west , as the tangent of the houre distance , is to the radius , so is the distance of the houre from 6 , to the height of the style . 2 as the radius is to the height of the style , so is the tangent of the houre distance from 6 , to the distance of the same houre from the substyle . the style must be equall in height to the semidiameter of the equator , and fixed on the line of 6. chap v. shewing the making a direct parallel polar plane , or opposite aequinoctiall . i call this a direct parallel polar plane for this cause , because all planes may be called by their scituation of their poles , and so an aequinoctiall parallel plane , may be called a polar plane , because the poles thereof lie in the poles of the world . the gnomon must be a quadrangled parallelogram , whose height is equall to the semidiameter of the equator , as in the east and west dials , so likewise these houres are tangents to the equator . arithmeticall calculation . draw first a line representing the meridian , or 12 a clock line , and another parallel to the said line for some houre which may have place on the line , say , as the tangent of that houre is to the radius , so is the distance of that houre from the meridian to the height of the style . 2 as the radius is to the height of the style , so the tangent of any houre , to the distance of that houre from the meridian . chap vi . shewing the making of a direct opposite polar plane , or parallel aequinoctiall diall . an aequinoctiall plane lyeth parallel to the aequinoctiall circle , making an angle at the horizon equal to the elevation of the said circle : the poles of which plane lie in the poles of the world . the making of this plane requires little instruction , for by drawing a circle , and divide it into 24 parts the plane is prepared , all fixing a style in the center at right angles to the plane . as the radins , is to the sine of declination , so is the co-tangent of the poles height , to the tangent of the distance of the sub-stile from the meridian . if you draw lines from 7 to 5 on each side , those lines so cut shall be the places of the houre lines of a parallel polar plane , now if you draw to each opposite from the pricked lines , those lines shall be the houre lines of the former plane . chap vii . shewing the making of an erect verticall declining diall . if you will work by the fundamentall diagram , you shall first draw a line , such is the line ab , representing the meridian , then shall you take out of the fundamentall diagram the secant of the latitude , viz. ac , and prick it down from a to b , and at b you shall draw a horizontall line at right angles , such is the line cd , then you shall continue the line ab toward i , and from that line , and where the line ab crosseth in cd , describe an arch equall to the angle of declination toward f if it decline eastward , and toward g if the plane decline westward . then shall you prick down on the line bf , if it bean easterly declining plane , or from b to g if contrary ; the secant complement of the latitude , viz. ag in the fundamentall diagram , and the sine of 51 degrees , viz. da , which is all one with the semidiameter of the equator , and therewithall prick it down at right angles to the line of declination , viz. bf , from b to h and g , and from f towards k and l , then draw the long square kikl , and from b toward h and g , prick down the severall tangents of 15 , 30 , 45 , and prick the same distance from k and l towards h and g : lastly , draw lines through each of those points from f to the horizontall line cd , and where they end on that line to each point draw the houre lines from the point a , which plane in our example is a verticall declining eastward ▪ 45 degrees , and it is finished . but because the contingent line will run out so far before it be intersected , i shall give you one following geometricall example to prick down a declining diall in a right angled parallelogram . now for the arithmeticall calculation , the first operation shall be thus : as the radius , to the co-tangent of the elevation , so is the sine of the declination , to the tangent of the substiles distance from the meridian of the place . then , ii operation . having the complement of the declination and elevation , finde the styles height above the sub-stile , thus , as the radius , to the co-sine of the declination , so the co-sine of the elevation , to the sine of the styles height above the substyle . iii operation . as the sine of elevation , is to the radius , so the tangent of declination , to the tangent of the inclination of the meridian of the plane to the meridian of the place . iv operation . having the styles height above the substyle , and the angle at the pole comprehended between the houre given and the meridian of the plane say . as the radius , to the sine of the styles height above the substyle ; so is the tangent of the angle at the pole , comprehended between the houre given and the meridian of the plane , to the tangent of the houre distance from the substyle . thus the arithmeticall way being laid down , another geometricall follows . you shall first on the semidiameter of the horizon , viz. ab , describe the arch bc the declination of the plane , and bd the complement of the elevation of the pole , then shall you draw the lines ac and ad , and at b you shall raise the perpendicular dcb . the figure of an upright plane declining from the south eastward 30 degrees . now good reader , labour to understand my plaine meaning in this , labouring only not to confound thy memory or capacity , & therefore give you also to understand that such are the houre distances of a westerly declining plane , as are those of an easterly , only changing the side of the plane , and naming it by the complementall houres , the complemental houres i call those that added together make 12 , as followeth . forenoon houres of the declining east plane . 6 complemental houres are 6 are afternoon houres of a declining west plane . 7 5 8 4 9 3 10 2 11 1 so that if the houres of the easterly declining plane be 6 , 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , the houres of the westerly declining diall is 6 , 5 , 4 , 3 , 2 , 1 , 12 , 11 , 10 , 9 , stil keeping the same distances of the houre lines in one as the other , so that if an easterly declining be but turned the back side , it represents a westerly declining dial as much , and the style must stand over his substyle , and whereabouts the houre lines are closest or neerest together , thereabout is the substyle . now having shewed you the making of all horizontall and verticall , whether direct or declining , polar or aequinoctiall , i shall proceed to shew the projecting of those which are oblique , whether declining reclining , or inclining , reclining , &c. whereto , for the more ease , i have calculated to every degree of a quadrant the houre arches of the horizontall planes , from one degree of elevation till the pole is in the zenith . the table and use followeth in severall chapters . here followeth the table of the arches of the houre lines distance from the meridian in all horizons , from one degree of elevation , till the pole is elevated 30 degrees , by which is made all direct murall , whether upright , or reclining dials .   1 11 2 10 3 9 4 8 5 7 6 6 1 0 16 0 35 1 00 1 44 3 43   2 0 32 1 9 2 00 3 27 7 25   3 0 48 1 44 3 00 5 11 11 3   4 1 5 2 19 4 00 6 54 14 36   5 1 20 2 53 4 59 8 65 18 1   6 1 36 3 27 5 58 10 16 21 19   7 1 52 4 1 6 57 11 55 24 27   8 2 8 4 35 7 54 15 9 30 3   9 2 24 5 9 8 54 15 9 30 3   10 2 40 5 44 9 51 16 45 32 57   11 2 56 6 18 10 48 18 18 35 27   12 3 11 6 51 11 44 19 48 37 49   13 2 27 7 24 12 41 21 17 40 1   14 3 46 7 57 13 36 22 44 42 5   15 3 59 8 30 14 31 24 9 44 0   16 4 14 9 2 15 25 25 31 45 49   17 4 28 9 35 16 17 26 51 47 30   18 4 44 10 7 17 10 28 9 49 4   19 5 15 11 10 18 53 30 39 51 55   20 5 15 11 10 18 53 30 39 51 55   21 5 29 11 41 19 43 31 50 53 13   22 5 44 12 13 21 20 34 5 55 34   23 5 58 12 43 21 20 34 5 55 34   24 6 13 13 13 22 8 35 10 56 37   25 6 28 13 43 22 54 36 12 57 37   26 6 42 14 12 23 40 37 13 58 34   27 6 57 14 41 24 25 38 11 59 27   28 7 10 15 00 25 9 39 11 60 37   29 7 24 15 39 25 52 40 2 61 4   30 7 38 16 6 26 36 40 54 61 49   the continuation of the arches of the horizontall planes , from 30 to 60 deg. of elevation of the pole   1 11 2 10 3 9 4 8 5 7 6 6 31 7 51 16 34 27 15 41 44 62 30   32 8 5 17 1 27 55 42 32 63 11   33 8 19 17 27 28 37 43 20 63 49   34 8 31 17 54 29 13 44 5 64 24   35 8 44 18 20 29 50 44 49 64 58   36 8 57 18 45 30 27 45 31 65 30   37 9 10 19 9 31 2 46 12 66 ●0   38 9 22 19 34 31 37 46 50 66 29   39 9 24 19 58 32 11 47 28 66 56   40 9 47 20 22 32 44 48 4 67 23   41 9 58 20 45 33 16 48 39 67 47   42 10 10 21 7 33 47 49 13 68 10   43 10 21 21 30 34 18 49 45 68 33   44 10 32 21 51 34 47 50 16 68 55   45 10 44 21 45 35 16 50 46 69 15   46 10 54 22 33 35 53 51 15 69 34   47 11 6 22 54 36 11 51 43 69 53   48 11 16 23 14 36 37 52 9 70 11   49 11 26 23 33 37 2 52 35 70 27   50 11 36 23 51 37 27 53 0 70 43   51 11 46 24 10 37 52 53 24 71 13   52 11 55 24 57 38 14 53 46 71 24   53 12 5 24 45 38 37 54 8 71 27   54 12 14 25 2 38 58 54 30 71 40   55 12 23 25 19 39 19 54 49 71 53   56 12 32 25 35 39 39 55 9 72 5   57 12 40 25 51 39 59 55 28 72 17   58 12 48 26 5 40 18 55 45 72 28   59 12 56 26 20 40 36 56 2 72 39   60 13 4 26 33 40 54 56 19 72 49   the continuation of the arches of the horizontall planes , from 60 deg. of elevation , till the pole is in the zenith .   1 11 2 10 3 9 4 8 5 7 6 6 61 13 11 26 48 41 10 56 34 72 58   62 13 18 27 1 41 26 56 49 73 7   63 13 25 27 13 41 42 57 3 73 16   64 13 32 27 26 41 57 57 17 73 24   65 13 39 27 37 42 11 57 30 73 32   66 13 45 27 49 42 25 57 42 73 39   67 13 52 27 59 42 38 57 54 73 46   68 13 56 28 9 42 50 58 5 73 53   69 14 3 28 19 43 2 58 16 73 59   70 14 8 28 29 43 13 58 26 74 5   71 14 13 28 38 43 24 58 36 74 11   72 14 18 28 46 43 34 58 44 74 16   73 14 22 28 55 43 43 58 53 74 21   74 14 26 29 2 43 52 59 1 74 25   75 14 30 29 9 44 0 59 8 74 29   76 14 34 29 5 44 8 59 15 74 34   77 14 38 29 22 44 15 59 21 74 37   78 14 41 29 27 44 22 59 26 74 41   79 14 44 29 32 44 28 59 32 74 44   80 14 47 29 37 44 34 59 37 74 47   81 14 49 29 42 44 39 59 41 74 49   82 14 51 29 44 44 43 59 43 74 50   83 14 53 29 49 44 47 59 49 74 53   84 14 55 29 52 44 51 59 52 74 55   85 14 57 29 54 44 53 59 54 74 57   86 14 58 29 56 44 56 59 57 74 58   87 14 59 29 58 44 58 59 58 74 59   88 14 59 29 59 44 59 59 58 74 59   89 14 59 30   44 59 59 59 75     90 15   30   45   60   75     chap viii . shewing the use of this table both in verticall and horizontall planes . for an horizontall diall enter the table with the elevation of the pole on the left hand , and the arches noted against the houres and the elevation found , are the distance of the houres from the meridian . for a verticall or direct south or north , enter the table with the complement of the elevation on the right side , and the common meeting of the houres at top , and the complement of elevation , is the distance of the houres from the meridian in the said plane . for every horizontall plane is a direct verticall in that place whose latitude or distance of their zenith from the aequator , is equall to the complement of the elevation of the horizontall planes axis or style . as to make an horizontall diall for the latitude of 51 degrees , i enter the table and finde these arches for 1 and 11 , for 2 and 10 , &c. now the same distances are the distances of the houre lines of a direct south plane , where the pole is elevated the complement of 51 degrees , that is 39 degrees , for 51 and 39 together doe make 90. so to make a verticall diall , i enter the table with 39 , the complement of the elevation of the pole , and finde the arches answering to 1 and 11 , to 2 and 10 , &c. thus much in generall of the use of the table , now followeth the use in speciall . chap ix . shewing the use of the tables in making any declining or inclining direct dials . let the great circle abcd represent the meridian , a the north , and c the south , then the line ef represents a south reclining plane , while it fals back from the south northward , and represents an inclining plane while it respects the horizon . this is sufficiently discussed before . so much as the plane reclines northward beyond the complement of the elevation of the pole , so much is the north pole elevated above the plane , as here the plane is represented by ef , the elevation of the style or axis the arch eg , therefore in this case substract the complement of the reclination of the plane from the elevation of the elevated pole , and the remainer is the arch of the poles elevation above the plane , with which elevation enter the table in the left margent , and there are the houre arches from the meridian . if the reclination of the plane be lesse then the complement , as is ik , substract the arch of reclination from the complement of the elevation , there is left the elevation of the south pole above the plane , and with the complement of the elevation of the pole above the plane enter the table on the right margent , and there shall you finde the distance of the houres : and herein mr. faile failed , for instead of substracting one from the other , he addeth one to another , causing a great errour . the distance of every houre of the north incliner on the back side of the south incliner as much are equall , saving that the houres on the north side must be named by the complement houres to 12 , and as the north pole is above one plane , so is the south pole above the other , you may also conceive the like in making of all south incliners and recliners , by framing the position of the plane on the south side as the figure is on the north : and in north recliners lesse then the elevation of the pole , adde the reclination of the flat , which is the elevation of the north pole above the plane : herein mr. fail failed also , as depending on the former , following the doctrine of contraries , which formost well examined would have saved the opening of a gap to this second errour : with the said elevation found enter the table for the horizontal arches , and thereby make a horizontall ▪ plane as is shewed , so is the diall also prepared . if it recline that it lie between the horizon and the equator , then to the elevation of the pole adde the complement of the reclination , which is the height of the style above the plane , and finish it as a horizontall plane for that latitude , and not as a verticall , as mr. faile would have it , because every reclining plane is a horizontall plane where the pole is elevated according to the style . in a given plane oblique to the meridian , and to the horizon , and to the prime verticall , that is , a given plane inclining declining , to finde as well the meridian of the place as of the plane , and the elevation of the pole above the plane : prob. 3 , petici , liber gnomonicorum . to give you the parallel of pitiscus his example , we will prosecute it according to the naturall tangents in his example , and give you his words . let the meridian of the place be abcd ▪ the horizon aec , the prime verticall bed , the orientall point e , the verticall declined bkd , and right angled at k , the poles of the world g and i : the poles of the planes h , the meridian ghi , the angle of declination ebf , the arch of inclination bk . but before all things the arch k , or the distance of the meridian of the place nl is from the vertical plane kl should be sought by the second axiome , then the arke bn by the third or fourth axiome , after these the angle bkn , that is , in one word , the triangle bkn is found , by which discharged , the arke bn is found either equal to the poles elevation , or greater or lesser . if the arke be equal to the complement of the poles elevation , by it is a token the plane is oblique under the meridian , to be inclined unto the pole , in that case the meridian of the place and of the plane , and also the axis doe concur in the same line g l ▪ if the plane be supposed to fall in the same great circle kn , but if the plane be not supposed , but in some parallel of the same , and the axis be somwhat carryed away , as necessarily it is done if the sciotericall be absolved , the meridian of the plane and place are two lines parallel between themselves , and are mutually joyned together according to the difference of longitude of the place and of the plane , which difference is according to the angle hgc , which is the complement of the angle bnk late found , because the angle kgh is right by 57. p. 1. yea , forasmuch as the meridian of the plane may goe by the poles of the plane , but concurring at g or n are equall to two right , by 20 p. 1. example , let the plane meridionall declined to the right hand 29 de . 59 m. inclining toward the pole artick 23 de . 3 m. the elevation of the pole 49 de . 35 m. and there are to be sought in the same the meridian of the place & the plane , and the elevation of the pole or axis above the plane . the calculation shall be thus . to 67874 the tangent of the arke kn the distance of the meridian of the place from the verticall of the plane , 34 de . 10 m. per ax . 2 ▪ the sine of the arke nc 49de . 35 m. whose complement is the arke bn 40de . 25 m per axi. 4. to 60388 the sine of the angle bnk 37d 9m . whose complement is the angle hnc , or hgc 52 de . 51. m. the difference of the longitude of the plane from the longitude of the place , or the distance of the meridians of the place and plane . therefore let the horizon of the place be lc , the verticall of the plane kl , the circle of the plane of the horizon knc , in which there is numbred from k towards c 34 de . 10m . and at the terme of the numeration n , draw the right line l n e , which shall be the meridian of the plane and place , if the center of the sciotericie l or f is taken for the center of the world , and the right line l n f for the axis , but because in the perfection of the diall , ig remaineth the axis , with e the center of the world , not in the right line l n f , but above the same , with props at pleasure , but notwithstanding it is raised equall in height with ei and og , and moreover the plane is somwhat withdrawn frō the axis of the world , therefore the line l n f is now not altogether the meridian of the place , but only the meridian of the plane , or as vulgarly they speake , the substilar . but you may finde the meridian of the place thus , draw ih at right angles to the meridian of the plane , which they vulgarly call the contingence to the common section of the equator , which in the plane let e the center of the world be set from the axis ig in the meridian of the plane l n f. then to the center e , consisting in the line l n e , le the circle of the equator fk be described , and in the same toward the east , because the horizon of the plane is more easterly then the horizon of the place , and moreover the beame is cast sooner or later upon the meridian of the plane then the place , let there be numbered the difference of longitude of the place and plane 52 de . 51 m. and by k the end of the numeration let a right line be drawn , as it were the certain beams of the equator ekh , which where it toucheth the common section of the equator with the plane , to wit , the right line fh , by that point let c the meridian of the place be drawn perpendicular . the second case of the third probleme of pitiscus his liber gnomonicorum . sivero arcus bn , repertus fuerit , &c. but if the arke bn shall be found lesse then the complement of the poles elevation , it is a signe the plane doth consist on this side the pole artick , and moreover above such a plane not the pole artick , but the pole antartick shall be extolled to such an angle as ilm is , whose measure is the arke im , to which , out of the doctrine of opposites , the arke go is equall , which you may certainly finde together with the arke no thus . as mog the right angle , to ng the difference between bn and bg , so ong the angle before found , to og , per axi. 3. as the tangent ong to radius , so the tangent og , to the sine o n , by axi. 2. example ; let the plane be meridionall declined to the right hand 34 de . 30 m. inclined toward the pole artick 16 de . 10 m. and again , let the elevation of the pole be 49 de . 35 m. and there are sought : the meridian of the place : the longitude of the countrey the meridian of the plane : the longitude of the plane ? the elevation of the pole above the plane . the calculation . 1. as bf radius , 100000 , to fc tangent complement of declination 55 de . 30 m. 14550 , so 27843 the sine of the inclination 16 de . 10 m. to 40511 , the tangent of k n 22 de . 31 / 3 m. the distance of the meridian of the place from the verticall of the plane , per axi. 2. the sine of the arke n c 62 de , 532 / 3 m. whose complement is b n 27 de . 61 / 3 m. by which substracted from bg the complement of the poles elevation 40 de . 25 m. there is remaining the arke n g 13 de . 182 / 3 m. by axi. 4. to 61108 the sine of the angle b n k , or o n g 37d . 40 m. per axi. 3. & comp. 1. to 14069 the sine of the arch og the distance of the axis gl from the meridian of the plane ▪ ol 8de . 51 / 3m . by ax . 3. to 18410 the sine of the arch n o , the distance of the meridian of the plane ol , from the meridian of the place n l 30 deg. 36½ m , by axi. 2. the calculation being absolved , let there be drawn the horizon of the place ac , secondly , the verticall of the plane bq , thirdly , the horizon of the plane abcq , in whose quadrant aq , to wit , according to the pole antartique , which alone appeareth above such a plane . first , let be numbred the distance of the meridian of the place from the verticall of the plane 22 de . 3 m. and by the ende of the numeration at p , let the meridian of the plane lp be drawn , then from the point p , let the distance of the meridian of the plane from the meridian of the place be numbered , by the terme of the numeration m , let the meridian of the plane lm be drawn . finally , from the point m , into whatsoever part , let the proper elevation of the pole be numbered , or the distance of the axis from the meridian of the plane 8 de ▪ 51 / 3m . and by the term of the numeration i , let the axis ▪ li be drawn , to be extolled or lifted up on the meridian of the plane lm , to the angle mln . the third case of the third probleme of pitiscus his liber gnomonicorum . si denique arcus bn repertus fuerit major , &c. lastly , if the arke bn be found greater then the complement of the poles elevation bg , it is a token the plane to be inclined beyond the pole artique , and moreover the pole artique should be extolled above such a plane to so great an angle as the angle glo , which the arke go measureth , which arke , together with the arke on in the end you may find in such sort as in the precedent case . example , let there be a meridian plane declining to the right hand 35 de . 54 m. inclining towards the pole artique 75 de . 43 m. and let the elevation of the pole be 49 de . 35½ m. but there is sought the meridian of the plane and place , together with the elevation of the pole above the plane , the calculation shall be thus . to 133874 tangent of the arke kn , the distance of the meridian of the place from the verticall of the plane , 53 de . 14½ m , by axi. 2. the sine of the arke nc 8 de . 29● m. whose complement is bn 81 de . 30½ m. from whence if you substract bg 40 de . 25 m. there remaineth the arke gn 41 de . 5½ m. to 97982 , the sine of the angle bnk , or ong , by axi. 3. to 64399 the sine of the arch og , the distance of the axis from the meridian of the plane 40 de . 51 / 3 m. by axi. 3. to 17483 the sine of the arke o n the distance of the meridian of the plane from the meridian of the place , 10 de . 4 m. by axi. & comp. 2. the calculation being finished , let the horizon of the place be ac , the verticall of the plane kd , the horizon of the plane akcd , in which let be numbered from the vertical point k toward c the distance of the meridian of the place from the vertical of the plane 53 de . 14½ m. and by the end of the numeration let be drawn the meridian of the place ln , then from the meridian of the place , to wit , from the point n backward , let the distance of the meridian of the plane 10 de . 4m . be numbred , and by o the end of the numeration , let lo the meridian of the plane be drawn , from which afterwards let the proper elevation of the pole be numbred , or the distance of the axis from the meridian of the plane 48d . 5½m . and by the term of the numeration g , let the axis lg be drawn , being extolled above the plane bo , to the angle glo . chap x. in which is shewed the drawing of the houre-lines in these last planes not there mentioned , being also part of pitiscus his example in the fourth probleme of his liber gnom . so then , saith he , si axis , &c. if the axis be oblique to the plane , as the foregoing are , as in any plane oblique to the equator many of the houre-lines doe concur at the axis with equal angles , but they are easily found thus . but because pitiscus is mute in defining which part he takes for the right hand and which the left , we must search his meaning . pitiscus was a divine is evident by his own words in his dedication , celsitudini tuae tota vita mea prolixe me excusarem quod ego homo theologus ▪ &c. if we take him as hee was a divine , we imagine his face to be towards the east , then the south is his right hand , and the north is his left hand . that he was an astronomer too , appeareth by his books both of proper and common motion , then we must imagine his face representing the south , the east on his left hand , which cannot be , as shall appear . neither must we take him according to the poets , whose face must be imagined toward the west . in short , take him according to geographie , representing the pole , and this shews the right hand was the east , and left the west , as is evident by the diall before going , for it is a plane declining from the south to the right hand 30 degrees , that is , the east , because it hath the morning houres not the evening , because the sun shines but part of the afternoon on the plane . thus in briefe i have run throngh all planes , and proceed to shew you farther conclusions : but i desire the reader to take notice that in these examples of pitiscus . i have followed his own steps , and made use of the naturall sines and tangents . chap xi . shewing how by the helpe of a horizontall diall , or other , to make any diall in any position how ever . having prepared a horizontall diall as is taught before : on the 12 houre , as far distant as you please from the foot of the style , draw a line perpendicular to the line of 12 , on that describe a semicircle , plasing the foot of the compasses in the crossing of the lines , this semicircle divide into 180 parts , each quadrant into 90 , to number the declination thereon , let the arch of the semicircle be toward the north part of the diall . then prepare a plane slate , such as will blot out what hath been formerly made thereon , and make it to move perpendicularly on the horizontal plane on the center of the semicircle , which wil represent any declining plane by moving it on the semicircle . now knowing the declination of the plane turn this slate towards the easterly part , if it decline towards the east , if contrary to the west , if toward the west , and set it on the semicircle to the degree of declination , then taking a candle and moving the diall till the shadow fall on all the houres of the horizontall plane , mark also where the shadow falls on the declining plane , that also is the same houre on the plane so scituated , drawn from the joyning of the style with the plane . it is so plain it needs no figure . so may you doe in all manner of declining reclining , or reclining and inclining dials , by framing your instrument to represent the position of the plane . note also that the same angle the axis of the horizontal dial makes with the plane , the same elevation must the axis of that plane have , and where it shadows on the representing plane when the shadow of the horizontal axis is on 12 , that is the meridian of the place . by the same also may you describe all the conclusions astronomicall , the almicanthers , circles of height : the parallels of the sun , shewing the declination : the azimuthes , shewing the point of the compasse the sun is in : and all the propositions of the sphere . seeing this is so plain and evident , nay a delightful conclusion , i will not give you farther directions in a matter of so great perspicuity , as to lay down the severall wayes for projecting the sphere on every severall plane , but proceed to shew the making of a general dial for the whole world , which we will use as our declinatorie to finde the scituation of any wall or plane , as shall be required to make a diall thereon , as followeth in the next chapter . chap xii . shewing the making of a diall on a crosse form , as also a universall quadrant drawn from the same projection , as also to describe the tropicks on meridian or polar planes . this universall diall is described by clavius in his eighth book de gnomonicis : but because the artists of these times have found out a more commodious contrivance of it in the fabrique , i shall describe it according to this figure . now to know the houre of the day , you shall turn the plane by the helpe of the needle , so as the end a shall be toward the north , and e toward the south , and elevate the end e to the complement of the elevation , then bringing the box to stand in the meridian , the shoulder of the crosse shall shew you the houre . upon this also is grounded the universall quadrant hereafter described , which instrument is made in brasse by mr. walter hayes as it is here described . prepare a quadrant of brasse , divide it in the limbe into 90 degrees , and at the end of 45 degrees from the center draw the line a b , which shall represent the equator , divide the limbe into 90 degrees , as other quadrants are usually divided , then number both wayes from the line ab the greatest declination of the sun from the north and south , at the termination whereof draw the arch cd which shall be the tropicks , then out of the table of declination , pag. 45 , from b both wayes let there be numbered the declinatiō of the signes according to this table .   g m   ♈ 00 00 ♎ ♉ ♍ 11 30 ♏ ♓ ♊ ♌ 20 30 ♐ ♒ ♋ 23 30 ♑ now the plane it selfe is no other then an east or west diall , numbred on one side with the morning houres , and on the other with the evening houres , the middle line ab representing the equator . and to set it for the houre , you shall project the tropicks and other intermediate parallels of the signes upon them as is hereafter shewed , but that the plane may not run out of the quadrant you shal work thus , opening the compasses to 15 degrees of the quadrant , prick that down both wayes , at which distance draw parallels to the line ab , and with the same distance , as if it were the semidiameter of the equator , describe the semidiameter of the equator on the top of the line ab , which divide into 12 parts , and laying a ruler through the center and each of those divisions in the semicircle to those parallel lines on each side of ab , marke where they cut , and from side to side draw the parallel houre lines as is taught in the making of an east and west diall , make those parallel lines also divided as a tangent line on each side ab , so if this quadrant were held on an east or west wall , and a plummet let fall from the center of the equator where the style stands ( which may be a pin fitted to take out and in , fitted to the height of the distance between the line a b and the other parallels , which is all one with the radius of the small circle ) it shall i say , be in its right scituation on the east or west wall if you let the plummet and threed fall on the elevation of the pole in that place . but because we desire to make it generall , we must describe the tropicks and other parallels of declination upon it , as is usuall to be done on your polar and east and west diall , which how to doe is thus . having drawn the houre lines and equator as is taught from e the height of the style , take all the distances between it and the houre lines where they doe crosse the line ab , and prick them down on the line representing the equator in this figure from the center b. then describe an occult arch of a circle , whereon describe a chorde of 23 degrees 30 minutes , with such other declinations as you intend on your plane . then on the line representing the equator , noted here with the figures of the houres they were taken from , 6 , 7 , 8 , 9 , 10 , 11 , at the marks formerly made , that was taken from e the height of the style , and every of the houres , from these distances i say raise perpendiculars to cut the other lines of declination , so those perpendiculars shall represent those houre lines from whence they were taken , and the distances between the equator and the severall lines of declination shall be the same distances from the equator , and the other parallels of declination upon your plane , through which marks being pricked down upon the severall hourelines from the equinoctiall . if you draw those hyperbolicall lines , you shall have described the parallels of declination required . but if you will performe the same work a second and easie way , worke by this table following , which is universall , and is composed out of the table of right & versed shadow . put this table before thee , & for the point of each houre line whereby the severall parallels of the signes shall pass worke thus . the style being divided into known parts ▪ if ▪ into 12 , take the parts of shadow out of the table in the same known parts by which the style is divided , & prick them down on each houre line as you finde it marked in the table answering the houre both before and after noon . as suppose that a polar plane i finde when the sun is in aries or libra at 12 a clock the shadow hath no latitude , but at 1 and 11 it hath 3 parts 13 min. of the parts of the style , which i prick from the foot of the style on the houres of 1 and 11 both above and beneath the equator : and for 2 and 10 i finde 6 parts 56 min. which i prick down also from the center to the houre lines of 10 and 2 , and so of the other houre lines and parallels , through which if i draw those lines they shall represent the parallels of the declination . a table of the latitude of shadows .   cancer . gemini leo virgo taurus libra aries   p m p m p m p m p m a m 12 5 13 4 25 2 26 0 0 12 1 6 17 5 35 4 5 3 13 11 2 8 11 8 35 7 27 6 56 10 3 14 5 13 31 12 39 12 0 9 4 23 15 22 45 21 21 20 27 8 5 49 6 47 57 45 45 44 47 7 6 vmbra infinita . 6 having promised in the description of the use of this instrument , to shew how to finde the inclination and reclination of a plane , i shal proceed to give you some cautions ; first then , the quadrant is divided in the limbe , as other quadrants are into 90 degrees , by which is measured the angles of inclination or reclination , for if it be a declining plane onely , the declination is accounted from the north or south toward the east or west , if it decline from the north , the north pole is elivated above it , and the meridian-line ascendeth , if it decline from the south , the south pole is elivated above that plane , if it decline from the south eastward , then is the style and sub-style refered toward the west side of the plane , if to the contrary the contrary , and may have the line of 12 except north decliners in the temperate zone , you may make use of the side of the quadrant to finde the declination , as is taught before page 33 , observing the angle as is cut by the shadow of the thred held by the limbe , & through the center , and that side that lieth perpendicular to the horizontal line which shal be the angle , as is before taught : and if the south point is between the poles of the plane and the azimuth , then doth the plane decline eastward , if it be the afternoon you take the azimuth in , if it be the forenoon you take the azimuth in , and the south point be between it and the poles of the planes horizontal line , it doth decline westward , if contrary it is in the same quarter where the sun is : for an inclining plane , which is the angle that it maketh with the horizon ▪ draw a horizontall line and crosse it again with a square , or verticall line , then apply the side of the quadrant to the vertical line at the beginning of the numeration of the deg. on the quadrant , and the angle contained between the thred & plummet , and the applyed side is the inclination ; in all north incliners the north part of the meridian ascendeth , in south incliners the south part , and in east and west incliners , the meridian lyeth parallel with the horizon . and for the reclination it being all one with the inclination , considered as an upper and under face of the same plane , if you cannot apply the side of the quadrant , you may set a square or ruler at right angles with the verticall line drawn on the upper face and apply the side of the quadrant to the edge of the ruler , and measure the quantity of the angle by the thred and plummet : but this is of direct , howsoever these are subject to another passion of declining and inclining together , which must be sought severally , and such are those whose horizontal line declineth toward the north or south and inclination from north or south , towarde the east or west , which must be sought severally . here followeth the tables of right and contrary shadows . a table of right and contrary shadow , to every degree and tenth minute of the quadrant . ☉ alt 0 1 2 3 4 5 6 7 8 9 ☉ alti● s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizontal shadow 0 41378 , 54 687 34 143 44 229 0 171 37 137 10 114 11 97 44 85 23 75 46 60 verticall shadow 10 4137 , 53 589 16 317 14 216 54 164 44 132 43 111 4 95 26 83 37 74 22 50 20 2065 , 23 515 46 294 31 206 3 158 23 128 33 108 7 93 15 81 55 73 1 40 30 1376 , 6 458 22 274 54 196 13 152 29 124 38 105 19 91 9 80 18 71 43 30 40 1031 , 45 412 29 257 40 187 16 147 1 120 56 102 40 89 9 78 44 70 27 20 50 825 , 13 374 55 242 28 179 6 141 56 117 28 100 8 87 14 77 13 69 14 10 60 687 , 34 343 54 229 0 171 37 137 10 114 11 97 44 85 23 75 46 68 3 0   10 11 12 13 14 15 16 17 18 19   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 68 3 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 57 34 51 60 verticall shadow . 10 66 55 60 47 55 40 51 18 47 32 44 16 41 24 38 51 36 34 34 31 50 20 65 49 95 52 54 53 50 38 46 58 43 46 40 57 38 27 36 13 34 12 40 30 64 45 85 59 54 8 49 59 46 24 43 16 40 31 38 4 35 52 33 53 30 40 63 43 85 7 53 24 49 21 45 51 42 47 40 5 37 41 35 31 33 35 20 50 62 43 57 16 52 41 48 44 45 19 42 19 39 40 37 18 35 11 33 16 10 60 61 44 56 27 51 59 48 8 44 47 41 51 39 15 36 56 34 51 32 58 0   20 21 22 23 24 25 26 27 28 29   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 32 58 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 60 verticall shadow . 10 32 40 31 0 29 27 28 3 26 45 25 52 24 25 23 23 22 25 21 ●0 50 20 32 23 30 44 29 13 27 49 26 32 25 21 24 15 23 13 22 15 21 21 40 30 32 6 30 28 28 58 27 36 26 20 25 10 24 4 23 3 22 6 21 13 30 40 31 49 30 12 28 44 27 23 26 8 24 58 23 54 22 53 21 57 21 4 20 50 31 32 29 57 28 30 27 10 25 56 24 47 23 43 22 44 21 48 20 56 10 60 31 16 29 42 28 16 26 57 25 44 24 36 23 33 22 34 21 39 20 47 0   30 31 32 33 34 35 36 37 38 39   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 20 47 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 60 verticall shadow . 10 20 ●9 19 50 19 5 18 21 17 41 17 2 16 25 15 50 15 16 14 44 50 20 20 31 19 43 18 57 18 15 17 34 16 56 16 19 15 44 15 11 14 39 40 30 20 22 19 35 18 50 18 8 17 28 16 49 16 13 15 38 15 5 14 33 30 40 20 14 19 27 18 43 18 1 17 21 16 43 16 7 15 33 15 0 14 28 20 50 20 6 19 20 18 36 17 54 17 15 16 37 16 1 15 27 14 54 14 23 10 60 19 58 19 12 18 29 17 47 17 8 16 31 15 55 15 22 14 49 14 18 0   40 41 42 43 44 45 46 47 48 49   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 14 18 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 60 verticall shadow . 10 14 13 13 43 13 15 12 48 12 21 11 56 11 31 11 8 10 45 10 22 50 20 14 8 13 39 13 10 12 42 12 17 11 52 11 27 11 4 10 41 10 19 40 30 14 3 13 34 13 6 12 39 12 13 11 48 11 23 11 0 10 37 10 15 30 40 13 58 13 29 13 1 12 34 12 8 11 43 11 19 10 56 10 33 10 11 20 50 13 53 13 24 12 57 12 30 12 4 11 39 11 15 10 52 10 30 10 8 10 60 13 48 13 20 12 52 12 26 12 0 11 35 11 11 10 48 10 26 10 4 0   50 51 52 53 54 55 56 57 58 59   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtal shadow 0 10 4 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 60 verticall shadovv . 10 10 1 9 40 9 19 8 59 8 40 8 21 8 3 7 45 7 27 7 10 50 20 9 57 9 36 9 16 8 56 8 37 8 18 8 0 7 42 7 24 7 7 40 30 9 54 9 33 9 12 8 53 8 34 8 15 7 57 7 39 7 21 7 4 30 40 9 50 9 29 9 9 8 50 8 30 8 12 7 54 7 36 7 18 7 1 20 50 9 47 9 26 9 6 8 46 8 27 8 9 7 51 7 33 7 15 6 59 10 60 9 43 9 23 9 3 8 43 8 24 8 6 7 48 7 30 7 13 6 56 0   60 61 62 63 64 65 66 67 68 69   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 6 56 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 60 verticall shadow . 10 6 53 6 36 6 20 6 4 5 49 5 33 5 18 5 3 4 48 4 34 50 20 6 50 6 34 6 17 6 2 5 46 5 31 5 16 5 1 4 46 4 32 40 30 6 47 6 31 6 15 5 59 5 43 5 28 5 13 4 58 4 44 4 29 30 40 6 45 6 28 6 12 5 56 5 41 5 26 5 11 4 56 4 41 4 27 20 50 6 42 6 26 6 10 5 54 5 38 5 23 5 8 4 53 4 39 4 24 10 60 6 39 6 23 6 7 5 51 5 36 5 21 5 6 4 51 4 36 4 22 0   70 71 72 73 74 75 76 77 78 79   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtal shadow 0 4 22 4 8 3 54 3 40 3 26 3 13 3 0 2 46 2 33 2 20 60 verticall shadovv . 10 4 20 4 6 3 52 3 38 3 24 3 11 2 56 2 44 2 31 2 18 50 20 4 17 4 3 3 49 3 36 3 22 3 8 2 55 2 42 2 29 2 16 40 30 4 15 4 1 3 47 3 33 3 20 3 6 2 53 2 40 2 26 2 13 30 40 4 13 3 59 3 45 3 31 3 17 3 4 2 51 2 37 2 24 2 11 20 50 4 10 3 56 3 42 3 29 3 15 3 2 2 48 2 35 2 22 2 9 10 60 4 8 3 54 3 40 3 22 3 13 3 0 2 46 2 33 2 20 2 7 0   80 81 82 83 84 85 86 87 88 89   s s s s s s s s s s p m p m p m p m p m p m p m p m p m p m horizōtall shadow 0 2 7 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 60 verticall shadow . 10 2 5 1 52 1 39 1 26 1 14 1 1 0 48 0 36 0 23 0 10 50 20 2 3 1 50 1 37 1 24 1 11 0 50 0 46 2 34 0 21 0 8 40 30 2 0 1 48 1 35 1 22 1 9 0 57 0 44 0 31 0 19 0 6 30 40 1 58 1 45 1 33 1 20 1 7 0 55 0 42 0 29 0 17 0 4 20 50 1 56 1 43 1 31 1 18 1 5 0 32 0 40 0 27 0 15 0 2 10 60 1 54 1 41 1 28 1 16 1 3 0 50 0 38 0 25 0 13 0 0 0 chap xiii . of the generall description and use of the preceding tablein , the pricking down and drawing the circles of declination and aximuths in any planes . the table you see consisteth of 11 columns , the first being the minutes of the suns altitude , and the greater figures on the top are the degrees of altitude , all the other columns consist of the parts of shadow , and minutes of shadow , noted above with s for shadow , and p m for parts and minutes of shadow , answerable to a gnomon divided into 12 equall parts , and it is , as the sine of a known altitude of the sun , is to the sine complement of the same altitude ; so the length of the gnomon in 10 or 12 parts , to the parts of right shadow : or for the versed shadow , as the sine complement of the given altitude of the sun , to the right sine of the same altitude ; so the style in parts , to the length of the versed shadow so if we enter the table with the given altitude of the sun in the great figures , and if we seeke the minutes in the sides , either noted with horizontall or verticall shadow , according as your plane is , it shall give you the length of the shadow in parts and minutes in the common angle of meeting together . as if we look for 50 de . 40 m. the meeting of both in the table shall be 9 parts 50 min. for the length of the right shadow on a horizontall plane : but for the versed shadow , take the complement of the altitude of the sun , and the minutes in the right side of the table , titled verticall shadow , and the common area of both shall give your desire . by this table it appeareth first , that the circles of altitude either on the horizontall or verticall planes are easily drawn , consicering they are nothing else but circles of altitude , which by knowing the altitude you will know the length of the shadow , which in the horizontall diall are perfect circles , and have the same respect unto the horizon , as the parallels of declination have to the equator , but in all upright planes they wil be conicall sections , and by having the length of the style , the altitude of the sun may be computed by the foregoing table with much facility , but for the more expediating of the work in pricking down the parallels of declination with the tropicks , i have here added a table of the altitude of the sun for every houre of the day when the sun enters into any of the 12 signes . a table for the altitude of the sun in the beginning of each signe , for all the houres of the day for the latitude of london . hours . cancer . gemini leo taurus virgo aries libra pisces scorpio aquar sagitta . capric . 12 62 0 58 43 50 0 38 30 27 0 18 18 15 0 11 1 59 43 56 34 48 12 36 58 25 40 17 6 13 52 10 2 53 45 50 55 43 12 32 37 21 51 13 38 10 30 9 3 45 42 43 6 36 0 26 7 15 58 8 12 5 15 8 4 36 41 34 13 27 31 18 8 8 33 1 15     7 5 27 17 24 56 18 18 9 17 0 6         6 6 18 11 15 40 9 0                 5 7 9 32 6 50                 11 37 4 8 1 32                     21 40 this table is in mr. gunters book , page 240 which if you desire to have the point of the equinoctiall for a horizontall plane on the houre of 12 , enter the table of shadows with 38 de . 30 m. and you shall finde the length of the shadow to be 15 parts 5 m. of the length of the style divided into 12 , which prick down on the line of 12 for the equinoctiall point , from the foot of the style . so if i desire the points of the tropick of cancer , i finde by this table that at 12 of the clock the sun is 62 de . high , with which i enter the table of shadows , finding the length of the shadow , which i prick down on the 12 a clock line for the point of the tropick of cancer at the houre of 12. if for the houre of 1 , i desire the point through which the parallel must pass , looke for the houre of 1 and 11 , in this last table under cancer , and i finde the sun to have the height of 59 de . 43 m. with which i enter the table of shadows , and prick down the length thereof from the bottome of the style reaching till the other foot of the compasses fall on the houre for which it was intended . doe so in all the other houres , till you have pricked down the points of the parallels of declination , through which points they must be drawn hyperbolically . proceed thus in the making of a horizontall diall , but if it be a direct verticall diall , you shall then take the length of the verticall shadow out of the said table , or work it as an horizontal plane , only accounting the complement of the elevation in stead of the whole elevation . for a declining plane you may consider it as a verticall direct in some other place , and having found out the equator of the plane and the substyle , you may proceed in the same manner from the foot of the style , accounting where the style stands to be no other wayes then the meridian line or line of 12 in a horizon whose pole is elevated according to the complement height of the style above the substyle , and so prick down the length of the shadows , from the foot of the style , on every one of the houre lines , as if it were a horizontal or verticall plane . but in this you must be wary , remembring that you have the height of the sun calculated for every houre of that latitude in the entrance of the 12 signes , in that place where your plane is a horizontall plane , or otherwayes , by considering of it as a horizontall or verricallplane in another latitude for the azimuths , or verticall circles , shewing one what point of the compasse the sun is in every houre of the day it is performed with a great deale of facility , if first , when the sun is in the equator , we doe know by the last table of the height of the sun for every houre of the day and by his meridian altitude with the help of the table of shadows , find out the equinoctiall line , whether it be a horizontall or upright direct plane , for having drawn that line at right angles with the meridian , and having the place of the style , and length thereof in parts , and the parts of shadow to all altitudes of the sun , being pricked down from the foot of the style , on the equinoctiall line , through each of those points draw parallel lines to the meridian , or 12 a clock line on each side , which shall be the azimuths , which you must have a care how you denominate according to the quarter of heaven in which the sun is in , for if the sun be in the easterly points , the azimuths must be on the western side of the plane , so also the morning houres must be on the opposite side . there are many other astronomical conclusions that are used to be put upon planes , as the diurnall arches , shewing the length of the day and night , as also the jewish or old unequal houres together with the circles of position , which with the meridian and horizon distinguisheth the upper hemispheare into 6 parts commonly called the houses of heaven : which if this i have writ beget any desire of the reader , i shall endeavour to inlarge my self much more , in shewing a demonstrative way , in these particulars i have last insisted upon . i might heare also shew you the exceeding use of the table of right and versed shadow in the taking of heights of buildings as it may very wel appear in the severall uses of the quadrant in diggs his pantometria , & in mr. gunters quadrant , having the parts of right and versed shadow graduated on them , to which books i refer you . chap xiv . shewing the drawing of the seeling diall . it is an axiom pronounced long since , by those who have writ of opticall conceipts of light and shadow , that omnis reflectio luminis est secundum lineas sensibiles , latitudinem habentes . and it hath with as great reason bin pronounced by geometricians , that the angles of incidence and reflection is all one ; as appeareth to us by euclides catoptriques ; and on this foundation is this conceipt of which we are now speaking . wherefore because the direct beams cannot fall on the face of this plane , we must by help of a piece of glasse apt to receive and reflect the light , placed somwhere horizontally in a window , proceed to the work , which indeed is no other then a horizontall diall reversed , to which required a meridian line , which you must endeavour to draw and finde according as you are before taught , or by the helpe of the meridian altitude of the sun , your glasse being fixed marke the spot that reflects upon the seeling just at 12 a clock , make that one point , and for the other point through which you must draw your meridian line , you may finde by holding up a threed and plummet till the plummet fall perpendicular on the glasse , and at the other end of the line held on the seeling make another mark , through both which draw the meridian line . now for so much as the center of the diall is a point without , and the distance between the glasse and the seeling is to be considered as the height of the style , the glasse it selfe representing the center of the world , or the very apex of the style , wee must finde out those two tangents at right angles with the meridian , the one neere the window , the other farther in , through severall points whereof we must draw the houre-lines . let ab be the meridian line found on the seeling , now suppose the sun being in the highest degree of cancer should shine into the glasse that is fixed in c , it shall again reflect unto d , where i make a mark , then letting a plummet fall from the top of the seeling till it fall just on c the glasse , from the point e , from which draw the line a b through d and e , which shall be the meridian required , if you do this just at noon : now if you would finde out the places where the hour-lines shall crosse the meridian , the center lying without the window ec , you may work thus chap x. shewing the making and use of the cylinder dial , whose hour-lines are straight , as also a diall drawn from the same form , having no style . this may be used on a staff or other round , made like a cylinder being drawn as is here described , where the right side represent the tropicks , and the left side the equinoctial : or it may be used flat as it is in the book ; the instrument as you see , is divided into months , and the bottom into signs , and the line on the right side is a tangent to the radius of the breadth of the parallelogram , serving to take the height of the sun , the several parallels downward running through the pricked line , in the midle , are the lines of altitude , and the parallels to the equator are the parallels of declination , numbred on the bottom on a sine of 23 de . and a half . for the altitude of the sun . the use of it is first , if it be described on the head of a staff , to have a gnomon on the top , equal to the radius , and just over the tangent of altitudes , to turn it till you bring the shadow of it at right angles to it self , which shal denote the height required . for the houre of the day . seek the altitude of the sun in the midle prick't line , and the declination in the parallels from the equator , and mark where the traverse lines crosse ; through the crossing of the two former lines , and at the end , you shal finde the figures of 2 or 10 , 3 or 9 , &c. only the summer houres are sought in the right side ▪ where the sun is highest , and the traverse lines longest ; and in the winter , the hour is sought on the left side , where the traverse lines are shorter . for the declination and degree of the signe . seek the day of the moneth on the top marked with j. for january , f for february , &c. and by the help of a horse hair or threed extended from that all along of parallel of declination , till it cut on the bottom where the signes are numbred : the down right lines that are parallel to the equator counted toward the right hand , is the degree of the declination of that part of the ecliptick which is in the bottom , right against the day of the moneth sought on the top . the pricked line passing through the 18 degree of the parallel of altitude , is the line of twy-light ; this projection i had of my very good friend john hulet , master of arts ▪ and teacher of the mathematicks . you may also make a dyal , by preparing of a hollow cylinder , and if you doe number on both ends of the circle , on top and bottom , 15 de . from line to line ; or divide it into 24 parts , and if from top to bottom you draw streight lines , first , by dividing the cylinder through the middle , and only making use of one half , it shal have 12 houres upon it . lastly , if you cut off a piece from the bottom at an angle according to the elevation , and turn the half cylinder horizontal on that bottom , til the shadow of one of the sides fal parallel with any one of those lines from top to bottom : which numbred as they ought , shal shew the hour without the use of a style ; so also may you project a dyal on a globe , having a round brim on the top , whose projection will seem strange to those that look upon it , who are ignorant of these arts . chap xvi . shewing the making of a universall dyall on a globe , and how to cover it , if it be required . if you desire to cover the globes , and make other inventions thereon , first learn here to cover it exactly , with a pair of compasses bowed toward the points , measure the diameter of the globe you intend to cover , which had , finde the circumference thus ; multiply the diameter by 22 , and divide that product by 7 , and you have your desire . that circumference , let be the line a b , which divide into 12 equal parts , and at the distance of three of those parts , draw the parallel c d , and e f , a parallel is thus drawn , take the distance you would have it asunder , as here it is ; three of those 12 divisions : set one foot in a , and make the arch at e , & another at b , and make the arch with the other foot at f , the compasses at the wideness taken , then by the outward bulks of those arches , draw the line e f , so also draw the line c d. and to divide the circumference into parts as our example is into 12 , work thus , set your compasses in a , make the ark b f , the compasses so opened , set again in b , and make the ark a c , then draw the line from a to f , then measure the distance from f to b , on the ark , and place it on the other arch from a to c , thence draw the line c b , then your compasses open at any distance , prick down one part less on both those slanting lines ; then you intend to divide thereon , which is here 11 : because we would divide the line a b into 12 , then draw lines from each division to the opposite , that cuts the line a b in the parts of division . but to proceed , continue the circumference at length , to g and h , numbring from a toward g9 of those equal parts , and from b toward h as many , which shal be the centers for each arch. so those quarters so cut out , shall exactly cover the globe , whose circumference is equal to the line a b. thus have you a glance of the mathematicks , striking at one thing through the side of an other : for i here made one figure serve for three several operations , because i would not charge the press with multiplicity of figures . chap xvii . shewing the finding of the elevation of the pole , and therewithall a meridian without the declination of sun or starre . this is done by erecting a gnomon horizontal , and at 3 times of the day to give a mark at the end of the shadows : now it is certain , that represents the parallel of the sunne for that day ; then take three thin sticks or the like , and lay them from the top of the gnomon , to the places where the shadows fell , and on these three so standing , lay a board to ly on all three flat , and a gnomon in the midle of that board points to the pole : because every parallel the sun moves in , is parallel to the equinoctial , and that is at right angles , with the pole of the world . now the meridian passeth through the most elevated place of that board or circle so laid , neither can the sun's declination make any sensible difference in the so small proportion of 3 or 4 houres time . chap xviii . shewing how to finde the altitude of the sun , only by scale and compasses . with your compasses describe the circle a b c d place it horizontal , with a gnomon in the center , crosse it with two diameters ; then turn the board till the shadow be on one of the diameters , at the end of the shadow , mark , as here at e , lay down also , the length of the gonmon from the center on the other diameter to f , from e to f drawe a right line : then take your compasses , and on the chord of 90 , take out the radius the ark of 60 , set the compasses so in e , describe an arch , then take the distance between the line e f , and the diameter d b ; which measure on the chord of 90 , and so many degrees as the compasses extend over ; such a quantity is the height of the sun , in like manner any angles being given , you must measure it by the parts of a circle . here followeth the problematical propositions of the office of shadow , and the benefit we receive thereof . prop. 1 by shadow , we have a plain demonstration that the sphere of sol is higher than the sphere of luna , to confirm such as think they move in one orbe . let the sun be at a , in the great circle , and the moon at b , in the lesser , let the horizon be c d , now , they make one angle of height , in respect of the center of the earth , notwithstanding though they so equally respect the earth , as one may hinder the sight of the other : yet the shadow of the sun shall passe by the head of the gnomon e , and cast it to f , and the beames of the moon shall passe by e to g much longer , which shewes shee is much lower , for the higher the light is , the shorter is the shadow . i call the moon a feminine , if you ask my reason , shee is cold and moist , participating of the nature of women ; and we call her the mother of moisture , but that 's not all , for i have a rule for it , nomen non crescens . prop. 2. by shadow , we are taught the earth is bigger then the moon ; seeing in time of a total obscurity , the moone is quite overshadowed ; for the shadow is cast in this manner . by the same we learn also , that seeing the shadow comes to a point , the earth is less then the sun : for if the opacous body be equal to the luminous body , then like two parallels they will never meet , but concurre in infinitum , as these following figures shew . or if the luminous body were less then the opacous body : then the shadow would be so great in so long a way , as from the earth to the starry firmament , that most of the starres as were in opposition to the sun , would not appear : seeing they borrow their light of the sun . it is also sufficiently proved by shadow , in the praecognita philosophical , that the earth is round , and that it possesseth the middle as proprius locus from which it cannot passe , and to which all heavie things tend in a right line , as their terminus ad quem . from which the semidiameter of the sun 15 min. substracted doth remain the altitude of the center of the sun 50 de . 3 m. the altitude required , or from this or the former proposition we may take notice that there is no dial can shew the exact time without the allowance of the suns semidiameter : which in a strict acception is true , but hereto mr. wells hath answered in the 85 page of his art of shadows , where saith he , because the shadow of the center is hindered by the style , the shadow of the hour-line proceeds from the limbe which alwayes precedeth the center one min. of time answerable to 15 min. the semidiameter of the sun ( which to allow in the height of the style were erroneous ) wherefore let the al●owance be made in the hour-lines , detracting from the true equinoctial distances of every 15 deg. 15 primes , and so the arches of the horizontall plane from the meridian shall stand thus . prop. 4. by shadow we may finde the natural tangent of every degree of a quadrant , as appeares by the former example . houres . equinoctial distances . true hour distances . 12 0 de . m. de . m. se . 11 1 14 45 11 38 51 10 2 29 45 24 6 31 9 3 44 45 37 4 2 8 4 59 45 53 19 12 7 5 74 45 70 48 6 6 6 89 45 89 40 51 for the sun being 46 deg , 13 min. of altitude makes a shadow of 95. parts of such as the gnomon is 100 , so then multiply the length of the gnomon 100 by the radius , and divide by 95 , and it yeelds 105263 the natural tangent of that ark . prop. 5. by shadow we may take the height of any building , by the rule of proportion ; if a gnomon of 6 foot high give a shadow of 10 foot : how high is that house whose shadow is 25 foot ? resolved by the rule of three . prop. 6. by shadow also we learn the magnitude of the earth , according to eratosthenes his proposition . prop. 7. by shadow we learne the true equinoctial line , running from east to west , which crossed at right angles is a true meridian , where note , that in the times of the equinoctiall that the shadows of one gnomon is all in one right line . prop. 8. by shadow we know the earth to be but as a point , as may appear by the shadow of the earth on the body of the moon . prop. 9. by shadow we may learn the distance of places , by the quantity of the obscurity of an eclipse . prop. 10. by gnomonicals we make distinctions of climates and people , some hetorezii , some perezii , some amphitii . prop. 11. by shadow the climates are known , in the cold intemperate zones the shadow goes round . in the hot intemperate zones the shadow is toward the west at the rising sun , and toward the east at the setting sun , and no shadow at noones to them as dwel under the parallels . and to them in the temperate zones always one way , toward the north , or toward the south . prop. 12. by shadow we are taught the rule of delineating painting , according to the perspective way , how much is to be light or dark , accordingly drawn as the center is disposed to the eye : so the office of shadow is manifold , as in the optical conclusions are more amply declared ; therefore i referre you to other more learned works , and desist to speak . but for matter of information , i will here insert certain definitions taken out of optica agulion ii lib. 5. first , saith he , we call that a light body from whence light doth proceed ; truly saith he , the definition is plain , and wants not an expositor , so say i , it matters not whether you understand the luminous body : only that which doth glister by proper brightness as doth the sun , or that which doth not shine but by an external overflowing light , as doth the moon . 2. that we call a diaphon body , through which light may pass , and is the same that aristotle cals perspicuous . 3. it is called adiopton , or opacous ; through which the light cannot pass , so saith he , you may easily collect from a diaphon body the definition of shadow : for as that is transparent through which the light may pass : so also is that opacous , or of a dense nature wherein the light cannot pass . 4. that is generated from a shining body , is called the first light , that hath his immediate beginning from the luminous body , it is called the second light , which hath his beginning from the first , the third which hath his beginning from the second , and so the rest in the same order . whence we make this distinction of day and light , day is but the second light , receiving from the sun the first , so that day is light , but the sun is the light . 5. splendor is light repercussed from a pure polished body ; and as light is called so from the luminous body : so this is called splendent from the splendor . theor. light doth not onely proceed from the center , but from every part of the superficies . theor. light also is dispersed in right lines . theor ▪ light dispersed about every where , doth collect into a spherical body . 6. the beames of light , some are equi distant parallels , some intersect each other , and some diversly shaped . let a be the light , a beam from a to b , and another from c to d are parallel , a d and c b intersect ; and the other two doe diversly happen , one ascending , the other descending : its plaine . 7. that is called a full and perfect shadow , to which no beam of light doth come . 8. that is called a full and perfect light ; which doth proceed from all parts of that which gives light ; but that which giveth light but in part , is imperfect : this he exemplified by an eclipse , the moon interposing her self between the sun and earth , doth eclipse the perfect light of the sun : whereby there appeares but a certaine obscure , dim , glimmering light , and is so made imperfect . hence we may learn to distinguish day from night ; for day is but the presence of the sun by a perfect light received , which we count from sun rising to sun setting . twy-light is but an imperfect light from the partial shining or neighbourhood with sun : whereas night is a total deprivation or perfect shadow , to which no beam of light doth appettain . yet from the over-flowing light of the sun , the starres are illuminated ; yet because shadow is always in the opposite , those stars that are in direct opposition to the sun , are obscure for that season , and hence proceeds the eclipse of the moon . hence it is with the sciothericalls as it is with the dutch emblamist , comparing love to a diall , and the sun with the motto , nil sine te , and his comparison to coelestis cum me sol aspicit ore sereno , protinùs ad numeros mens reddit apta suos . implying that as soone as the sun shines it returnes to the number , so a lover seeing his love on a high tower , and a sea between , yet ( protinùs ad numeros ) he will swim the sea and scale the castle to return to her : so here lyes the gradation , first , from the suns light , from the light by the opoacus body , interposition , shadow , and from the shadow of the axis is demonstrated the houre . adde also , the beam and shadow of a gnomon , have one and the same termination or ending , toward which i now draw my pen ; desiring you to take notice that the whole method of dialling , as may appear by the former discourse , doth seem to be foure-fold , viz. geometrical , arithmetical , or by tables mechanically , or by observation . so that the art of shadowes is no other then a certain and demonstrative motion of the heavens in any plaine or superficies , and a gnomonical houre is no other then a direct projecting of the hour-lines of any plain ; so as that it shal limit a style so to cast its shadow from one line to another , as that it shall be just the twenty-fourth part of the natural day , which consisteth of 24 houres ; and this i have laid down after a most plain manner following : a gnomonical day is the same that the artificial day is ; which the shadow of a gnomon maketh from the rising of the sun , till the setting of the same in a concave superficies : which length of the day is also projected from the motion of the shadow of the style , a gnomonical moneth is also described on planes , which is the space that the shadow of a gnomon maketh from one parallel of the signe , to an other succeeding parallel of a signe , again , a gnomonical year is limited by the shadow of a gnomon , from a point in the meridian of the tropick of cancer , till it shall revolve to the same meridian altitude and point of the tropick , and is the same as is a tropical year : wherefore , above all things we ought first tobe acquainted with the knowledge of the circles of the sphaere ▪ secondly , to have a judicious and exact discerning of those planes in which we ought to project dials . thirdly , to consider the style , quality , and position of the axis or style , with consideration of the cause , nature and effects in such or such planes as also an artificial projecting of the same , either on a superficies by a geometricall knowledge , and reducing them to tables by arithmetick , which we have afore demonstrated , and come now to the conclusion : so that as i began with the diall of life , so we shall dye-all , for , mors ultima linea . to abraham chambrelan esq . s m. consecrateth his court of arts . sir . if the originall light be manifestatiu , by it i have made a double discovery , your genius did so discover it self according to the quality of the sun , that i am umbrated and passive like the eclipsed moon , yet cannot but reflect a beame which i have received from the fountain of light ; 't is you which i make the patron to my fancy ( which perhaps you may wonder at the idleness of my head , to tell you a dream , or a praeludium of the several arts : howsoever knowing you are a lover of them , i did easily believe you could not but delight in the scaene ; though in most i have written , i have in some sort imitated nature it self , which dispenseth not her light without shadows , which will truly follow them from whom they proceed , and shall sir , in time to come render me like pentheus whose curiosity in prying into secrets makes me uncertain . et solem geminum duplices se ostendere thebas , & while i know neither copernicus , nor ptolomies systeme of the world , dare affirmatively reject neither , but run after both ; and submitting my wisdome to the wisest of men , must conclude , that cuncta fecit tempestatibus suis pulchra , and hath also set the world in their meditation : yet can not man find out the work that god hath wrought . sir , pardon my boldness , in fastning this on your patronage , who indeed are called to this court of arts , as being nobly descended , whom only it concernes ; and only whose vertue hath arrived them to the temple of honour , who are all invited as appeareth in the conclusion of this imaginary description , wherein , whilst i seem to be in a dream ; yet sir , i am certain , i know my selfe to be yours in all that i am able to serve you , s. m. topothesia . or an imaginary description of the covrt of art . comming into a librarie of learning , where there was more languages then i had tongues , that if i had been asked to bring brick i should have brought morter , and going gradually along , as then but passus geometricus , there i met minerva , which said unto me ( vade mecum ) & had not the expression of her gesture be-spoke my company , i should have shunned her ; she then taking me by the hand , led me to the end , where sat one which was called as i did inquire , clemency , the name indeed i understood , but the office i did not , whose inscription was custos artis , i being touched now with a desire to understand this inscription ; began with desire , & craving leave , used diligence to peruse the library , and found then a booke intituled the gate of languages , by that i had perused it , i understood the fore-named inscription , and craving leave of clemency in what respect she might be called the keeper of arts , who answered with claudanus thus ; principio magni custos clementia mundi , quae jovis incoluit zonam quae temper at aethrum , frigoris & flammae mediam quae maxima natu , coelicolum : nam prima chaos clementia solvit , congeriem miserata rudem , vultuque sereno , discussus tenebris in lucem saecula fundit . and arising from a globe which was then her seat , she began to discourse of the nature and magnitude of the terrestiall body , and propounded to me questions : as first , if one degree answerable to a coelestiall degree yield 60 miles , what shall 360 degrees yield , the proportion was so plainly propounded , that i resolved it by the ordinary rule of proportion , she seeing the resolution , propounded again , and said , if this solid body were cut from the center how many solid obtuse angles might be cut from thence , at this i stumbled , and desired , considering my small practise , that she would reduce this chaos also , and turne darknesse into light : seeing then my desire and diligence bid me make observation for those three were the wayes to bring me to peace , and resolved , that as from the center of a circle but three obtuse angles could be struck , so from the center of a globe , but three such angles could be struck and from thence fell to another question , & asked what i thought of the motion of that body : i answered , motion i thought it had none , seeing i had such secretaries of nature on my side , and was loth to joyn my forces with the copernicans . she answered , it was part of folly to condemn without knowing the reasons , i said it should stil remain a hypothesis to me , but not a firme axiome : for the resolution of which i wil onely sing as sometimes other poets sang concerning the beginning of the world , and invert the sense onely , as that in another case , so this for our purppse . if tellus winged bee the earth a motion round , then much deceiv'd are they that it before nere found . solomon was the wisest , his wit ner'e this attain'd ; cease then copernicus , thy hypothesis vain . and began to discourse of the longitude of the earth , and then i demanded what benefit might incurre from thence to a young diallist , she answered above all one most necessary probleme , which we may finde in petiscus his example , and propounded it thus ; the difference of meridians given , to finde the difference of hours . if the place be easterly , adde the difference of longitude converted into time to the hours given : if it be westerly , substract the easterly places , whose longitude is greater & contra , as in petiscus his example , the meridian of cracovia is 45 deg. 30 min. the longitude of the meridian of heidelberge is 30 degrees , 45 minutes , therefore heidleberg is the more westerly . one substracted from 45 30 30 45 the other sheweth the difference of longitude , to which degrees and minutes doth answer o ho . 59 m. for as therefore when it is 2 hours post merid. at cracovia ; at neidelberg , it is but 1 hour , 1 minute past noon . for , there is left 1 houre 1 minute . thus out of the difference of meridians , the divers situation of the heavens is known , and from the line of appearances of the heavens , the divers hours of divers places is known , and this is the foundation of observing the longitude : if it be observed what houre an eclipse appears in one place , and what in another , the difference of time would shew the longitude , and hereby you may make a dyall that together with the proper place of elevation , shall shew for any other country ; for this proposition i did hartily gratifie geographia , and turning , said astronomy , why stand you so sad ? she answered , art is grown contemptible , and every one was ready to say ( astrologus est gastrologus ) then i said , what though vertue was despised , yet let them take this answer : thou that contemnest art and makes it not regarded , in court of art shal have no part none there but arts rewarded . gnashing the teeth as if ye strive to blame it , yet know i 'le spare no cost for to obtein it . perceiving your willingnesse said astronomy , i will yet extend my charity and lay down the numbers , so that if you add the second and third and substract the first , it shall give the fourth ; the question demanded , and then i being careful of the tuition of what she should say , took a table-book and writ them as follows . 1 the sine comp. elevation pole 38½ , sine 90 ; sine of the decl. of the sun yields the sine of the amplitude ortive : which is the distance of the suns rising from due east . 2 the sine 90 , the sine ele . pole 51d½ ; the sine of decli. yields the sine of the suns height at six a clock . 3 sine comp. of altitude of the sun , sine comp. declina . sine 90 ; the sine of the angle of the vertical circle , and the meridian for the azimuth of the sun at the hour of 6 : the azimuth is that point of the compasse the sun is on . 4 sine comp. decli. of the sun : sine compl. eleva . pole 38d½ , sine altitude of the sun ; the houre distance from six . 5 sine compl. of decli. sine 90 ; compl , of sine suns amplitude to sine compl. of the assentional difference . 6 the sine of the difference of assention , tang. decli. sun ; sine 90 : tangent complement of the elivation . 7 sine altitude of the sun , sine declina . of the sun ; sine 90 : elevation of the pole . 8 sine 90 , sine com . of distance from 6 ; sine com . declination of the sun : sine comp. of the altitude sun . 9 sine 90 , sine eleva . pole ; sine alti . of a star : sine decli. of that star . 10 the sine of a stars altitude in an east azimuth , sine amplitude ortive ; sine 90 : sine of the elevation . 11 the greatest meridian altitude , the lesse substracted sines ; the distance of the tropicks , whose halfe distance is the greatest declination of the sun ; which added to the least meridian altitude , or substracted from the greater , leavs the altitude of the equator : the complement whereof , is the elevation of the pole . 12 tang. eleva . pole , sine 90 ; tang. decli. of the sun , to the co-sine of the hour from the meridian , when the sun will be due east or west . by these propositions said astronomy , you may much benifit your selfe ; but let us now go see the court of art : i liked the motion , and we went and behold the sight had like to made me a delinquent , for i saw nought but a poor anatomy sitting on the earth naked exposed to the open ayre , which made me think on the hardnesse of a child of art , that it had neither house nor bed , and now being at a pitch high enough resolve never to follow it : this anatomy also it seems was ruled by many , both rams , and buls and lions , for he was descanted thus on . anatomy why do'st not make thy moane , so many limbes , and yet can'st govern none ; thy head although it have a manly signe , yet art thou placed on watry feminine . 't is true , yet strong , but prethee let me tell yee , let not the virgin always rule your belly : for what , although the lion rule your heart ; the weakest vessell will get the strongest part . then be content set not your foot upon a slippery fish , that 's in an instant gone ; a slippery woman , who at cupids call will slip away , and so give you a fall : and if rams horns she do on your head place ; it is a dangerous slip , may spoil your face . here at i smiled , then said astronomy , what is your thought ? then said i , do men or artists so depend on women , as that their strength consists in them ? she said , i misunderstand him , for the ram that rules the head is a signe masculine , because it is hot and dry , the fish that rules the feet is cold and moist is therefore called feminine . pisces the fish you know's a watery creature , 't is slippery , and shews a womans nature ; so women in their best performance fail , there 's no more hold then in a fishes tail . but the more to affect the beholder , i will typigraphe this court of art . under was written these lines , to shew mans misery by the fall , which i will deliver you , as followes : when chaos became cosmos , oh lord ! than how excellent was microcosmus , man when he was subject to the makers will , stars influence could no way worke him ill : but since his fall his stage did open lye , and constellations work his destiny . thus man no sooner in the world did enter , but of the circumference is the center . and then came in vertue , making a speech , and said ; honour to him , that honour doth belong : you stripling artist , coming through this throng , have found out vertue that doth stand to take you by the hand , and gentleman you make . for geometry , i care not who doth hear it , may bear in shield coat armor by his merit : we respect merit , our love is not so cold , we love mens worth ( not in love with mens gold ) not herald-like to sel , an armes we give ; honour to them that honourably live . the noble professours of the sciences , may bear as is here blazoned , ( viz. ) the field is jupiter , sun and moon in conjunction proper , in a chief of the second , saturn , venus , mercury in trine or perfect amity ; and mars in the center of them ; mantled of the light , doubled of the night , and on a wreath of its colours a helitropian or marigold of the colour of helion with this motto , quod est superius , est sicut inferius ; then did i desire to know , what did each planet signifie in colour , she then told me as followeth . ☉ or gold ☽ argent silver ♂ gules red ♃ azure blew ♄ sable black ♀ vert green ☿ purpre purple and by mantled of light , she meant argent and of the night she meant an azure mantle , powdered with estoiles , or stars silver . i indeed liked the blazon , and went in , where also i found a fair genealogie of the arts proceeding from the conjunction of arithmetick and geometry collected by the famous beda dee in his mathematicall praeface . both number and magnitude saith he have a certain originall seed of an incredible property ; of number a unit , of magnitude a point . number , is the union and unity of unites , and is called arithmetick . ☽ magnitude is a thing mathematicall ▪ and is divisible for ever , and is called geometry . geodesie , or land measuring geographia , shewing wayes either in spherick , plane , or other the scituation of cities , towns , villages , &c chorographia , teaching how to describe a small proportion of ground , not regarding what it hath to the whole &c. hydrographia , shewing on a globe or plane the analogicall description of the ocean , sea-coasts , through the world , &c. n●vigation , demonstrating how by the shortest way , and in the shortest time a sufficient ship , betweene any two places in passages navigable assigned , may be conducted , &c. perspective , is an art mathematicall which demonstrateth the properties of radiations , direct , broken , & reflected astronomie demonstrates the distance of magnitudes and naturall motions , apparances and passions proper to the planets and fixed stars . cosmographie ; the whole & perfect description of the heavenly , & also elementall part of the world & their homologall & mutuall collation necessary . stratarithmetrie . is the ki●● appertaining to the war●● , to set in figure any number of men appointed : differing from tacticie which is the wisdome & foresight . musick , saith plato , is sister to astronomie , & is a science mathematicall , which teacheth by sence & reason perfectly , to judge & order the diversity of sounds high & low . astrologie , severall from , but an off-spring of astronomie , which demonstrated reasonably the operation and effects of the naturall beams of light , and secret influence of the stars . statick ; is an art mathematicall , demonstrating the causes of heavinesse and lightnesse of things . ●●thropographie , being the description of the number , weight , figure , s●●uation and colour of every diverse thing conteined in the body of man . trochilike , descended of number and measure , demonstrating the properties of wheel or circular motions , whether simple or compound , neer sister to whom is holicosophie , which is seen in the describing of the severall conicall sections and hyperbolicalline in plants of dyals or other by spirall lines , cylinder , cone , &c. pneumatithmie , demonstrating by close hollow figures geometricall , the strange properties of motion , or stay of water , ayr , smoak , fire in their continuity . menadrie , which demonstrateth how above natures vertue and force , power may be multiplyed ▪ hypogeodie , being also a child of mathematicall arts , shewing how under the sphaericall superficies of the earth at any depth to any perpendicular assigned , to know both the distance and azimuth from the entrance . hydragogie , demonstrating the possible leading of water by natures law , and by artificiall help . h●rometrie , or this present work of horologiographia , of which it is said , the commodity thereof no man would want that could know how to bestow his time . ●ographie , demonstrating how the intersection of all visuall pyramids made by any plane assigned , the center , distance and lights , may be by lines and proper colours represented . then followed architecture , as chief master , with whom remained the demonstrative reason and cause of the mechanick work in line , plane and solid , by the help of all the forementioned sciences . thaumaturgike , giving certain order to make strange works , of the sence to be perceived , and greatly to be wondred at . arthemeastire , teaching to bring to actuall experience , all worthy conclusions by the arts mathematicall . while i was busied in this imployment which indeed is my calling , i questioned caliopie , why she put the note of illegitimacy upon astrologie ; she said , it indeed made astronomy her father , but it was never owned to participate of the inheritance of the arts , and therefore the pedegree doth very fitly say , doth reasonably not , quasi intellectivè ▪ but imperfectivè ; then did i ask again , why arithmetick had the distinction of an elder brother the labell , she told me , because it was the unity of units , and hath three files united in one lambeaux , and did therefore signifie a mystery , then said i , why do you represent magnitude by the distinction of a second brother , to which she said , because as the moon , so magnitude in increasing or decreasing is the same in reason , then did she being the principall of the nine muses , and goddesse of heralds summon to urania , and so to all the other to be silent , at which silence was heard harmonicon coeleste by the various motions of the heavens , and fame her trumpeter sounded forth the praise of men , famous in their generation ; and concluded with the dedication and consecration of the court of arts in these words of the learned vencelaus clemens . templum hoc sacrum est , pietati , virtuti , honori , amori , fidei , semi deûm ergò , & coelo ductum genus , vos magni minoresque dei , vos turba ministra deorum vos inquam . sancti davides , magnanimi hercules , generosi megistanes bellicosi alexandri , gloriosi augusti , docti platones , facundi nestores , imici jonathanes , fidi achatae . uno verbo boni huc adeste , praeiste , prodeste vos verò orbis propudia impii holophernes , dolosi achitopheles , superbi amanes , truculenti herodes , proditorus judae , impuri nerones , falsi sinones , seditiosi catilinae , apostatae juliani . adeoque , quicunque , quacunque , quodcunque es malus , mala , malum , exeste procul hinc procul ite prophani . templi hujus pietas excubat antefores , virtute & honore vigilantibus amore & fide assistentibus reliqua providente aedituo memoria , apud quam nomin● profiteri fas & jura sin●nt . quantum hoc est ? tantum vos caetera , quos demisse compellamus , praestabitis , vivite , vincite , valete , favete . et vos ô viri omnium ordinum , dignitatum , honorum , spectatissimi amplissimi , christianissimi , &c. which being done , the muses left me , and i found my self like memnon , or a youth too forward , who being as the learned sir francis bacon saith , animated with popular applause , did in a rash boldnesse come to incounter in single combate with achilles the valiantest of the grecians , which if like him i am overcome by greater artists , yet i doubt not but this work shall have the same obsequies of pitty shed upon it , as upon the sonne of aurora's bright armour , upon whose statue the sun reflecting with its morning beames , did usually send forth a mourning sound . and if you say , i had better have followed my heraldry ( being it is my calling ) henceforth you shall find me in my own sphear . finis . notes, typically marginal, from the original text notes for div a89305e-52990 ☞ ☞ horologiographia, or, the art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by john brown, philomath. brown, john, philomath. 1671 approx. 407 kb of xml-encoded text transcribed from 196 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2004-05 (eebo-tcp phase 1). a29762 wing b5042 estc r17803 12547379 ocm 12547379 63104 this keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the early english books online text creation partnership. this phase i text is available for reuse, according to the terms of creative commons 0 1.0 universal . the text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. early english books online. (eebo-tcp ; phase 1, no. a29762) transcribed from: (early english books online ; image set 63104) images scanned from microfilm: (early english books, 1641-1700 ; 955:8) horologiographia, or, the art of dyalling being the second book of the use of the trianguler-quadrant : shewing the natural, artificial, and instrumental way, of making of sun-dials, on any flat superficies, with plain and easie directions, to discover their nature and affections, by the horizontal projection : with the way of drawing the usual ornaments on any plain : also, a familiar easie way to draw those lines on the ceiling of a room, by the trianguler quadrant : also, the use of the same instrument in navigation, both for observation, and operation : performing the use of several sea-instruments still in use / by john brown, philomath. brown, john, philomath. 305 [i.e. 304], [6] p. : 40 ill. printed by john darby, for john wingfield ... and by john brown ... and by john seller ..., london : 1671. reproduction of original in cambridge university library. "an appendix to the use of the trianguler-quadrant in navigation ..." (p.[225]-305 [i.e. 304]) has special t.p. errata: p. 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ascii text with mnemonic sdata character entities); displayable xml (tcp schema; characters represented either as utf-8 unicode or text strings within braces); or lossless xml (tei p5, characters represented either as utf-8 unicode or tei g elements). keying and markup guidelines are available at the text creation partnership web site . eng dialing -early works to 1800. 2004-01 tcp assigned for keying and markup 2004-02 spi global keyed and coded from proquest page images 2004-03 judith siefring sampled and proofread 2004-03 judith siefring text and markup reviewed and edited 2004-04 pfs batch review (qc) and xml conversion horologiographia : or , the art of dyalling , being the second book of the use of the trianguler-quadrant . shewing the natural , artificial , and instrumental way , of making of sun-dials , on any flat superficies : with plain and easie directions , to discover their nature and affections , by the horizontal projection . with the way of drawing the usual ornaments on any plain : also , a familiar easie way to draw those lines on the ceiling of a room , by the trianguler quadrant . also , the use of the same instrument in navigation ; both for observation , and operation . performing the use of several sea-instruments still in use . by iohn brown , philomath . london , printed by iohn darby , for iohn wingfield , and are to be sold at his house in crutched-fryers ; and by iohn brown at the sphear and sun-dial in the minories ; and by iohn seller at the hermitage-stairs in wapping . 1671. to the courteous reader . thou hast here presented to thy view ( courteous reader ) in this second part , a plain discourse of dialling , both natural , artificial , & instrumental . natural i call it , from the plain illustration thereof , by the armilary sphear of brass herein described , or by the poor-man's dial-sphear , as i fancy to call it , being only a moving horizontal-dial , and a moving plain , according to the figure thereof in the book annexed , whereby all the arks , angles , scituations and affections , are very plainly represented to an ordinary capacity . artificial i call it , from the lively delineation of the horizontal-projection , the fittest in my opinion for the making plain the mystery of dialling . instrumental i call it , from the applying of the trianguler-quadrant , to the ready resolving all the arithmetical and astronomical work , needful thereunto ; and that to competent exactness as in the first part , and also in this second part is sufficiently seen , in finding the requisites and delineating the hour-lines to small parts , exactly & speedily by the natural sines , tangents and secants on the sector and quadrant . also , the ready way of finding the suns altitude , hour , azimuth , angle of the plain , and any such business relating to dyalling , as in the first part is largely treated on . further , in this second part you have tables of the suns declination to every day of the years , 1 , 2 , & 3 , after the bissextile , as near as any extent . also , a short , but plain direction , how to use the trianguler-quadrant , at any manner of way of observation used at sea ; as backward or forward , as the davis-quadrant , and the cross-staff is used ; also , as gunter's bow is used both for the sun or stars . a figure of the stereographick projection pag ● the prints of the lines of numbers , as you see here inserted , are in part according to mr. windgates , as to a single and broken line of numbers : but the addition of the line of the fractional parts of a pound , and the several gage-points , were never before used as i know of ; but do much ease & expedite the operations by the line of numbers , sines and tangents . also , these scales of reduction are convenient for the finding the decimal-fraction , equal to the other sexagenary-fraction , and are agreeable to those tables in mr. windgates book of arithmetick , pag. 82. also note , that the figure of the rule at the beginning of the book , pasted on a board , is the very same with that spoken of chap. xv. use 28. pag. 397 , of the first part , and will work all questions wrought by the trianguler-quadrant , to exercise them that are out of the way to have them made , and may serve as good directions to the young instrument-maker , though these are made too too small a radius to arrive at exactness . the like may i say of the gunters-lines in the figures annexed , yet as large as the book will bear . thus i have given you a brief account of my present thoughts about this matter , and somewhat more particularly in the first part , disclaiming all boasting or vain ostentation , knowing that at the next impression it may be amended in many places ; i shall rest and remain , ready to make amends in the making of these , or any other mathematical instruments , at my house at the sphear and sun-dial in the great minories , john browne . february 16. 1670. chap. i. the use of the trianguler quadrant , in making of dials . svn-dials may be made on any plain , and all kind of plains are either flat , as horizontal ; or vpright , or leaning . the horizontal hath two faces , the one beholding the zenith , called the horizontal-plain ; the other , beholding the nadir , as the ceiling of a room is . the upright plains , are those that make right angles with the horizon , and do behold neither the zenith or nadir , but are parallel to them . the leaning plains are of two sorts generally ; the one called recliners , beholding the zenith ; the other sort called incliners , beholding the nadir , as the outside , and in-side of a roof of a house , may represent . the two last sorts , viz. upright and leaning , may be direct , or declining , viz. beholding the south , or north , or east , or west point of the horizon ; or declining therefrom , viz. declining from south , or north , toward the east or west . all which plains , are lively represented by a sphear , made for that purpose , in brass or pasteboard , or by the projection of the sphear in plano , thus ; equal to the radius of the smaller tangents , describe the circle eswn , representing the horizon , crossing it precisely in the the center z , with the lines sn and ew , denoting the points of south and north , east and west . then counting the smaller tangent on the sector-side doubly , as thus , calling 5 , 10 ; & 10 , 20 ; & 20 , 40 ; & 30 , 60 ; & 40 , 80 ; & 45 , 90 ; &c. lay off from z , towards s , the complement of the suns meridian altitude , in ♋ , in ♈ , and ♑ ; for those points on the meridian-line , between z and s ; and consequently the half tangent of the complement of the suns meridian altitude in every degree of declination , ( if you proceed so far ) . then for the intersections of all those lines and parallels of declination on the north-side of the meridian , observe , that the same number of degrees and minuts , that any point is above the horizon on the south part of the meridian in summer , just so many degrees and minuts is his opposite parallel in winter below the horizon . as thus for example . the sun being in ♋ , or 23 deg . 31 min. of declination north , hath for his meridian altitude 62 degrees , and so many degrees is his opposite parallel of 23-31 , or ♑ , below the north part of the horizon , at midnight . as thus ; let the center , at the beginning of the line of tangents , represent the center z ; and let the tangent of 45 , represent the horizon in the scheam , viz. s. and n. then , as the distance from s. to ♑ , is 15 deg . taken from 45 toward 0 , and laid from s. to ♑ inwards toward the center z , as the distance was taken from the tangent of 45 , toward the beginning of the line of tangents , that represents the center ; so the point cancer from n. is 15 deg . counted beyond 45 , toward the end , below or beyond the horizon . again , as s. ♋ is 62 degrees from 45 towards 00 ; so is the other point 62 degrees below n , taken from 45 , viz. at 76 degrees ; which being laid from n , doth over-reach this little page . so that to draw the tropick of ♑ , the point ♋ being his opposite , is 28 degrees from z , or 62 deg . from s ; and the other point of ♑ , on the north part of the meridian , is 62 degrees , counting from 45 doubly also ; or 28 degrees from 90 , the supposed end of the tangent , which is naturally infinite , being the tangent of 76 degrees , or the semi-tangent of 152 , reading the tangents doubly from the center ; which distance from the center , to the tangent of 76 ; or as half-tangents , 152 , laid from z , gives the point ♑ on the north-part of the meridian , below the horizon ; the midst between which two points of ♑ on the south and north part of the meridian , is the center to draw the tropick of capricorn . again , to illustrate this difficulty , to draw the tropick of cancer , the suns meridian-altitude in ♑ , his opposite sign is 15 degrees above the horizon on the south part of the meridian , and 15 degrees below the horizon , on the north-part of the meridian , viz. the extent from the center to the tangent of 52 deg . 30 min. or the semi-tangent of 105 , reading it doubly ; being laid from z , gives the point ♋ below the horizon ; the middle between which two points is the center to draw the tropick of cancer . again , for the equinoctial or parallel of ♈ ; the meridian altitude in ♈ , is 38-28 ; and the meridian altitude likewise in ♎ , his opposite parallel is 38-28 also ; so that if you count 38-28 doubly beyond 45 , which will be at the tangent of 64 degrees and 14 minuts , and take from thence to the center ; this distance laid from z , shall give the point ae below the horizon , and the the middle between the two points ae , is the center to draw the aequinoctial . then for the hour-lines ; first , set off the semi-tangent of 38-28 from z to p ; and the secant of 38-28 to the same radius from z to l , and draw the line l 45 parallel to ew ; then make pl a tangent of 45 degrees , and lay off the tangents of 15-30 , and 45 , from l both-wayes , as you see in the figure . also , as the sector stands , take out the = tangents of 60 and 75 severally , and turn them four times from l both-wayes , and note those points with 6 , 7 , 8 , 9 , 10 , 11. lastly , set one point of the compasses in l , and open the other to p , and draw the line wpe for the hour of 6. again , set one point in 7-15 degrees from l , and open the other to p , and draw the hour-line 5 p 5 ; set the same extent also in 7 , or 5 , on the other side of l , and draw the hour-line 7 p 7 , as the figure sheweth . then , set one point of the compasses in 8 , 30 degrees from l , and open the other point to p , and draw the hour-line 8 p 8 , and remove it to the other side of l , and draw the hour-line 4 p 4 : and so for all the rest in order . thus having drawn the figures ; to draw lines therein , which shall truly represent any plain whatsoever , observe the following rules . 1. the horizontal-plain , is represented by the circle e.s.w.n. 2. a direct south or north-diall , is represented by the line e.z.w. 4. an east or west plain , is represented by the meridian-line of 12 , viz. s. & n. 5. a polar plain , is represented by the hour of 6 , viz. the line e.p.w. 6. an equinoctial plain , is represented by the equinoctial-line e.ae.w. 7. any direct reclining , or inclining-plain , between the two last , is called , a direct recliner , whose poles are alwayes in the meridian , and are represented by any reclining circle , as the two circles w. ♋ . e. and e. ☉ . w. do shew . 8. an east or west recliner or incliner , represented by the circle n.f.s. 9. a declining and reclining , or inclining polar-plain ; that is , it so declines and reclines , or inclines , as to lie parallel to the pole , as the circle 8 p 8 doth represent . 10. a declining reclining-plain , that so declines and reclines , as not to fall in the pole or equinoctial , as generally they will do , as the circle 60 g 60 doth represent , which declines from the south-eastwards , and reclines 62 deg . which kind of plains are various and infinite , yet confined to six varieties , as afterward . now , the way of drawing these scheams , to represent these varieties , is briefly thus , by the sector . first , to the radius of the small tangents , draw the circle n. e. s. w. observing this method , if it be a south recliner , to set the letter n above , and e on the right hand ; and contrarily , in north recliners ; for we meddle not with incliners till afterwards , ( and alwayes observe , that a south incliner is the same with a north recliner , and the contrary ) then cross that circle with two diameters , precisely in the center , as the letters shew ; then according to your plains declination from north or south , toward either east or west , set off the declination with a line of chords or sines , as before is shewed ; and draw that line for the perpendiculer line of the plain , and laying the same distance as much from e. and w. draw another line perpendiculer to the former , representing the plain ; then , on the first line , viz. the plains perpendiculer , lay off from z , the half tangent of the plains reclination from z to e , and the half tangent of the complement thereof from z to q the contrary way ; and the whole tangent of the complement thereof from z , contrary to e , on the same line , extended for a center , to draw the reclining circle that represents the plain . lastly , you must draw a circle through q and p , ( p being alwayes the semi-tangent of the complement of the latitude laid alwayes from z toward n for the north pole ) so as to cut the primitive circle n.e. s.w. into two equal parts , as is shewed in the 10th proposition of the third chapter ; part of which line , doth represent the stile-line of the dial ; which last work shall be again shewed in the example . example . to draw the scheam for a plain , declining from the south to the west 35 degrees ; and reclining 20 degrees , for the latitude of 51-30 . first , to the radius of your small line of tangents , being the latteral distance from the center to 45 , ( or larger if you please ) draw the circle n.e.s.w. representing the horizon , crossing it in the center with the lines n.s. & w.e. for the north and south , and east and west lines . then , take out the latteral tangent of half the latitude , viz. 19-15 , for 38-30 , calling the tangent of 10 , the half tangent of 20 ; and lay it from z at the center , to p for the pole-point . then consider the declination of your plain , and which way , as here 35 deg . − 0′ from the south towards the west ; take out the chord of 35 deg . and lay if from s to c , and from w to a , and from n to d , and from e to b , for the more exact drawing of the lines ab , cd ; the lines cd representing the poles of the plain , and the line ab the declining plain it self ; then from z towards d , lay off the tangent of 10 deg . ( being the half tangent of 20 degrees , the given declination ) to e. also , take out the secant of 70 degrees , the complement of 20 , to the same radius ; and that laid from the point e , on the line dc produced , shall be the center to draw the circle afeb , that represents the declining , reclining plain , that declines 35 degrees , and reclines 20 degrees . also , lay off half the tangent of the complement of the reclination , viz. 35 degrees ( for the reclination is 20 , the complement whereof is 70 , and the half of 70 is 35 ) from z to q. then to draw the line qp , do thus ; observe how many degrees you count from z to the point e , counting from the center , count so many in the manner of half tangents from 45 ; and the latteral distance from thence to the center , laid from the center z , on the line cd , gives a third point , viz. the point i ; which three points , qpi , brought into a circle , will cut the circle n.e.s.w. into two equal parts . or thus ; the semi-tangent of the complement of the reclination to 180 degrees , laid from z on the line cd , will find the point i. as thus ; the reclination is 20 , the complement 70 , being taken from 180 rests 110 , whose half is 55 , the tangent of zi . or more briefly thus ; set one point of the compasses in the small tangent of 45 , and count the reclination from thence in the way of semi-tangents , both wayes , both above and under 45 ; and lay one , viz. that under 45 , from c to q ; and the other , viz. that above 45 from d to i ; then on the middle point , between q and i , last found , raise a perpendiculer to cd , and in that line will be the center to draw ipq. also , chap. ii. to draw the hour-lines on all ordinary dials ; the easiest in the first place . 1. and first for the first equinoctial dial . an equinoctial plain , as before is shewed , is that whose plain or flat superficies lieth parallel to the equinoctial , and is represented by the line waee in the general scheam , and therefore needs no other scheam to represent it ; in which dial all the hour-lines are equal one to the other , being just 15 degrees assunder ; so that to draw the hour-lines here , describe a circle as the circle 12.6.12.6 . and fit the radius in the sine of 30 degrees ( or the chord of 60 ) and take out the parallel sine of 7 degrees 30 minuts , the half of 15 degrees , and number it from 12 round about , and that shall divide the whole circle into 24 equal parts , for the 24 hours , for the true hour-lines on the equinoctial-plain , and is the same in all latitudes ; only in the setting of it , the poles of it are to be set due north and south ; the horizontal-line on the plain , lying parallel to the east and west-points of the horizon , and the stile thereof , only a wyre or sharp edge standing perpendiculerly on the center ; which being so set , must point directly to the north ( and south ) poles of the world. the reclining dials-stile pointing to the north-pole , and the inclining dials-stile pointing to the south pole ; then is the dial truly placed . to set a plain ( or to try whether a plain be set ) polar , or parallel to the equinoctial , do thus ; but to try the inclining plain , apply the loose-piece to the plain with the head-end downwards ; or else apply the head-leg to the plain with the head-end downwards , and the thred shall cut on 38-30 in london latitude , if the plain be set parallel to the equinoctial . 2. to draw a direct polar dial. the next dial , shall be a direct polar-dial , which is represented in the general scheam by the hour-line of 6 , viz. the line epw ; and here also the horizontal-line on the plain , is parallel to the east & west-points of the horizon ; and the pole ( or point opposite to the plain ) is in the equinoctial-point . the hour-lines , in this plain , are all parallels , because the axis , or stile-line , in all plains , is parallel to the poles of the world ; and this plain it self , being so parallel , the stile or axis therein makes no angle ; therefore the hour-lines must needs be parallels also . and the way of drawing those hour-lines , is thus ; first draw the perpendiculer-line on the plain , which is done thus by the trianguler-quadrant ; hang a plummet and thred on the center , and apply the moveable-leg to the plain , to and fro , till the thred falls neatly on 600 , and draw that line along by the moveable-leg , which shall be a true horizontal-line on any reclining plain ; and a perpendiculer-line thereunto , is the perpendiculer line on the plain . or else ; when the sun shineth ( the sun begins in the pole of the plain ) hold up a thred and plummet , till the shadow of the thred fall on the plain , making two points in that shadow at the remotest distance asunder ; then a line drawn through those two points shall be a true perpendiculer-line , ( this shall need no more repetition ) . then , the several = tangents of 60-45 , 30 & 15 , laid both wayes from 12 on both the horizontal-lines , shall give you po●n●s whereby to draw all the hour-lines in their true places . also , the = tangent of 45 , shall be the true breadth of the plate that must be a s●●●e to this dial ; or the length of an upright wyre set any where in the line 12. note , that for the hours under 45 , you may take = 45 from the small tangents , and make it a = tangent of 45 in the great tangents ; and then take o● = tangent of 30 & 30 , for 2 & 10 ; an● the = tangent of 15 for 11 & 1 ; and if you want them above 45 , then take the = tangent of 60 & 60 from the small tangents , and turn that extent 4 times from 12 both wayes , on both the horizontal-lines , and those shall be the points for 8 in the forenoon , and 4 afternoon . and lastly , the = tangent of 75 , taken and turned 4 times from 12 to 7 in the morning , and to 5 in the afternoon , will fit and fill a plain of 4 foot in breadth , with a sector of one foot , shut . 3. to draw a direct east or west-dial . the next dial , in the third place , is the direct east or west-dial , which is represented in the general scheam by the line nzs , whose poles are in the line ezw , whose plain also is = to the pole , & drawn in the same manner as the polar dial was ; yet with this difference , the equinoctial-line , whereon to prick the hours , is not the horizontal-line , but is thus found . thus ; you may take the tangents under 45 , when the sector is set to the small tangent , and by turning 4 times , you have the remainder of the great tangent above 45 ; when the sector is set to the great line of 45 , as in the polar dial. or else ; alter the sector to the radius of 45 in the great tangent that goes but to 45 , and take out the = tangents of 30 , and lay it from 6 both wayes , for 4 & 8 ; and the = tangent of 15 , and lay it from 6 , both wayes , for 7 & 5. and lastly , by all those points , draw lines = to 6 , for the hour-lines required ; and number the east-dial with the morning-hours , and the west with the afternoon-hours , the stile is to be a plate , or an upright point ; the top of whose edge , or point , is to be equal to the tangent of 45 , as the sector stood to prick down the hour-lines . 4. to draw the horizontal-plain . the fourth plain next , in a natural order of easiness to apprehend as i judge , is the horizontal dial , that lies with its plain = to the horizon ; and the zenith is the pole thereof , represented by the primitive circle s.n.e.w. in the general scheam , wherein only the hour , arks , and stile is required . the stiles elevation is alwayes equal to the latitude , and therefore given ; the substile is alwayes in the hour 12 , being the meridian-line . the hour-lines are found by this general canon ; as the sine of 90 , the right angle pn 1 , to the sine of pn , a side alwayes equal to the latitude or stiles elevation 51-30 ; so is the tangent of the angle np 1 , 15 ; or np 2 30 , &c. the angles at the pole , to the tangent of n 1 a side , or n 2 a second side , the several hour-arks on the plain required ; found by the artificial sines and tangents , as fast as one can write them down . thus ; the extent of the compasses , from the sine of 90 , the sine of the latitude 51-30 , being laid the same way from the tangent of 15 , shall reach to the tangent of 11-50 ; and if you turn the compasses the other way from the tangent of 15 , it shall give the tangent of 71-6 , for the hour of 5 as well as 11 ; which numbers being gathered into a table , and laid off by chords or sines in a semi-circle , shall be the true hour-points to draw the lines by . but i shall not insist further thereon , but shew how to draw it more readily , and as truly by the sector , thus ; first , draw a streight line ( in the meridian , if the plain be fixed ) for 12 , as the line ab ; then design a point in that line to serve for a center , as at c ; then on the center c , erect a perpendiculer-line to ab , and draw it through the center c , for the two 6 a clock hour-lines , as the line de ; then draw two lines equally distant from , and = to the first line ab , on either side , as large as the plain will give leave , as df and eg ; ( which may commonly serve for margents to put the figure in ) . then , take the distance cd , and make it a = secant of 00 , and take out the = secant of the complement of the latitude , and lay it from d to f , and from e to g , on the two parallel-lines , and draw the line fg. then lastly , for pricking down the stile , note , that the = tangent of 38-30 , the complement of the latitude , as the sector stands for the noon hours , laid from d to h , gives a point to draw it truly by ; or the sine of 51-30 the latitude , laid from b at nearest distance about h , as the sector stood for the morning-hours will do as well . the stile is to be a plate , or a bended wyre , cut or bended according to the angle hcb , and erected perpendicularly on the line 12 , so long , as the sun being 62 degrees high may cause the shadow thereof to reach the hour of 12 ; and then set duly north & south , and horizontal , the shadow will shew the true hour of the day . note the figure . note also , that a horizontal dial drawn for any one latitude , may serve for any other latitude north or south , elevating or depressing the stile , till it look to the pole-point ; that is , by making it to recline northward , or southward , as much as the difference of the latitudes , viz. that the dial was made for , and that wherein it is to be used , shall be . 5. to draw a north or south plain . the next plain to this , and most like it , is the direct north and south dial ; whose plain lies = to the prime virtical , or circle of east and west , and its poles in the south and north part of the horizon , and represented by the line ezw , in the general scheam , whose stile is alwayes equal to the complement of the latitude , as the horizontals was equal to the latitude , and consequently given . the hour arks on the plain , are found by the former canon , viz. as the sine of 90 , viz. the angle pze , is to the sine of the side pz , the co-latitude or stiles elevation ; the arks on the plain , found as before by artificial sines and tangents ; and being drawn into a table , to be laid off by chords , or sines , or by the sector , thus ; draw a perpendiculer-line for the substile , or 12 a clock line ; and in that line design a point for a center , as the point a in the line ab ; through which point a , draw another line , crossing the former at right angles , for a horizontal-line , and the two sixes , as you did in the horizontal ; then , on each side , and equi-distant from 12 , make two lines = to ab , as marginal-lines , as cf and de ; the distance ad , of the parallel make a = secant of 00 , and take out the = secant of 51-30 , the latitude of the place , and lay it from c to f , and from d to e , and draw the line fe ; then make df a tangent of 45 , and lay off the hours and quarters as you did in the horizontal in all respects . also , make bf a = tangent of 45 , and lay off the = tangents of every hour and quarters ( if you please ) from b , both wayes , toward e and f ; and by those points draw lines for the hours required . the angle of the stile may be laid off by sines , tangents , or chords , as before is shewed , to the quantity of the complement of the latitude , and may be a plate or wyre , as you please , as the angle gab. the north dial is the same with the south , for manner of making , only the noon-hours are neglected , and the morning and evening-hours , both before and after 6 on each side only inserted ; and the center of the dial for that cause appointed in the middle of the plain , and not on the upper-part , as in the south , and the stile-points upwards ( as in the south it points downwards ) . note the figures . 6. to draw the hours on a direct recliner . the next plain to be considered , being also direct , but not erect , or upright , but leaning from you ; and may be either a north or south recliner ; that is thus ; as the poles , of a direct south plain , are in the meeting-point of the meridian and horizon , viz. the point s. in the general scheam ; and the point n , in the same scheam , is the pole of the north plain ; and the point z , is the pole of the horizontal plain . so the pole of these plains is a point in the meridian , elevated as many degrees above the horizon , as the plain shall recline from the zenith , or upright toward the horizon . as thus ; suppose the hour-circle of 6 in the general scheam , to represent a reclining plain , the point ae , in the meridian , is the pole of it , being as many degrees above the horizon s , as p is below the zenith z. so also is p the pole to the north reclining plain waee ; for the point p , is as much above the horizon n , as ae is below the zenith z. thus you see what the pole of a plain means , viz. a point 90 degrees every way from it . now therefore north direct recliners have their poles any where between z & n , and south direct recliners have their poles any where between z & s , according to the degrees and minuts of their reclination . this being premised , for drawing the hour-lines , observe , that for south recliners direct , the difference between the reclination and the complement of your latitude , is alwayes the stiles height for that reclining plain . but note , that when the reclination is more than the complement of the latitude , that then the contrary pole is elevated , viz. the north pole on south recliners . but for north recliners , the sum of the co-latitude , and the plains reclination is alwayes the stiles elevation ; but note , when the sum is above 90 , then the complement to 180 is the stiles elevation ; but it must be turned the other way , viz. contrary to the nature of a north recliner , for the stile will look downwards in the north recliner , and upwards in his opposite south incliner . note also by the way , that when the south recliner reclines equal to the complement of the latitude , it is called a direct polar dial , or rather an equinoctial in respect of his poles , ( but i mind not to be singular ) . and when the reclination of a north recliner , is equal to the latitude , then the stiles height is just 90 degrees ; and the plain , called an equinoctial-plain , or rather polar , in respect of his poles , ( being first in order treated on ) . thus understanding , and right conceiving what the plains are , the drawing of any of them is the same with the north and south ; for the stiles height is alwayes to be counted the complement of the latitude , and by consequence you have the latitude . as thus for example . comparing the reading , and figure vii . suppose a plain recline from the zenith toward the north part of the horizon 10 degrees , his pole is 10 degrees above the horizon ; and then 10 taken from 38-30 , there remains 28-30 , for the stiles elevation ; or the latitude to draw it as a horizontal dial ; and 61-30 for the latitude for which place you are to draw a direct erect south dial. again , suppose a south recliner , recline 50 degrees , being more than 38-30 , the co-latitude ; then take the co-latitude from thence , and there remains 11-30 for the latitude or stiles height , to draw a horizontal dial by ; and 78-30 for a latitude to draw a south dial by ; but the cock must look up to the north , therefore must be turned the other way . again , for a north recliner , reclining 60 degrees ; 60 & 38-30 , added , makes 98-30 , whose complement to 180 , is 81-30 , the stiles height ; but the contrary way , as you may well perceive by the horizontal . as once more thus ; suppose a north plain recline 85 degrees ; that , and 38-30 , added , makes 123-30 , whose complement to 180 , is 56-30 , the stiles height : but put the con●●ary way , as a south incliner , being almost a horizontal dial ; so that to draw this dial , let 56-30 be the stiles height , or co-latitude ▪ then , 33-30 , is the latitude to draw a direct south dial by . or , you may count the stiles height the latitude , and then draw it as a horizontal-dial , by taking out the secant of the co-latitude , and the work will be the same ; as in the figure north reclining 60 degrees . 7. to draw a direct east or west-recliner . as thus ; with 45 degrees of the small tangents , draw the circle n.e.s.w. crossing it in the center with the lines we , sn ; then lay off the half tangent of 38-30 , from z to p ; and the half tangent of the reclination 45 , from z to e , and from z to q , and draw the circles nes , and fpq ; in which scheam , pf represents the stile , fg the distance of the substile from the meridian , and gpf the angle between the two meridians , viz. zpn of the place , qpf of the plain . all which requisites are thus found out by the artificial , or natural sines and tangents . 1. and first for the stiles elevation . as the sine of 90 nz , to the sine of ze , the reclination 45 ; so is the sine of np the latitude 51-32 , to the sine of pf , the stiles height , 33-37 . to work this by the trianguler quadrant , or sector , do thus ; as — sine of np , the latitude 51-32 , to = sine 90 ze ; so is = sine of ze , the reclination 45 , to — sine of pf 33-17 , the stiles height . 2. for the distance of the substile from 12 , thus , by artificial sines and tangents . as the tangent of the reclination , ze 45 , to sine of en 90 ; so is the tangent of the stiles elevation , pf 33-37 , to the sine of fg 41-40 , the distance of the substile from 12. or , by natural sines and tangents , thus ; as — sine 90 , to = tangent of the reclination 45 ; so is = tangent of the stiles height 33-37 , to — sine of the substile from 12 , 41-40 . for , if you only take the tangent of 33-37 , from the moveable-leg , and measure it on the sines from the center , it shall reach to the sine of 41-40 , the substiles distance from 12. 3. for the inclination of meridians , thus ; as the sine of the latitude np 51-32 , to the sine of pfg 90 ; so is the sine of the substile from 12 , gf 41-40 , to the sine of gpf 58-7 , the angle between the two meridians . by natural lines thus ; as — sine of the substile 41-40 gf , to = sine of the latitude 51-32 np ; so is = sine of 90 pfg , to — sine of gpf 58-7 , the incliner . if this rule fails , use a less radius , or a = answer , as is largely shewed before . thus having found the requisites , proceed to draw the dial thus ; first consider the scheam , where you shall find the reclining plain to be represented by the line sen , as the upper-edge thereof ; szn is the horizontal-line and meridian-line also ; n or g is the place where the meridian cuts the plain , being in the horizon ; therefore here the hour-line of 12 is a horizontal-line , and the sun being in the south part of the meridian , doth cast his shadow northwards ; and being in the east , casts his shadow westwards : therefore laying the scheam before you , as the plain reclines from you , you shall see that the meridian must lie to the right-hand from z , toward n ; and the substile upwards from n , towards e , at f. and the the stile must look upwards , as the angle gpf doth plainly shew ; and the morning-hours are chiefly fit for the plain , because the sun rising eastward , is opposite to the plain . thus the affections and scituations of the cardinal-lines are naturally and demonstratively shewed , the delineation followes . first , draw the horizontal-line sn ; and on z , ●s a center , describe a semi-cirle as sen , ●nd from n toward e lay off 41-40 , the distance of the substile from 12 , and draw ●he line zf for the substile ; also beyond ●hat , from f to e , prick off 33-17 , and ●raw that line for the stile-line . then for drawing the hour-iines , you ●ust first make the table of equinoctial-●istances , or angles at the pole , thus ; first , in all direct plains , it is orderly ●●us ; 3-45 , for the first quarter of an hour ●●om 12 ; 7-30 , for half an hour ; 11-15 , ●or three quarters ; and 15 degrees for an ●our ; and so successively to 90 : so also ●ill it be in all plains , whose inclination of meridians is just 15 , 30 , 45 , 60 , 75 , or ●0 degrees , being even whole hours ; and ●ear as well , when it falls on an even quar●er of an hour also . but when it doth not 〈◊〉 here , then the best rule or method i ●now is thus ; first , set down 12 , 11 , 10 , 9 , 8 , 7 , 6 , ● , 4 , 3 , 2 , 1 , & 12 , as in the table follow●●g . then right against 12 , set down the in●ination of meridians ; then substract 15 degrees for every hour ; and 3-45 for every quarter , as often as you can , setting down the remainder ; then draw a line a-cross , and what the last number remaining wants of 15 , or 3-45 ( for hours , or hours and quarters ) set down on the other side below the line , as you see in the table following ; and so proceed , adding of 3-45 to that sum , for every quarter ; or , 15 degrees for every whole hour , till you come to 90 both wayes ; so is the table of hour-arks at the pole , compleated for all hours that can come on this dial , or on any other . the table .   91 52 8 01 53 2 88 07   05 38   84 22   09 23   80 37   13 08   76 52 7 16 53 1 73 07   20 38   69 22   24 23   65 37   28 08   61 52 6 31 53 12 58 07   35 38   54 22   39 23   50 37   43 08   46 52 5 46 53 11 43 07   50 38   39 22   54 23   35 37   58 08   31 52 4 61 53 10 28 07   65 38   24 22   69 23   20 37   73 08   16 52 3 76 53 9 13 07   80 38   09 22   84 23   05 37   88 08   01 52 2 91 53 thus you see that the substile falls on near a quarter past 8 , or 3 hours 3 quarters and better from 12 ; then if you will , by the former canon , you may find all the hour-arks on the plain . thus ; as the sine 90 , to the sine of the stiles height 33-37 ; so is the tangent of the hours , in the table last made , called arks , at the pole 31-53 for 6 , to the tangent of the respective hour-arks at the plain from the substile 90-0 for 6. more br●●f thus ; as the sine of 90 , to the sine of the stiles elevation ; so is the tangent of the hour from 12 , to the tangent of the hour from the substile . which being brought into a table , may be pricked down in a semi-circle by sines , or a line of chords , from the substile on the plain . but i prefer this geometrical way before it thus ; having drawn 12 , the substile and stile , ●●aw also a line at any convenient distance ●●rallel to 12 , as di ; then at any conve●●●nt distance from the center z , draw a 〈◊〉 perpendiculer to the substile quite ●●ough the plain , as the line kl . then , take the nearest distance from that meet●●● point at f , to the stile-line , and make 〈◊〉 = tangent of 45 ; then the = tangent 〈◊〉 every hour and quarter , as in the table●●●ken ●●●ken from the sector , and laid from f the 〈◊〉 way , as the hours go , shall be the true 〈◊〉 whereby to draw the hour-lines re●●●red . but in regard that this way will some●●●e be troubled with excursions in some of 〈◊〉 hours , you may help it thus ; having 〈◊〉 some hours , as suppose 6 & 3 ; or 〈◊〉 9 ; or indeed any 2 hours , 3 hours di●●●●ce assunder , as here 6 & 9 ; take the di●●●nce between 6 and 9 , and lay it from the 〈◊〉 to n on the meridian , and draw the 〈◊〉 9 n = to 6 at length , beyond 12 ; 〈◊〉 , as before , make 6-9 a = tangent of 〈◊〉 , and lay off every hour and quarter , as 〈◊〉 the south erect dial , both wayes from also , make n 9 a = tangent of 45 , ●nd do likewise laying the hours both wayes ●rom n , and you shall have points enough to draw the dial by . otherwise , make these dials thus . count the complement of the latitude where the dial is to stand , for the latitude ; and the complement of the reclination for a new declination ; and then draw them as upright decliners by the following rules , and you shall do as well and speedily as any way . but note , that all east recliners , are north-east decliners ; and west recliners , are north-west decliners ; and east and west incliners ( being the under faces ) are south-east , and south-west decliners . also note , that if you draw your scheam true , and large , you may from thence geometrically find the substile , stile , inclination of meridians , and every hours distance on the plain , by scale and compass , thus ; as captain lankford hath shewed . first , for the stiles elevation set off a quadrant or quarter , as w.n. from b to i ; then a rule laid from z to i , cuts the plain at a ; then a rule laid from a , to p & f , cuts the circle at o and l ; the ark ol , is the stiles elevation , and measured by fit chords , gives 33-17 . secondly , for the distance of the substile from 12 , a rule laid from q to f , cuts the limb or circle at m , the ark mn measured on fit chords , gives 41-40 , the substile from 12. thirdly , for the inclination of meridians , a rule laid from p to a , in the limb gives c ; the ark wc 58-7 , is the angle between the two meridians . fourthly , to find the hour-arks on the plain ; a rule laid from q to the intersecting of every hours ark ( in the scheam ) and the plain as here , a rule laid from q to 6 , cuts the limb at r , the ark mr 19-0 is the distance of 6 from the substile on the plain ; and so for all others , as 12 & 6 is nr 60-40 , both hours and quarters if you have them truly drawn on a large general scheam , as mr. lankford hath done . thus much for direct plains , both erect and reclining , before i come to speak of decliners ; it will not be amiss to shew how to find the declination of a plain , both by the sun-shine , or without , by a magnetical-needle , as followeth . as — sine of the substile 41-40 gf , to = sine of the latitude 51-32 np ; so is = sine of 90 pfg , to — sine of gpf 58-7 , the incliner . chap. iii. to find the declination of any plain . for finding the declination of a plain , the most easie way is by a magnetical-needle , fitted according to mr. failes way , in the index of a declinatory ( as he calls it ) being 180 degrees of a semi-circle , divided on an oblong-board , or quadrant , or a longer needle in a square box , ( or ) fitted with hinges and a cover ; after all which wayes , you may have them made at the sign of the sun-dial in the minories , by iohn brown ; or of any other manner you shall think fit . but , to our trianguler quadrant , is a box and needle also to be fitted of another form , in some things more convenient . whose form is thus ; first , in a piece of box 5 inches long , 2 ½ broad , and 6 tenths of one inch thick , is a hole made near 4 inches long , 1 inch ¾ broad , and 4 tenths deep for a needle to play in ; about 50 degrees at each end ; with brass-hinges , and a cover , and a brace to keep the lid upright , & an axis of th●ed , and a plummet playing in the lid , and a horizontal and a south-dial , drawn on the box and cover ; also a hasp and glass to keep the needle close covered , and on the bottom a grove one tenth of an inch dee● , made just as broad as one leg of the sector is . the use whereof is thus ; put your box and needle , on that leg of the sector , as will be most convenient for your purpose ; the north or cross-end of the needle toward the wall , when it is a south decliner ; and the contrary when it is applyed to a north decliner , as the playing of the needle will tell you better than many words ; then open or close the rule , till the needle play right over the line in the bottom of the box , ( unless there be variation , then you must allow for it eastwards or westwards what it is ) . then , i say , the quantity of the angle in degrees and minuts the sector stands at , above or under 90 , is the degrees and minuts of declination ; being counted from 00 in the little semi-circle , as complements to the angle of opening ; as in the 4th use of the 5th chapter is largely and plainly shewed . thus you have the quantity of degrees and minuts of declination : but to determine which way , consider thus ; if the needle will stand still in the middle , when the north-end is toward the wall , then the first denomination is south , if not north. again , when you know where north and south is , you may resolve which way the east and west is ; for , observe alwayes , if the north be before you , then the east is on the right-hand , and the west on the left ; and contrarily , if the south be before you , the west is on the right-hand , and the east on the left . then , if the sun , being in the east-point of the horizon , can look on the plain , it is a south-east plain ; but if it beholds it when in the west-point , it is a south-west plain . likewise , if the cross-end of the needle will not stand toward the wall ( the needle playing well ) and the sun being due east , beholds the plain , then it is so many degrees north-east ; but if it cannot look on the plain , being due east , then it is a north-west plain , declining so many deg . as the sector stands at , under or above 90 , being alwayes the complement of the angle the legs of the sector stand at , and found by taking the angle the legs stand at , from 90 , when the angle is less than 90. or , taking 90 out of the angle , when it stands at an angle above 90 degrees as a look at the little semi-circle on the head sheweth . example . suppose i come to a wall , and putting the box and needle on the leg of the sector , and applying the other leg to the wall ( or on a streight piece of wood , applied to the wall , because of the walls unevenness ) ; and open or close the legs , till the needle playes right over the meridian-line , drawn on the bottom of the box ; then , i say , the complement of the angle the legs of the sector stands at , being alwayes what it wants of , or is above 90 degrees , is the degrees of declination ; and the coast which way , the needle and suns being east and west , tells you . for , if the north or cross-end of the needle be toward the wall , it is a south plain ; and if the sun , being in the east , can behold it , then it is south-east ; if not , a south-west plain . a ready way of counting the angle found , may be thus ; take the = distance between center and center , in the middle of the innermost-lines , and lay it latterally from the center , and co●nt two degrees more than the point sheweth , after the manner of chords from 90 ( at the sine of 45 ) toward the compass-point , and that shall be the degrees and minuts required . example . suppose the legs are so opened , that the = distance between the two centers , makes the — sine of 25 ; then , i say , the lines do stand at an angle of 50 degrees , and the legs at 48 , two degrees less , the complement whereof is 42 ; as if you count thus from 45 , you will find , 40 from 45 is 10 , 35 is 20 , 30 is 30 , 25 is 40 , and 2 degrees more makes 42 , the thing desired . but , if you like not the abating of two degrees , then the = distance taken just be●ween the two legs right against the cen●ers , shall be just the — sine of 24 degrees , ●r 42 , counting after the manner of chords , viz. every 5 degrees on the sines , for 10 on ●he chords backwards from 45 of the sines , which is 90 in chords . or , if you use the first rule , of the 4th use●f ●f the 5th chapter , viz. by taking the — ●ine of 30 , and put one point of the com●asses in the middle center in the tangent-●●ne , and apply the other to the line of ●ines , you shall find it reach to the sine com●lement of the angle the lines stand at , ●iz . 40 degrees and 2 degrees more , viz. ●2 , is the angle or thing desired ; as pra●tice with consideration will make easie . thus , by the needle , you may find the ●eclination of a wall , which in cloudy ●eather may stand you in good stead ; or 〈◊〉 prove a declination taken by the sun , to ●revent mistakes . and if nothing draw the ●eedle from its right position , but that it ●ay well , and you find the angle truly , ●ou may come to less than half a degree : and this convenience it hath , that it carries the needle a competent distance from the wall , to prevent that attraction ; but if it happen to be so near a meridian , or east and west-plain , that the angle , by the sector , cannot well be taken ; then you may only apply the side of the box and needle to the wall , and the needle it self will shew the declination , on the degrees on the bottom of the box. yet for exactness , the way by the sun is alwayes the best , where you may come to make a good observation , and then the needle only is not to be trusted to ; a better way with opportunity offering it self , to f●nd a declination of a wall by the sun. for this purpose you ( must or ) ought to have another thred and plummet , which thred may be a fine even small pack-thred , and it is convenient to have it ready hanged up near the wall , so far off , as the trianguler-quadrant may pass along between it and the wall , that you may not be troubled to hold it up , and lay it down , and be annoyed with the inconveniencies of your hand shaking , and time wasting , to more uncertainty than needs be . also , you must needs take notice of the two meridians , viz. one of the place which is the meridian , or 12 a clock ; to which place , when the sun or a star comes , it is said to be in the meridian . and the other is the meridian of the plain , in which line the pole-point of every plain is , being 90 degrees distant from the plain every way , and in all upright-dials their pole is in the horizon ; and that degree of azimuth in which the pole-point lies , counted from south or north toward east or west , is alwayes the declination thereof ; so that by finding the suns azimuth at any time , and the distance of the sun at the same time from the meridian of the plain , is gotten the declination . the azimuth of the sun from the meridian of the place , is found by the 26 , 27 , 28 , 30 , 32 , 34 , 39 uses of the 15th chapter . but the azimuth of the sun from the meridian of the plain , is found by applying the head-leg against the plain horizontally , slipping it to and fro , till the shadow of the thred , hung ( or held ) up , play right over the center of the trianguler-quadrant on the head-leg ; then what deg . soever the thred cuts , counted from 60●0 on the loose-piece ( being the perpendiculer or pole-point of the plain ) shall be the azimuth of the sun from the meridian of the plain . this is the operation ; the application or use is worded several wayes by several men ; i hope i shall do it as fully , and as briefly as some others . the sun , to our appearance , passeth from east , by the meridian , to the west every day ; therefore in the morning it wants of coming to the meridian ; at noon it is for a moment just in the meridian , and in the afternoon it is past the meridian of the place . even so it begins to shine on , and is directly against , and leaveth to shine upon most plains , when it begins to shine upon , or is not directly against ; i say , it wants of coming to the pole or meridian of the plain . when it is directly against the plain , then it is in the meridian or pole of the plain ; when it ●s past , it is past , or begins to leave the plain . which th●ee varieties i intend thus briefly to e●press ; azimuth want , or w in the morning only ; azimuth direct at noon ; azimuth past , or p being in the afternoon . the other three varieties let be shadow want , shadow direct , shadow past ; all which may be in several plains at several times ; that is to say , at morning , noon , and night . these observations , and cautions premised , the rule is thus ; 1. if the azimuth and shadow are both wanting , or both past ; substract the losser out of the greater , and the residue is the declination . but if one want , and the other be past , then the sum of them is the declination . 2. if the sun come to the meridian of the plain , before it come to the meridian of the place , it is an east plain . but if it come to the meridian of the place , before it come to the meridian of the plain , it is a west plain . 3. if the sum or remainder , after addition or substraction , be under 90 , it is a south-east , or south-west plain , declining so many degrees , as the sum or residue is . but if the sum or remainder be above 90 , it is a north-east or a north-west plain , and the complement of the sum or remainder to 180 , is the quantity of declination north-east , or north-west . 4. if the sum or remainder be 00 , it is just south ; if 90 , just east or west . but , if it be 180 , it is a direct north plain . it shall be further explained by two or three examples . suppose that on the first of may , in the forenoon , i come and apply the head-leg of the trianguler-quadrant to the wall , and holding of it level , the shadow of the thred , held up steady , cuts the center and 60 degrees on the moving-leg ; that is , 60 deg . want ; which i presently set down in a paper ready prepared , thus ; may 1 , 1669. forenoon . shadow 60 00 want . altitude 20 00   azimuth 94 00 want . substract . 34 00 south-east . then , as soon as possible , or rather by some body else , at the same moment , find the suns altitude , which suppose to be 20 degrees ; ( but if you are alone , and have a thred ready hanged up ; then take the altitude first , and the shadow will be had presently after , the thred hanging steadily ) and set that down also , as here you see . then by the 26th use of the 15th chapter , you shall find the suns azimuth at that time and altitude to be 94 degrees , and after substraction remains 34-0 , for the walls declination eastward , becau●e the remainder is under 90 , and the sun comes to the meridian of the plain , before it comes to the meridian of the place , or south . again , in a morning , iune 13 , i observe the altitude , and find it 15 degrees , and instantly the shadow , and find it to be 10 degrees past the plain , viz. on the loose-piece , toward the head-leg , i set both altitude and shadow , with the day and time down thus ; iune 13 , forenoon . altitude 15 0   shadow 10 0 past. azimuth 109 0 want. sum is 119 0     180 0     061 0 north-east . and then find the azimuth at that time and altitude to be 109 degrees ; here the terms being unlike , i add them together , and the sum being above 90 , i know it must be a north plain ; and because the sun comes to the plain before it comes to the meridian of the place , it is north-east ; and the complement of 119 to 180 , is 61-0 north-east . again , iune 13 , afternoon ▪ altitude 15 0   shadow 20 0 want. azimuth 109 0 past. sum is 129 0     180 0     051 0 north-west . but if you happen to come when the sun is in the meridian of the plain , then the suns azimuth is the declination , east or west , as the azimuth is . also , if you take the shadow , when the sun is just in the south , or meridian of the place , the shadow is the declination ; if it is past the plain , it is eastward ; if it wants , it is westwards . thus i have ( i hope ) shewed the true manner of finding the declination of a wall by the sun shining on the plain , as plainly and as briefly as the matter will bear , speaking to young tiroes therein . it may be done also , by observing when the sun just begins to shine on a chimny , or wall , or high place you cannot for the present come near , conceiving the sun to be then just 90 degrees from the meridian of the place wanting , or just when it leaves it being then 90 degrees past the plain , then take the altitude and azimuth , and work accordingly to the former rules . chap. iv. to draw a south , or north erect declining-dial ▪ for better illustrations sake , i will draw a particular scheam for this dial also , as i did for the east recliner ; whose declination let it be 20 degrees declining from the south toward the west , in the latitude of 51-32 for london . the scheam is drawn by the former directions ; the pole of the plain being at d , declining 20 degrees from s toward w , and the plain it self is represented by the line ab ; the circuler pricked line dhpc is a certain meridian drawn through the three given points dpc , whose center will be in the intersection of the plain ab , and the tangent line for the hours , which being drawn , whatsoever zh is in the half tangents , zq is the complement thereof , in the same half tangents . the scheam thus drawn , zh is the substile , ph is the stile , qz the distance of 6 from 12 , hpz the inclination of meridians , or angle between the two meridians , viz. of the place pz , and of the plain ph , found by the following canons . by artificial sines and tangents . 1. first for the substile from 12. as the sine of 90 zn , to the sine of the declination nc 20 degrees ; so is the co-tangent of the latitude pz 38-28 , to the tangent of zh 15-12 . 2. for the stiles elevation . as the sine of 90 zn , to the co-sine of the plains declination na 70-0 ; so is the co-sine of the latitude zp 38-28 , to the sine of ph 35-46 , the stiles elevation . 3. for the distance between 6 & 12. as the sine of 90 zw , to the sine of the plains declination wa 20-0 ; so is the tangent of the latitude np 51-32 , to the tangent of aq , the co-tangent of 6 from 12 , 23-18 . or thus ; as co-tangent latitude zp 38-28 , to the sine 90 zpq ; so is s. declination wza 20-0 , to the tangent of qa 23-18 . 4. for the inclination of meridians . as the sine of the latitude zae 51-32 , to the sine of 90 zs ; so is the tangent of the declination sd 20-00 , to the tangent of aek 24-56 , the inclination of meridians . or , as co-sine latitude 38-28 pz , to the sine 90 phz ; so is the sine of the substile zh 15-12 , to the sine of zph 24-56 , im . 5. then having made a table of arks at the pole , by this canon you may find the hour-arks on the plain . thus ; as the sine of 90 pk , to the sine of the stiles height ph 35 46 ; so is the tangent of the hour from 12 , 19-56 for i , aei , to the tangent of the hour from the substile on the plain , h 1 , 12-14 . but i prefer the way by tangents before it , as followeth . all these requisites may be found by the general scale and sector , the canons whereof in brief are thus ; by the trianguler-quadrant and sector . substile . as — co-tang . lat. 38-28 zp , to = sine 90 90-00 zn ; so = sine declination 20-00 nc , to — i substiler 15-12 zh . stile . as — co-sine lat. 38-28 zp , to = sine 90 90-00 zn ; so = co-sine declin . 70-00 na , to — sine stile 35-46 ph. distance between 6 & 12. as — sine declin . 20-0 wa , to = sine of 90 90-0 zw ; so is = c.t. of lat. 51-32 np , to — c.t. 6 & 12 23-18 aq . tangent 66-42 .   inclinations of meridians . as — i declination 20-0 sd , to = sine latitude 51-32 zae ; so = sine 90 90-00 zs , to — t. inclin . merid. 24-56 aek . these requisites are also found by the particular quadrant , very really and truly , for that latitude the rule is made for , in this manner . 1. first , for the substile . lay the thred to the complement of the plains declination , counted on the azimuth line , and on the degrees it giveth the substile from 12 , counting from 600 on the moveable-leg . example . the thred laid to 70 , the complement of 20 on the degrees , gives 15-12 for the substile . 2. for the stiles height . take the distance between 90 , and the plains declination on the azimuth-line , and measure it on the particular scale from the begin●ing , and it shall give the angle of the stiles-elevation above the substile , 35-46 . 3. for the inclination of meridians . take the substile from the particular scale of altitudes , and measure on the azimuth-line from 90 , and it shall give the complement of the inclination of meridians , or the angle counting from 90. viz. here 24-56 . 4. to find the angle between 12 & 6. take the plains declination from the particular scale of altitudes ( less by the sine of the declination , to a radius equal to 45 minuts of the first degree on the particular scale of altitudes ) , and lay it from 90 on the azimuth scale , and to the compass-point lay the thred , then on the line of degrees , the thred gives the complement of 6 from 12 , counting from 60 toward the end , as here it is in this dial 23-18 . also , the requisites may be found geometrically by the scheam , thus ; as , 1. a ruler laid from d to h in the limb , gives f ; the ark cf is the substile . 2. a ruler laid from q to p in the limb gives i , the ark ai is the stiles height . 3. a rule laid from p to q , cuts the limb at t ; the ark te is the inclination of meridians . or , a rule laid from p to k , cuts the limb at l ; then sl is the inclination of meridians . 4. a rule laid from d to q , cuts the limb at 6 , the ark c 6 , is the angle between 12 and 6. 5. a rule laid from d , to the intersection of any other hour-line , with the plain ab on the limb , gives points , whose distances from c , are their angles from 12 , or their distances from f , or their angles from the substile . to delineate the dial by the sector . thus by any of these wayes , having gotten the requisites , proceed to draw the dial thus ; then , take out the = tangent of 30 , & 15 , and the respective quarters , and &c. as before ; then make 6 g a = tangent of 45 , and do likewise as before , in the horizontal and south dials , and to those points draw the hour-lines required . 2. to draw the hour-lines on a north-declining dial. the requisites , as substile , and stile , inclination of meridians , 6 & 12 , are found the same way , and by the same rules , as the south decliners are done . but when you come to delineate the dial , there is some alteration ; which i conceive is best seen by an example , as northeast declining 35 deg . lat. 51-32 , at london . first , as be●ore , draw a perpendiculer-line for 12 a clock , as ab ; then about the middle , or toward the lower-part of that line , as at c , make a point for a center , as c ; then on the center c describe the arch of a circle , that way , from the line ab , as is contrary to the coast of declination , as if the plain declines eastward , as here , draw the arch westward from ab , as bd ; and the contrary way in north-west plains ; and on that ark lay down the substile from 12 , and the stiles height above the substile , and the hour of 6 , by the angle of 6 & 12 ; and then , by those points and the center , draw these lines . then , at any distance , draw a line = to 12 ▪ ( or ab ) as the line ef , and make that distance a = secant of 35 , the declination ; the sector so set , take out the = secant of the latitude 51-32 , and lay it on the parallel-line from 6 to 9 , then make 6-9 , the measure the compass stands at , a = tangent of 45 ; and take out the = tangent of 15-30 , &c. and lay them both wayes from 6 , upwards and downwards ; also , for the hour of 10 , as the sector stands , take out the = tangent of 60 , and turn it 4 times from 6 on the line ef ; and ( when you want it ) the = tangent of 75 , and turn that also 4 times from 6 , for 11 a clock-line ; and then by those points , draw lines for the hours required . chap. v. to draw the hour-lines on a dial falling near the meridian , whose stile hath but a small elevation , and therefore no center . the former examples may be sufficient to the considerate , to draw any erect declining-dial having a center ; but when the stile happens to be less than 15 deg . of elevation ; then , if it be not augmented by casting away the center , the usefulness , and handsomness of the dial is l●st ; now if you draw the dial by the former rules on a table , and cut off so much , and as many hours as you care for , the work is performed . lat. 51-32 d. mi.   s.w. 80-25   sub. 38-4   stile 5-56   ● & 12 38-51   i. m. 82-30 12   75●00     67-30 1   60-00     52-30 2   45-00     37-30 3   30-00     22-30 4   15-00     10-30 5   00-00 —   07-30 6   15-00     22-30 7   30-00     37-30 8 first , on or near the north edge of the plain , in far south decliners , ( but near to the south-edge of the plain , in far north-decliners ) draw a perpendicular-line , representing the hour-line of 12 , as the line ab in our example , being a southwest declining 80 deg . 25 min. then , in the upper-part of that line , in south-decliners ; or about the middle , or lower-part in north-decliners , appoint a center , as here at a ; then upon a , as a center , as large as you may , draw an arch as bd ; and in that arch , or rather by a tangent-line , lay 〈◊〉 the substile from b to d , and draw the ●ine ad , as an obscure line , for the present only to be seen ; and upon that , the stile-line , as before : then at any convenient places , as far from the center as you can , draw two lines perpendiculer to the substile , as the lines ce , fg , for two contingent lines , ( antiently and properly so called ) ; then by the inclination of meridians , by the directions in the east and west recliner , being the 7th dial in the 2d chapter , make the table of hour-arks at the pole , by setting down against 12 , 82-30 ; and taking out 7-30 for every half hour , till you come to 00 at the substile ; and then by adding 7-30 for every half hour , and 15 for every hour , to 8 ½ , as long as the sun shines ; which in regard it falls on an even half hour , is the most easie , and fits the points in the tangent ready made for hours and quarters . the next work , is to resolve what hours shall come on the plain , as will be best determined by the discreet orderer , or surveyor , or experimental dialist , as here 8 and 1 ; and for those two hours , mark the upper contingent line in two places where you would have them to be , as at e and c ; then take the — tangent of 37-30 for 8 , from the small tangents , and add it to the = tangent of 67-30 , the tangent for 1 ; and behold ! it makes the — tang. of 72-33 . then , take the whole space ce , and make it a = tangent of 72-33 ; then take out the = tangent of 67-30 , and lay it from c to h ; and take also in the same common-line , right against the small tangent of 37-30 , which is in the large tangent 10-50 , the = tangent of 10-50 , taken for 37-30 , being laid from e , the place for 8 , will meet just at h ; which point h , is the true place for the substile , to fit and fill the plain , with the hours determined . then , the sector so set , take out all the = tangents above 45 , as in the table , and lay them the right way from h , toward c , and e ; then take out = tangent of small 45 ; and setting one point in h , strike the touch of an arch , as at i ; then make hi a = tangent in great 45 , and take out the = tangents of the rest of the hours under 45 , as in the table , and lay them both wayes from h , because the substile falls on an even half hour . then , draw the line hk , = to the first line ad , for the true substile ; then make hk radius , or the tangent of 45 , and take out the parallel tangent of 5-56 , the stiles height , and lay it from k to l ; then take hi , the first radius , and setting one point in l , draw the touch of an ark as by m ; then draw a line by the convexity of the arches by i and m , for the true stile-line . then , take the nearest distance from the point k , to the line im , and make it a = tangent of 45 , the greater radius , and take out the = tangents , as in the table , and lay them from k both wayes ; and then lastly , by those points draw lines from the hours required . note , that if in striving to put too many hours , the sum of the two extream hours come to above 76 , it will make the hours too close together , and put you to much more trouble . also note , if your rule prove too small , then take the half of the sum of both the tangents , and turn the compasses twice . also , if you be curious , you may use the natural logarithm tangents , instead of the line of tangents , but this will serve very well . this is a general way of augmenting all manner of dials , when the stiles height is low , as under 15 degrees ; and as ready a way , as you meet with in any author whatsoever . chap. vi. to draw the hour-lines on declining reclining dials . for the compleat and true drawing of these dials , that you may plainly see their affections and properties , it will be necessary to have a scheam for every variety ; in doing whereof , i shall follow the method that mr. wells , of deptford , used in his art of shadows ; which will comprehend any sort of reclining and declining dial , under 6 varieties , viz. 3 south recliners , and 3 north recliners ( the inclining being their opposites , and no other , as afterwards is shewed ) . wherein i shall be very brief , yet sufficiently plain , to a mathematical genius , and render the canons , by artificial and natural sines and tangents , and draw the dial by the sector , the fittest instrument for that use ; with other occurrent observations , as they come in place , and the way by the scheam geometrically also . 1. and first for a south-declining , reclining dial , declining from the south toward the west 35 degrees , and reclining from the zenith 20 degrees , being less than to the pole , viz. falling from you , between the zenith and the pole : as the circle aeb , representing the reclining plain , plainly sheweth p being the pole , and z the zenith . the manner of drawing this scheam , is plainly shewed before , ( chap. 1. ) both generally and particularly for the drawing of dials , and the example there , is the very scheam for this dial ; wherein you may further consider , that the perpendiculer-line is right before you , and when you look right on this plain that declines southwest , the north is before you on the left hand , the south behind you on the right hand 35 degrees , the east on the right hand , the west on the left ; the line cd the perpendiculer-line right before you , representing the perpendiculer-line on the plain , ab the horizontal-line , ze the quantity of reclination , pf the stiles elevation above the plain , having the south pole elevated above the lower-part of the plain , because the north-pole is behind the plain , eg the distance on the plain , between the plains perpendiculer , and the meridian , ( being to be laid eastwards , as the dial-draught sheweth , besides that general rule before hinted , that whensoever a plain declines eastward , the substile line must stand westward , and the contrary ; for the arch whereon to prick the substile , and stile is alwayes to be drawn on that side of the plain , which is contrary to the coast of declination ) ef the distance from the substile and perpendiculer , to be laid the same way ; gf the distance on the plain , from the substile to the meridian , to be laid the same way also ; the angle fpg , is the inclination of meridians ; all which requisites are found by these canons arithmetically , or by the artificial and natural sines and tangents . 1. to find the distance of 12 , from the perpendiculer eg , or horizon ag , by the second axiome of mr. gellibrand , viz. that the sines of the base and tangent of the perpendiculer are proportional . by the sector and quadrant . as sine 90 radius , zd 90 — 00 to tang. of declin . plain nd 35 — 00 so sine of reclin . plain ze 20 — 00 to tang. of perp. & 12 eg 13 — 28 whos 's complement ac 76 — 32 is the distance from the east-end of the horizon to 12. as — tangent of nd 35 — 0 to = sine of zd 90 — 0 so = sine of ze 20 — 0 to — tangent of eg 13 — 28 , the distance of 12 , from the perpendiculer . 2. to find the distance on the meridian ; from the pole to the plain , pg , by the 3 propositions of mr. gellibrand , the sines of the sides are proportional to the sines of their opposite angles . by the quadrant and sector . as the sine of the perpendiculer from 12 ge 13-28 to the sine of declination gze 35-00 so is the sine of 90 gez 90-0 to the sine of the distance on the meridian , from the plain to the zenith — gz 23-55 as — sine 90-0 gep , to = sine 35-0 gze ; so = sine 13-28 ge , to — sine 23-55 gz , which taken from 38-28 gives pg 14-33 . which being taken from 38-28 , the distance on the meridian from the pole to the zenith , leaveth the distance on the meridian of the place , from the pole to the plain , viz. 14-33 , as a help to get the next . 3. to find the height of the stile above the plain pf . in the two triangles zge , and pgf , which are vertical , by the second consectary of mr. gellibrand ; if two perpendiculer arks subtend equal angles , on each side of the meeting , then the sines of their hypothenusaes , and perpendiculers are proportional , ( and the contrary ) ; for the angles zge , and pgf are equal angled at g , and ze , and pf , are both two perpendiculer arks on the plain ab . therefore , as the sine of the hypothenusa gz , to the sine of the perpendiculer ze ; so is the sine of the hypothenusa pg , to the sine of the perpendiculer pf , and the contrary . then thus ; by the quadrant and sector . as sine of the arch of the merid. from the zenith to the plain , zg 23-57 to sine of the reclination ze 20 00 so is the sine of the arch on the meridian , from the pole to the plain pg 14-33 to sine of the stiles height pf 12-13 as — sine of pg 14-33 to = sine of zg 23-58 so = sine of ze 20-00 to — sine of pf 12-13 4. to find the distance of the substile from the meridian , gf . in the same vertical triangle , having the same acute angle at the base , the tangents of the perpendiculers , are proportional to the sines of the base , by the second axiome of mr. gellibrand . therefore , by the quadrant and sector . as the tang. of the reclin . ze 20-0 to the sine of the distance on the plain , from the perpend . to the merid. ge 13-28 so is the t. of the stils height pf 12-13 to the s. of the subst . fr. 12 fg 7-58 as — sine of ge 13-28 to = tang. of ze 20-0 so = tang. of pf 12-13 to — sine of fg 7-58 5. to find the angle between the two meridians , of the place and plain , viz. the angle , pfg . by the third proposition of mr. gellibrand , it is proved , that the sines of the sides are proportional to the sines of their opposite angles , and the contrary . therefore , by the quadrant and sector . as the sine of the dist. on the merid. from the pole to plain pg 14-33 to the s. of 90 , the opp . angle pfg 90-00 so is the s. of the subst . fr. 12 , fg 07-58 to the s. of the inclin . merid. fpg 33-28 as = sine of the side 07-58 fg to = sine of the side 14-33 pg so = sine of the angle 90-00 pfg to — sine of the angle 33-28 fpg the angle between the 2 meridians . by angle of inclinations of meridians , make the table of the hour-angles at the pole , by the directions , chap. 2. which being made as in the table , draw the dial in this manner ; 12 33-28   1 18-28   2 3-28       sub. 3 11-32   4 26-32   5 41-32   6 56-32   7 71-32   8 86-32   9 78-28   10 63-28   11 48-28   12 33-28   upon ab , the horizontal-line of your plain , describe the semi-circle aeb , and from the perpendiculer-line ce of the plain , lay off 13-28 eastward for the 12 a clock line , on the plain , or the complement thereof 76-32 , from the east-end at b to + , & draw the line c + . again , set further eastward from 12 , 7-58 , the distance of the substile from 12 , to f , and draw the line cf for the substile ; and beyond that , set off from f 12-13 , the stiles height above the substile to g , and draw cg also . then , draw a contingent line perpendiculer to the substile cf , as far from the center as you can , as the line hi ; then take the nearest distance from the point f , to the line cg , and make it a = tangent of 45 ; then the sector being so set , take out the = tangents of all the hour-arks in the table , and lay them both wayes from f toward h and i , as they proceed ; then lines drawn from the center c , and those points shall be the hours required . or , having in that manner pricked down 12 , 6 & 3 ( or any other hours 3 hours distant ) draw two lines on each side 12 = to 12 , and measure the distance from 6 to 3 in the = , and lay it from c the center on the line 12 ; and by those two points draw a third line , = to the 6 a clock-line ; then 6-3 , and 12-3 , made a = tangent of 45 , shall be the two radiusses to lay off the hour-lines from 6 & 12 , as before in the former dials . and the = tangent of inclination of meridians , doth prove the truth of your work here also , as well as in the decliners erect . but note , that this dial is better to be augmented by the losing the hours of 8 and 9 in the morning , which makes the hours more apparent , as you see . also , the requisites formerly sound , may geometrically be found by the scheam , being large and truly drawn , as before is shewed in the other dials . thus , 1. a rule laid from q , the pole-point of the plain , to g the point of 12 on the plain , gives in the limb the point 12 ; d 12 , 13-28 , is the distance of 12 a clock-line on the plain from the plains perpendiculer-line zd , ( and to be laid from the perpendiculer-line on the plain eastwards in the dial ) ; and the distance on the limb from a to 12 , is the meridians distance from the east-end of the horizontal-line on the plain , namely 76-32 . 2. a rule laid from q to f , on the limb , gives the point sub , for the substile ; and the ark sub. 12 , 7-58 , is the distance from 12 , or the ark sub. d 21-26 , the distance from the perpendiculer . 3. a rule laid from q to 6 , the place where the 6 a clock hour-line on the scheam cuts the plain , gives on the limb the point 6 , the ark 6 12 , 25-38 , or 6 d , 38-56 , is the distance of the hour-line of 6 on the plain , from the hour-line 12 , or the perpendiculer . 4. a rule laid from y , the pole-point of the circle qfp , to p & f , on the limb , gives two points ik , and the ark ik is the stiles elevation 12-13 . 5. a rule laid from p to y on the limb , gives the point m ; em is the inclination of meridians : or , a rule laid from p , to the intersection of the circle pfq , and the equinoctial-line , gives a point in the limb near c , which ark cs , is more naturally the angle between the two meridians , 33-28 . or , if you like the way of referring this plain to a new latitude , and to a new declination in that new latitude , then thus by the scheam ; 6. a rule laid from e , to p and g , in the limb gives l and o ; the ark lo is the complement of the new latitude , being the ark pg , the second requisite , in the former calculation being 14-33 , the distance on the meridian from the pole to the plain . but note , that this dial is better to be augmented by the losing the hours of 8 and 9 in the morning , which makes the hours more apparent , as you see . also , the requisites formerly found , may geometrically be found by the scheam , being large and truly drawn , as before is shewed in the other dials . thus , 1. a rule laid from q , the pole-point of the plain , to g the point of 12 on the plain , gives in the limb the point 12 ; d 12 , 13-28 , is the distance of 12 a clock-line on the plain from the plains perpendiculer-line zd , ( and to be laid from the perpendiculer-line on the plain eastwards in the dial ) ; and the distance on the limb from a to 12 , is the meridians distance from the east-end of the horizontal-line on the plain , namely 76-32 . 2. a rule laid from q to f , on the limb , gives the point sub , for the substile ; and the ark sub. 12 , 7-58 , is the distance from 12 , or the ark sub. d 21-26 , the distance from the perpendiculer . 3. a rule laid from q to 6 , the place where the 6 a clock hour-line on the scheam cuts the plain , gives on the limb the point 6 , the ark 6 12 , 25-38 , or 6 d , 38-56 , is the distance of the hour-line of 6 on the plain , from the hour-line 12 , or the perpendiculer . 4. a rule laid from y , the pole-point of the circle qfp , to p & f , on the limb , gives two points ik , and the ark ik is the stiles elevation 12-13 . 5. a rule laid from p to y on the limb , gives the point m ; em is the inclination of meridians : or , a rule laid from p , to the intersection of the circle pfq , and the equinoctial-line , gives a point in the limb near c , which ark cs , is more naturally the angle between the two meridians , 33-28 . or , if you like the way of referring this plain to a new latitude , and to a new declination in that new latitude , then thus by the scheam ; 6. a rule laid from e , to p and g , in the limb gives l and o ; the ark lo is the complement of the new latitude , being the ark pg , the second requisite , in the former calculation being 14-33 , the distance on the meridian from the pole to the plain . 7. a rule laid from g to q on the limb , gives r , the ark sr is the new declination in that new latitude , 32-37 . or else find it by this rule ; as sine of 90 , to the co-sine of the reclination , or inclination ; so is the sine of the old declination , to the sine of the new , in this example , being 32-37 , and generally the same way as the old declination is . only observe , that when the north-pole is elevated on south recliners , you must draw them as north-decliners ; and north-west and north-east incliners , that have the south-pole elevated , you must draw them as south-east and west-decliners , which will direct as to the right way of placing the substile , and hour of 6 from 12. in this place i shall also insert the general way , by calculation , to find the new latitude , as well as new declination : which is thus ; as radius , or sine of 90 , to the co-sine of the plains old declination ; so is the co-tangent of the reclination , or inclin . to the tang. of a 4th ark. then , in south recliners , and in north incliners , get the difference between this 4th ark , and the latitude of your place , and the complement of that difference is the new latitude : if the 4th ark be less then the old latitude , then the contrary pole is elevated ; but if it be equal to the old latitude , it is a polar-plain . but in south incliners , and in north recliners , the difference between the 4th ark , and the complement of the latitude of the place ( or old latitude ) shall be the new latitude , when the 4th ark and old latitude is equal , it is an equinoctial-plain . thus in this example ; as sine 90 , to co-sine of 35 , the old declination ; so is co-tangent of 20 , the reclination to 66-03 , for a 4th ark ; from which taking 51-32 , the old latitude , rests 14-31 , the complement of the new latitude , which will be found to be 75-29 , the new latitude . by which new latitude , and new declination , if you work as for an erect dial , you shall find the same requisites , as by the former operations you have done ; and the distance of the perpendiculer and meridian will set all right . the second variety of south recliners , reclining just to the pole. 1. the scheam is drawn , as before , to the same declination , and the same way , viz. 35 degrees westward , and reclines 33-3′ , now , to try whether such a plain be just a polar-plain or no , use this proportion : by the sector ; as the sine of 90 da 90-0 to co-sine of declin . na 55-0 so co-tang . of reclin . de 56-57 to tang. of latitude np 51-32 as — co-sine declination na 55-00 to = sine of ad 90-00 so is = co-tang . of reclin . de 56-57 being taken from the small tangents , to — tangent of np 51-32 being measured from the center on the same small tangents . which 4th ark , if it hit to be right the latitude , then it is a declining polar-plain , or else not . 2. if you have a declination given , to which you would find a reclination to make it polar , then reason thus ▪ by the sector ; as the co-sine of the declin . an 55-0 to the radius or sine of ad 90-0 so is the tang. of the lat. pn 51-32 to the co-tang . of the reclin . de 56-57 as — tangent of np 51-30 to = sine of an 55-00 so = sine of ad 90-00 to — tangent of de 56-57 3. if the reclination were given , and the declination required to make it a polar , then the canon may be thus ; by the sector ; as the co-tang . of the reclin . de 56-57 to the radius , or sine of ad 90-00 so is the tang. of the lat. np 51-32 to the co-sine of the declin . na 55-00 as — co-tang . reclination . de 56-57 to = sine of ad 90-00 so — tang. of latitude nd 51-32 to = co-sine of declination na 55-00 but by the scheam , these three operations are found by drawing the scheam . 1. for if the line or circle , representing the plain , cut the pole p , it is a polar-dial . 2. if ab , the co-declination , be given , then draw the circle apb , and it gives e ; then ze is the reclination , measured by half tangents ; or a rule laid from a to e on the limb , gives an ark from b ; which measured on fit chords , is the reclination . 3. if p , the pole-point , and ze the reclinatin , be given ; then , with the distance ze , on z as a center , draw an ark of a circle in that quadrant which is contrary to the coast of declination , observing the letters in the scheam ; then by the convexity of that ark , and the pole-point p , draw the circle pe , cutting the limb into two equal parts , which are the points a & b , the declination required . this being premised , there are two things requisite to be found , before you can draw the dial. viz. the substile from the perpendiculer or horizon , and the inclination of meridians . 1. and first for the substile , by the sector . as the sine of pez 90-0 to the co-sine of the lat. pz 38-28 so the sine of the declination pze 35-00 to the sine of substile from perp. pe 20-54 as — sine of declination pze 35-0 to = sine of pez 90-0 so = sine of co-latitude pz 38-28 to — sine of substile from perp. fe 20-54 the distance of the substile from the perpendiculer , whose complement 69-06 , is the elevation above the horizon . or , a rule laid from q to p , gives i ; di is 20-54 . 2. for the inclination of meridians , say , by the sector . as the co-sine of the latitude pz 38-28 to the sine of pez 90-00 so the sine of the reclin . ze 33-03 to the co-sine of incl. mer. zpe 61-15 whose complement zpq 28-45 is the inclination of meridians required . as — sine of reclination ze 33-3 to = co-sine of latitude pz 38-28 so = sine of 90 pez 90-00 to — co-sine of incl. mer. zpe 61-15 whose complement qpz 28-45 is the inclin . of meridians required . or , a rule laid from p to y , gives m ; em is 28-45 , the inclination of meridians . again , 8 88 — 45   81 — 15 9 73 — 45   56 — 15 10 58 — 45   51 — 15 11 43 — 45   36 — 15 12 28 — 45   21 — 15 1 13 — 45   6 — 15 2 1 — 15   8 — 45 3 16 — 15   23 — 45 4 31 — 15   38 — 45 5 46 — 15   53 — 45 6 61 — 15   68 — 45 7 76 — 15 if i take 15 , the quantity in degrees of one hour , out of 28-15 , the inclination of meridians ; there remains 13-45 , for the first hour on the other-side of the substile . then again , by continual addition of 15 degrees to 13-45 , and the increase thereof , i make up the other half . or else , against 12 , set 28-45 , and add 15 successively to it , & its increase , till it come to 90 , then , to 13-45 , the residue of 15 , taken from 28-45 ; add 15 as often as you can to 90 , and thus is the table made . to draw the dial. first , draw a perpendiculer line on your plain , as cb , by crossing the horizontal-line at right angles ; then from the perpendiculer-line lay off from the upper-end , toward the left-hand ( as the scheam directs , zd being the perpendiculer , and zn the meridian , and ep on the plain , the distance between , being toward the left hand ) 20-54 , for the substile-line , as cd ; then on that line ( any where ) draw two perpendiculer lines quite through the plain , crossing the substile at right angles , for two equinoctial-lines , as ef , & gh . then consider what hours shall be put o● your plain , as here is convenient , from 10 in the morning , to 6 afternoon ; ( though the sun may shine on it from 8 to 7 , bu● then the lines will be too close together , and the radius too small ) . and also when you would have those two utmost hours 〈◊〉 be , as at e and f on the upper equinoctial-line ; or , at g & h on the lower contingent-line . then , then , take the whole distance ef , or gh , and make it a = tangent of 73-55 ; then the sector to set , take out the = tangent of 58-45 , and lay it from the point e to i , on the equinoctial-line ; also , take out the = tangent 61-15 , and lay it from the point f ; and if your work be true , it must needs meet in the point i ; then draw the line ik for the true substile , and from thence lay the = tangent of 45 , to draw a line near 5 , for the stiles elevation , parallel to ik the substile ; for being a polar-plain it hath no elevation , but what you please to augment it to ▪ as here from i to l. then , as the sector stands , prick on all the whole hours , halfs , and quarters , according to the numbers in your table , at least those that be above 45 ; and for those under 45 , make = tangent of 45 in small tangents , a = tangent of 45 in the great tangents , and then the sector shall be set to that radius , which is most convenient for your use . note , that this way of augmenting the stile , is general in all dials . 3. the third variety of south-recliners . the next and last kind of south recliners , are such as recline , or fall from you below the pole , viz have their plains lying between the pole and the horizon , as by the scheam is more apparent . in which work , the drawing the scheam , and the things required , are the same as in the first example , as the figure , and following words , do make make manifest . the example here , is of a plain that declines from the south toward the west 35 degrees , and reclines upon its proper azimuth ze , 60 degrees from the zenith . 1. having drawn the scheam , then first for the distance of the meridian from the perpendiculer , or horizon . by the sector , or quadrant . as the sine of zd 90-00 to the tangent of declination nd 35-00 so the sine of reclination ze 60-00 to the tang. of perp. & merid. eg 31-12 as — tangent of declination nd 35-00 to = sine of 90 zd 90-00 so = sine of reclination ze 60-00 to — tang. of perp. & merid. eg 31-12 whose complement is 58-48 ag , the distance between the west-end of the horizontal-line , and the meridian . or by the scheam ; a rule laid from q to g , cuts the limb at l ; the dl , and al , are the arks required ; dl from perpendiculer , and al from the horizon . 2. to find pg , the ark on the meridian from the pole to the plain . by the sector . as sine of ad 90-0 to co-tang . of the reclin . de 30-0 so co-sine of the declination an 55-0 to tang. of dist . plain & horiz . ng 25-19 as — co-tangent reclin . ed 30-0 to = sine of 90 ad 90-0 so = sine of reclination an 55-0 to — tang. dist . on mer. p. hor. ng 25-19 which being taken from np 51-32 , leaveth gp 26-13 , the distance on the meridian from the pole to the plain , or the complement of the new latitude . or , a rule laid from e , to p and g , gives on the limb 2 points , whose distance between , is ab 26-13 , the ark required . 3. to find the stiles elevation above the plain . by the sector . as sine dist . merid. horizon . ga 58-48 to co-sine declination an 55-00 so sine dist . pole to plain gp 26-13 to sine stiles elevation pf 25-02 as — sine of gp 26-13 to = sine of ga 58-48 so = sine of an 55-00 to — sine of pf 25-02 being found by the scheam , by laying a rule from y , to p and f , on the limb , gives the distance between being 25-02 , the stiles elevation , 4. to find the substile from 12. by the sector . as co-tang . of the declin . an 55-00 to s. dist . on mer. fr. pl. to hor. ng 25-19 so tang. of the stiles height pf 25-02 to s. of the substile from 12 fg 8-05 as — co-tang . of declin . plain an 55-00 to = s. dist . on mer. fr. pl. to hor. ng 25-19 so — tang. of the stiles height pf 25-02 to = s. of the substile from 12 fg 08-03 by the scheam , a rule laid from q , to g and f on the limb , gives l and m 8-3 ; or else , the ark md , is the distance of the substile from the perpendiculer 23-19 . 5. to find the inclination of meridians . by the sector . as the sine of the distance . on mer. from pole to plain pg 25-19 to the sine of the angle gfp 90-00 so the sine of dist . of sub. fr ▪ 12 gf 08-03 to the sine of the incl. of mer. gpf 18-27 as — sine gf 08-03 to = sine pg 25-19 so = sine gfp 90-00 to — sine gpf 18-27 by the scheam , a rule laid from p to y , on the limb , gives o , the ark eo is 18-27 ; the inclination of meridians , by help of which , to make the table of hour-arks at the pole , as before is shewed , and as in the table following . 12 18 — 27 8-3   10 — 57   1 3 — 27 1-27     subst .   4 — 03   2 11 — 33 4-58   18 — 03   3 26 — 33 11-55   34 — 03   4 41 — 33 20-35   48 — 03   5 56 — 33 32-45   64 — 03   6 71 — 33 51-45   78 — 03         7 86 — 33 81-52   85-57   8 78 — 27 64-10   70 — 57   9 63 — 27 40-20   55 — 57   10 48 — 27 25-36   40 — 57   11 33 — 27 15-33   25 — 57   12 18 — 27 8-3 to draw the dial. first , for the affections , consult the scheam , wherein , laying the perpendiculer-line cd right before you , you see that the substile , and the meridian , are to be laid from the perpendiculer toward the left-hand , the substile lying between the perpendiculer and the meridian , and the stile or cock of the dial must look upwards , the north-pole being elevated above this plain , which will guide all the rest . then , first , draw the horizontal-line ab , and on c as a center raise a perpendiculer , and set off by chords , sines , or tangents , the meridian or 12 a clock line , the substile , and stile , as exactly as you may ; and draw the lines 12 c , substile c , and stile c. then , as far from the center c , as you conveniently may , draw a long line perpendiculer to the substile , as the line ehf ; then setting one point of a pair of compasses in h , open the other till it touch the stile-line at the nearest distance . then , make this distance a = tangent of 45 , and take out the = tangents of every whole hour , as in the table , as far as the tangent of 76. will give leave ; and then from the center c , to those points draw lines for the even whole hours ; then to any one whole hour , as suppose the hour-line of 3 ; draw two = lines equally distant on both sides the line of 3 , as ik , lm . then , count any way 3 hours , and 6 hours from 3 , as here 12 , and 9 , so as the = line may cross the 3 remotest hours , as here you see 9 and 12 a clock hour-lines do cross the = line at i and k ; then take the distance ik , and lay on the hour-line of 3 from c to n , and draw inl = to 9 c ; which work doth constitute the parallellogram kilm . then lastly , make ki , and ni , = tangents of 45 , and p●ick off every hour , half , and quarter ( and minut if you please ) on the two lines ik , and il , from k and n both wayes , as before is already shewed in the erect decliners . note also , that to supply the defect on the other side , when the point m falls out of the plain , the distance from i , to the hour-point from 11 , will reach from l to 7 , and from i to 10 , from l to 8. this is general in all dials . also note , if you like not to lay off the ●irst hours by the tangents , having made the table , as before , you may soon find the hour-arks on the plain for 3 hours , as ●ere 3 , 12 , and 9 ; or , 4 , 1 , and 8 , which ●ould have made the parallellogram more ●●uare , and consequently more better , and ●●en to draw the rest by the sector . thus ●ou may see how your work accords ; the ●ay by the table and contingent-line , and 〈◊〉 way by the sector on the parallellogram , 〈◊〉 by calculation , & at last use the mystery 〈◊〉 dialling made plain and ready , to an ●●dinary capacity . of north declining recliners . the other kind , viz. north declining recliners , have also three varieties ; as those , ● . that fall back or recline between the zenith and equinoctial : 2d . those that recline to the equinoctial : and 3d. those that recline below the equinoctial . and first of the first variety , reclining less then to the equinoctial . the drawing the scheam , is the same as in the former , except in the placing of the points and letters ; for first , these plains behold the north-part of the horizon , and then when you look on the plain , the south is before you , and the west on your right-hand , and the east on the left ; then the south and north are alwayes opposite , and the point p , representing the elevated pole of the place , which with us being north , must be placed towards n downwards , as before in south recliners it was upwards . also , it is necessary in the scheam , to draw the equinoctial-line , by laying the half tangent of 51-32 from z to ae ; then the secant of 38-28 , the complement of ze , laid from ae on the line sn , shall be the center to draw eaew for the equinoctial-circle . thus the scheam being drawn , to find the requisites , thus ; 1. for the meridians elevation , or distance from the perpendiculer , ag , or ge. by the secctor . as sine 90 radius zd 90-0 to tangent declination plain sd 55-0 so sine reclination plain ze 20-0 to tangent merid. & perpend . ge 26-2 as — tangent of declin . sd 55-0 to = sine of radius zd 90-0 so = sine of reclination ze 20-0 to — tang. of 12 from perp. ge 26-02 whose complement ag , 63-58 , is the meridians elevation above the east-end of the horizon . by the scheam , a rule laid from q to g , on the limb gives l ; then dl and al are the arks required . 2. to find the distance on the meridian , from the pole to the plain gp . by the sector . as sine declin . of the plain gze 55-0 to sine dist . of mer. & perp. ge 26●02 so sine of the radius gez 90●00 to sine of dist . on merid. from pole to plain gz 32-03 as — sine of gez 90-0 to = sine of gze 55-0 so = sine of ge 26-2 to — sine of gz 32-03 which added to 38-28 zp , makes up gp to be 70-31 . or , by the scheam , a rule laid from e , to p and g , gives on the limb ab ; the ark ab is 70-31 . 3. to find the stiles height above the plain pf . by the sector . as sine of distance on mer. from zenith to the plain gz 32-03 to sine of the plains reclin . ze 20-00 so sine of dist . on mer. from pole to the plain gp 70-31 to sine of the stiles elevat . above the plain pf 37-01 as the — sine gp 70-31 to the = sine gz 32-03 so the = sine ze 20-00 to the — sine pf 37-01 by the scheam . a rule laid from y , to p and f , on the limb gives c and d , the stiles height . 4. to find the distance of the substile from the meridian gf when it is above 90 deg . take the comp . to 108 deg . ; by the sector . as tangent of the reclin . ze 20-00 to sine of dist . of 12 from perp. ge 26-02 so tang. of the stiles elevat . pf 37-01 to sine of the substile from 12 gf 65-24 as — sine eg 26-02 to = tangent ze 20-0 so = tangent pf 37-01 to — sine ge 65-24 by the scheam . a rule laid from q to g and f , gives on the limb lf , the ark required . 5. to find the inclination of meridians fpg . by the sector . as sine dist . on merid. from pole to plain gp 70-31 to sine radius opposite angle gfp 90-00 so sine dist . on plain from 12 to substile gf 65-24 to sine of the inclin . of mer. gpf 74-38 as — sine gf 65-24 to = sine gp 70-31 so = sine gfp 90-00 to — sine gpf 74-38 by the scheam . a rule laid from p to y , on the limb gives g , the ark eg is 74-38 , the inclination of meridians . or , a rule laid from p to k , gives h , sh is the inclination of meridians , by which to make the table as before is shewed , and as followeth . to draw the dial. 3 29 — 38 2 44 — 38 1 59 — 38 12 74 — 38 11 89 — 38 10 75 — 22 9 60 — 22 8 45 — 22 7 30 — 22 6 15 — 22 5 0 — 22 4 14 — 38 for drawing the dial , consult with the scheam , laying the plain aeb , and his perpendiculer cd right before you ; then note , sn is the meridian-line , ze the plains perpendiculer , with the meridian g on the left-hand , and the subtile f on the right-hand . also note , that the sun being in the south as s , casts ●is beams , and consequently the shadow of ●he stile into the north ; so that though g be the true meridian found , yet it is the north-part that is drawn as an hour-line ; ●ut the substile , and other hours , are coun●ed from the south-end thereof , as the table●nd ●nd the figure of the dial , do plainly make ●anifest ; being drawn in this manner . first , draw the horizontal-line ab , then 〈◊〉 c , as a center , draw a semi-circle equal 〈◊〉 60 of the chords , and lay off the meri●ian , substile , and stile , in their right sci●●ations , as last was declared ; then draw ●●ose lines , and to the substile erect a per●endiculer , as de ; then take the extent , or nearest distance from the place where the perpendiculer or contingent-line last drawn , cuts 12 and the stile-line , and make it a = tangent of 45 ; then is the sector set , to lay off all the hours by the = tangents of the arks in the table , except 11 and 10 , which do excur . for ▪ if you prick the nocturnal-hours 12 , 1 , 2 , 3 ; and draw them through the center , on the other side , they shall be the hours of 12 , 1 , 2 , 3 , 4 , &c. on the north-part of the plain , where they are only used . as for the hours of 10 and 11 , do thus ; draw a line = to any one hour , which = line may conveniently cut those hour-lines . as , suppose the line 6 12 , which is = to the hour-line of 3 ; then make the distance from 9 to 12 , or from 6 to 9 , in that line last drawn , a = tangent of 45 , and lay off hours and quarters , or else the whole hours , by the distances from 9 to 7 , and 8 for 10 and 11 , turning the compasses the other way from 9 ; then to all those points lines drawn , shall be the hour-lines required . or , having only the hours of 3 , 6 , & 9 , & 12 in a parallellogram , design the rest by sector . the second variety of north-recliners , reclining to the equinoctial . by the bare drawing of the scheam , you see , that the circle aeb , representing the reclining plain , doth cut the meridian just in the equinoctial ; now to try by arithmetick , whether it be a just equinoctial-plain , or no , say : 1. by the sector . as the sine of 90 ad 90-0 to tang. of the reclination de 54-10 so co-sine of declin . plain as 35-0 to co-tang . of the latitude sg 38-28 as — tangent reclination df 54-10 to = sine 90 ad 90-0 so = co-sine of declination as 35-0 to — co-tang . of the lat. sg 38-28 which happening so to be , it is a declining equinoctial , or polar in respect of its poles , which are in the poles of the world. 2. if the declination were given , and to it you would have a reclination , to make it equinoctial . by the sector ; as the co-sine of the declin . as 35-0 to the co-tang . of the lat. sg 38-28 so is the sine of 90 ad 90-00 to the co-tang . of the reclin . de 54-10 as the — co-tang . lat. sg 38 28 to the = co-sine declin . as 35-00 so the = sine radius ad 90-00 to the — co-tang . reclin . de 54-10 by the scheam . the points ab of declination , being given , and the point g on the meridian , if you draw the reclining circle agb , it will intersect the perpendiculer at e ; then the measure of ze is the reclination , measured by half-tangents , or by chords , by laying a rule from a , to e on the limb , gives a ; the chord b a , is the reclination 35-50 . 3. but on the contrary , if the reclination be given , and a declination required , to make an equinoctial plain ; then contrarily say thus , by the sector . as co-tang . of the reclin . ed 54-10 to sine of 90 ad 90-00 so co-tang . of the latitude sg 38-28 to co-sine of the declin . sa 35-00 as — co-tang . reclin . ed 54-10 to = sine ad 90-0 so — co-tang . latitude sg 38-28 to = co-sine declination sa 35-00 but by the scheam . by the point g , and the touch of an arch about e , draw the circle ge , to cut the limb into two equal parts , and you have the points ab . 4. the plain thus made , or proved to be equinoctial ; to find the meridians elevation above the horizon , ag or , his distance from the perpendiculer eg . ; by the sector . as sine of 90 zeg 90-0 to sine of dist . on the mer. from z , to the plain gz 51-30 so sine of declin . of the plain gze 55-0 to sine of dist . on the plain from perpend . to merid. ge 39-54 as — sine gze 55-0 to = sine zeg 90-0 so = sine gz 51-32 to — sine ge 39-54 whose complement is ag 50-06 , the elevation above the horizon . by the scheam . a rule laid from q to g , gives b on the limb , db is 39-54 , as before . 5. to find the stiles elevation above the substile on the plain . by the sector . as sine of the latitude gz 51-32 to sine of the reclination ze 35-50 so sine of dist . mer. pole to plain gp 90-00 to sine of the stiles elevation pf 48-24 as — sine 90 gp 90-0 to = sine latitude gz 51-32 so = sine reclination ze 35-50 to — sine stiles height pf 48-24 by the scheam . a rule laid from y to f on the limb , gives c , nc is 48-24 , the stiles height . the distance of the substile from 12 , in these equinoctial dials , is alwayes 90 degrees ; for a rule laid from q , the pole of the plain , to g , on the limb gives b ; a rule also laid from q to f , the substile , on the limb gives d ; the ark bd , is 90 degrees , both for the distance of the substile from 12 , and also for the inclination of meridians , for the substile stands on the hour of 6 , being part of the circle epw , which is the hour of 6 , 90 degrees distant from the hour of 12. or , a rule laid , as before , from y to p , on the limb , gives n ; the ark en , or wn , is 90 , for the inclination of meridians . which being just 90 , the table is easily made , viz. 15 , 30 ; 45 , 60 ; 75 , 90 ; twice repeated , from 12 to 6 both way s. to draw the dial. on the horizontal-line ab , draw an obscure semi-circle , and set off the meridian , as the scheam sheweth , viz. 50 degrees 6 min. above the east-end of the horizontal-line ; but make visible only the north-end thereof , as the line c 12 ; then 90 degrees from thence , toward the right-hand , as the scheam sheweth , when the perpendiculer-line is right before you , draw a line that serves both for 6 and the substile , as c 6. also , lay off the chord of 36-47 from 6 to 9 , and draw the line c 9 also , which is found by calculation , as before is shewed . or thus ; draw a line = to 12 , or perpendiculer to 6 , being in this dial all one , as the line feg ; then setting one point in e the substile , take the nearest distance to the stile-line , and it shall reach from e to g , the point for 9. the same distance eg lay also on the line 12 , from c to h , and draw the line ghi ; then make eg a = tangent of 45 , and lay off the = tangents of 15-30-45 , both wayes from e , as hath been often shewed . also , make the distance of hg a = tangent of 45 , and lay the same = tangents both wayes from h , and to those points draw the hour-lines required . the third variety of north-recliners . this third and last sort of north-recliners , are those that recline beyond the equinoctial , that is , lie between the equinoctial and the horizon ; and it differs somewhat from the other five before , in the scheam and operation also . for first , the ark of the plain is extended below the horizon , till it meet with the north-part of the meridian below the horizon at h ; and the center of the ark aqb , is in the line zd , as much distance from q , as the secant of 65 deg . to the radius of the scheam , being the complement of zq 25-0 ; here also the same requisites are to be found as in the other dials . 1. first , for the meridians elevation above the horizon , ag. by the sector ; as sine 90 zd 90-00 to tang. declin . plain sd 55-00 so sine reclin . plain ze 65-00 to co-tang . elevation merid. ge 52-18 as — tangent declin . sd 55-0 to = sine 90 zd 90-0 so = sine reclination ze 65-0 to — co-tang . merid. elev . ge 52-18 whose complement ga 37-42 , is the meridians elevation above the horizon . by the scheam ; a rule laid from q to g , gives on the limb a ; then d a is the distance from the perpendiculer 52-18 ; and a a the distance from the horizon 37-42 . 2. to find the distance on the meridian from the pole to the plain gp . by the sector . as sine of ad radius , or sine of ad 90-0 to co-tang . of reclin . plain de 25-0 so co-sine of declin . plain as 35-0 to co-tang . dist . on merid. from plain to zenith sg 14-58 as — tang. of ed 25-0 to = sine of ad 90-0 so = sine of as 35-0 to — tang. of gs 14-58 whose complement 75-02 gz , added to zp , the complement of the latitude , makes 113-30 , for the distance of the north-pole p , on the meridian of the place , from ( the north-pole p , to ) the plain below the equator at g ; which being more than 90 , find the complement thereof to 180 , viz. 66-30 , being the distance on the meridian from p the pole , to the plain on the north-part of the meridian , viz. ph , found on the scheam , by laying a rule from e or w , to p and h , on the limb gives b and c ; the ark bc is 66-30 , the distance on the meridian from the pole to the plain . 3. to find th● stiles height above the plain pe. by the sector . as the sine dist . mer. from zenith and plain gz 75-02 to the sine of the reclin . plain ze 65-00 so the sine dist . mer. from the pole to the plain ph 66-30 to the sine of the stiles elev . pf 59-21 as — gz 75-0 to = ze 65-0 so — ph 68-0 to = pf 59-21 by the scheam . a rule laid from y , to p and f on the limb , gives d and e ; the ark de is 59-21 , the stiles elevation . 4. to find the substile from 12 , viz. fg from the south part , or hf from the north part . by the sector . as tang. reclin . of the plain ze 65-00 to co-sine dist . mer. & horiz . eg 52-18 so tang. of the stiles elevat . pf 59 21 to sine of the substile from north part merid. fh 38-30 as sine dist . mer. from perp. eg 52-18 to tang. of the reclin . ze 65-00 so tang. of the stiles height pe 59-21 to sine of the substile from 12 fh 38-30 by the scheam . a rule laid from q , to h and f , on the ●imb , gives f and e ; the ark f and e , is the substiles distance on the plain from 12. 5. to find the angle between the two meridians , viz. pf , and ph. by the sector . as sine dist . mer. fr. pole to plain ph 65-00 to sine of 90 radius pfh 90-00 so sine of dist . from subst . & 12 fn 38-30 to sine of inclin . merid. fpn 42-45 as — sine of pfn 90-00 to = sine of pn 65-00 so = sine of fn 38-30 to — sine fph 42-45 by the scheam . a rule laid from p to y , on the limb gives g , then w g is the angle of the inclination of meridians , viz. 42-45 ; by which make the table , as is several times before shewed , and as followeth . to draw the dial. 12 42-45   1 57-45   2 72-45   3 87-45   4 77-15   5 62-15   6 47-15 42-55 7 32-15   8 17-15   9 02-15 1-56 10 17-15   11 2-15   12 42-45   then prick down the hours of 9 & 6 , by the table on the contingent-line , as before , or on the semi-circle , having calculated only those two hour-lines , by the general canon . then , draw a line = to 12 , at any convenient distance from it , as gh ; then , take the distance between 6 and 9 in that = line , and lay it from the center to i , on the 12 a clock hour-line , and draw the line hi ; then make the distances gh , and ih , severally one after another , = tangents of 45 ; and take out the = tangents of 45 , 30 , 15 ; and lay them both wayes from 12 and 6 , on those two lines , as hath been often shewed , in the former dials ; then lines drawn to those points , shall be the hours required . chap. vii . of declining and inclining-plains . inclining plains are but the under faces of recliners , beholding the nadir , at the same angle that the recliners behold the zenith . also , if you turn the paper , and look against the light , and then the north-east becomes a north-west decliner 55 , and reclining 35-50 ; and the south-east becomes a south-west , declining and inclining as much . thus you see , that every draught of a dial will serve for 4 plains , that is for the place you draw it , and his opposite ; and for another plain , declining as many degrees the contrary way , and reclining as much also , and for the opposite thereunto , as by the two draughts of the two sides , may plainly be seen to appear . and the like holds in all sorts , as upright decliners also . as a north-east and a north-west , a south-east and a south-west , declining 30 ; one dial drawn round about , serves all 4 dials ; but note , that no south erect or inclining dial , can have the sun to shine on any hour-line that falls above the horizontal-line ; and those hours on the north-recliners , that fall below the horizontal-line , belong also to the south dials . but for a plain general rule , to know what hours belong to any plain whatsoever in any latitude , do thus . to know what hours belong to any plain . first , draw a general scheam to your latitude , as this is done for 51-32 ; and mark the 4 cardinal-points with e.w.n. & s. as is usual for setting the scheam right before you . then , for all declining upright dials , draw only a streight line for the plain , perpendiculer to the line that doth represent the pole of the plain , counting so many degrees as the declination of the plain shall happen to be from s. or n. toward e. or w. then all the hour-lines of the scheam that that line of the plain shall intersect , are the hour-lines proper to that plain . 1. example . therefore , if you conceive the sun to be in cancer , and going of his diurnal motion , at his rising about a quarter before 4 , beholds the north-side of the line ew , and continueth so to do till 25 minuts after 7 ; and then it shines on the south-plain till 35 minuts after 4 , and then begins again to shine on the north-plain , and so continues till sun setting . but when the sun is in the equinoctial , it beholds the south-plain at the rising , being at 6 a clock in the morning ; and shines on it all day , till sun set , being at 6 at night ; and then the north dial is useless . 2. for a declining-plain . suppose 30 degrees south-east ; first set the scheam in his right scituation for a south-east plain ; then if you count 30 degrees from s toward e , for the pole of the plain ; and 30 degrees from w toward s , or from e toward n , and draw that line that shall represent the plain ; then you shall find that the sun being in cancer will begin to shine on this plain , just a quarter before 5 in the morning , and continue till near half an hour after 2. but about the middle of ianuary , it will shine on it till a quarter after 4 , viz. till sun set ; and all the hours after 2 , belong to the north-west plain that declines 30 degrees , and one hour in the morning also , viz. from a quarter before , till three quarters after 4. the like work serves for any decliner whatsoever , in any latitude . 3. but for decliners and recliners . draw a long line , as ab , and cross it with a perpendiculer in the center c , and lay off from c , toward a and b , the tangent of 45 ; or the semi-tangent of 90 , equal to the largeness of your scheam ; then lay off the semi-tangent of the reclination from c to d , up and down , both wayes ; then take out the secant of the complement of the reclination , which will be a radius to draw the arks adb , which paper you must cut out , and apply the two points of the paper adbd , to the two points of declination of the plain , noted in the scheam with a and b ; that is , put a to a , and b to b ; then the round or convex-edge of the paper , represents the reclining plain ; and the same edge , on the other part next the horizon southwards , represents the south-west incliner . example . suppose i make the paper adb , to recline 35-50 , the reclination of the equinoctial-plain ; then , first set the scheam right before you in its right scituation , and putting the points a , in the paper , on a on the scheam ; and b in the paper ▪ to b on the scheam ; i shall find it to be even with the reclining circle aeb ; then following the tropick of cancer , i find that it shines on the north recliner from the rising till near 2 , at which time it leaves the north-recliner declining eastward , and begins to shine upon the opposite plain , viz. the south-west incliner , declining 55-0 , and reclining 35-50 , and so continues till sun-set . but note , that if the line that represents the plain , cuts the tropick twice , as the line ew for a north-plain ; then , though the sun leave the plain in the morning , it will shine on it again in the afternoon . note also , that a north-east recliner , is represented by the other convex-edge of the paper , as here a north-east decliner 55 , and inclining 35-50 , the sun will shine but till 3 quarters after 8 in cancer ; but in capricorn it shines till half an hour after 9 , and comes no more on it that day : and note alwayes , that when it leaves any plain , that then it begins to shine on his opposite , as here the opposite to this north-east incliner , is the south-west recliner , being represented by the same line or circle adb , that the north recliner was : only , you must count that side of the line next to the horizon , the inclining-plain ; and that side next the zenith , the reclining-plain ; for , the line that represents it , having no bredth , can be no otherwise distinguished , unless you will make a material , armilary sphear , of pastboard or brass , as the following discourse doth plainly demonstrate , in these several operations , for the better conceiving of these mathematical excercitations . thus you have the way of making all manner of sun dials , upon any plain superficies , the axis of the world being the supposed stile to all these plains ; as for those curiosities of upright stiles , and eliptical dials , and drawing of dials by the horizontal , or equinoctial dials , you have them in the works of mr. samuel foster , and others ; and in kerkers ars magna , &c. but i intended not a volumn of shadows , but only a further improvment of the trianguler-quadrant , as you will see in the next chapter , of drawing the furniture or ornament of dials ; which being but seldom used , i shall here crave an apology for the brevity therein , fearing , lest that to the young practitioner it may seem somewhat hard to conceive , though to the exercised in these matters it may be plain enough . then for a conclusion , you shall have an easie mechanick way , to draw a dial on the ceiling of a room , that lieth flat or horizontal , which will be very good for painters or plaisterers , to ornament a room withal , and is not yet treated on that way , as ever i read of . chap. viii . to furnish any dial , with the usual mathematical ornaments by the trianguler-quadrant , as parallels of the suns declination , or the suns place , or length of the day , to find the horizontal and virtical lines , and points , to draw the azimuths , and almicanters ; the iewish , italian & babylonish hours , and 12 houses on any plain before mentioned . 1. to draw the tropicks , or parallels of the suns declination , or the length of the day artificial , on any dial. but note , that if it be a perpendiculer stile , whose upper point , or apex , is to be the nodus to give the shadow ; then you must strain a thred very hard , or apply a rule for the present whereon to rest the moving-leg on , instead of the axis ; or , else you may do it thus , as mr. gunter sheweth . first , to make the trygon , if the rule or quadrant prove too large for your small dial. on a sheet of pastboard , or slate , draw a long streight line , as ab ; to which line erect two perpendiculers , one at the upper , and the other at the lower end , as cd , and ef ; then make ab a tangent of 45 degrees , ( then having first made these little tables that follow , by the trianguler-quadrant , which is only the suns declination , at his entrance into the whole signs , or at an even half-hour of rising ) ; lay of both wayes from b , the tangents of the suns declination at ♈ ♉ ♊ ♋ , as in the table following ; and draw lines to these points from the center a , as in the figure annexed ; and then set the marks to them , and this is the trigon . figure i. suns declinations for the parallels of the length of the day . hours declin 16-26 23-31 16-0 21-41 15-0 16-55 14 11-37 13 5-53 12 0-00 11 5-53 10 11-37 9 16-55 8 21-41 7-34 23-31 for the signs of the zodiack . signs . declin ♋ 23-31 ♌ ♊ 20-14 ♉ ♍ 11-31 ♈ ♎ 0-00 ♓ ♏ 11-31 ♒ ♐ 20-14 ♑ 23-31 declinations . 5-0 10-0 15-0 20-0 23-31 both ways then from the center a , any way on the line cd , at such a convenient distance as you think may fit the plain , set off the point g ; then making ga radius of 45 tang. set off on ab from a , the tang. of the stiles elevation to f , and draw the line fg , as an obscure line . then come to the dial plain , and measure from the center to the place on the substile-line , where you would have your remotest line of the sign ♋ or ♑ to pass ; and take this distance between your compasses , and carry it in , above , or below the line fg , first drawn and produced to ♋ , or ♑ , till you find one point to stay in a ♋ , and the other in ag , so as to draw a line = to fg first drawn ; if that doth not fit , then dele fg , and draw this = to it in its stead , to fit and fill the plain with the tropicks to your mind , to make them large and yet convenient . then note , the point g represents the center of the dial ; ag is the length of the stile from the center to the nodus ; a perpendiculer let fall from a to fg , shews the point h ; gh is the measure on the substile-line on the plain from the center to the horizontal-line , ha is the perpendiculer height of the stile ; a the apex or top of the stile or nodus to give the shadow . then , draw a line from g , = to ab , as lk ; and any where between ab , draw lm = to ag ; and wheresoever fg cuts lm , make a mark as at m ; then make lm a = sine of 90 degrees , and the sector so set , take out the sine complements of the arks at the pole for every hour , and lay them from l towards m , on the line lm , and to all those points , draw lines from g , and mark them with 12 , 1 , 2 , 3 , 4 , &c. as in the table . or else , take the measure from g to f , and lay it on the dial from the center on the substile , and draw that line precisely perpendiculer to the substile , for the true equinoctial-line on the plain . then , the measure from the center of the dial , to the crossing of every hour-line , and the equinoctial-line , taken and laid from g , to the line ab , gives points to draw the hour-lines on the trygon ; as in the figure . wherein you may note , that if the substile happens to fall on an even , whole , or half hour , then one line will serve on both sides of the substile ; but if not , you must draw as many more , and set figures to them , to avoid confusion . then , i say , that the several distances from g , to the crossings of those hour-lines last drawn on the trygon ; and the signs being laid on their correspondent hour-lines from the center of the dial , shall give points in those hour-lines , to draw the signs of the zodiack , with a thin rule that will bend to those hyperbolick sections . the same way serves to draw the parallels of the length of the day , if you lay the distunce from g , the crossings of the pricked lines and hours on the trygon , and is as true as any other way by calculation , which must afterward be performed by protraction in this manner . thus you have the way to proportion the height of the stile , to fit the plain , and the place of the horizontal-line in all erect-dials , which is alwayes perpendiculer to 12 , and drawn through that point a-cross the plain ▪ and this way of drawing the signs , is general in all plains whatsoever , that will admit them . ii. to find the horizontal-line in all manner of plains . first , the horizontal-plain can have none , nor many other both reclining and inclining , whose reclination or inclination is above the complement of the suns meridian altitude in ♑ , if the stile have any considerable altitude . in all other plains , the best mechanick way is thus ; the dial being set in ( or , as in ) his place , apply the moveing-leg to the top of the stile & one corner of it to the plain ; and at the same time let the thred play evenly on 600 , and the corner at the plain will make as many marks as you please to draw it by . otherwise note , that wheresoever the hour-line of 6 , and the equinoctial-line do meet , there is one point : then find at what hour and minute the sun doth rise or set at , in the beginning of any other whole sign , most remote from the first point , and that shall be another ; and so as many as you please to draw that line by : this is general for all plains : to find that point by the trianguler-quadrant . lay the thred to the sign given , and in the hour-line is the hour and minut required ; thus the sun being in ♈ , riseth and setteth at 6 , or 1 quarter of a minut before , or after ; and in ♉ at just 5 , and sets at 7 ; in ♊ at 9 minuts after 4 , or sets 9 minuts before 8 : the like for winter signs . iii. to draw the old unequal hours . the unequal , jewish , or planetary hours , divide the day , be it long or short , into 12 equal hours ; for the drawing of which , in the equinoctial the common hours gives points . for the tropicks do thus ; divide the number of minuts in the longest and shortest dayes by 12 ; viz. divide 986 , the minuts in one day in ♋ at london , by 12 , the quotient is 82 ½ ; and divide 454 , the number of minuts in one day at london in ♑ , and the quotient shall be 37′ ¾ ; then if you fasten an index , or lay a rule to the center , and to every 1 hour and 22′ ½ in ♋ from 12 ; and to every 37′ ¾ in ♑ , it shall give points to draw the jewish or planetary hours required , according to this table , thus made for london , by the line of numbers ; against 12 set 6 , and the rest in order as the day proceeds , for our 12 is the 6th hour , according to the jewes . to make this table readily by the line of numbers . a table to divide the planetary hours in ♋ and in ♑ , for london , 51-32 latitude . h ♑ m hou h ♋ m 8 43 1 5 10 9 28 2 6 31 10 16 3 7 52 10 44 4 9 15 11 22 5 10 37 12 00 6 12 00 12 37 7 1 22 1 15 8 2 44 1 53 9 4 06 2 31 10 5 29 3 8 11 6 51 3 47 12 8 13 extend the compasses from 16-26 , the length of the longest day in hours and minuts to 1 , the same extent shall reach the contrary way from 60 , to 986 , the number of minuts in one day . or rather , as 1 hour , to 60 minuts ; so is 16 hours 26′ , to 986 minuts . then , as 1 , to 82 minuts ¼ ; so is 2 , to 164 minuts ½ ; so is 3 , to 246 minuts ¾ . or you may say , as 12 , to 1 ; so is 986 to 82-2 , the minuts in 1 hour . which properly is one hour 22 minuts , the length of one hour in cancer ; then the second hour , is 2 hours 44 ½ ; the third hour is 4 hours and 6′ ¾ from 12 ; and so for the rest , as in the foregoing table for london . but if you draw the parallels , of the length of the day in the dial , you shall find these hours to cross the even hour-lines and quarters in the parallels for 15 and 9 hours , as well as in the equinoctial . iv. to draw the italian or babylonish-hours . first , draw the common hours , and the parallels of the signs , or rather the length of the day ; then note , that these hour-lines meet with the common hours in the equinoctial ; only the italians who account from sun-setting , call our 12 in the equinoctial 18 ; and the babylonians , who reckon from the sun-rising , call our 12 in the equinoctial 6 hours . then to mark these in the tropicks , do thus ; but , if you draw the parallels of the length of the day , then you shall find the 18th hour after sun setting , to cut the hour-line of 10 in the parallel of the day , being 8 hours long , and 12 in the parallel of 12 hours long ; and the common hour-line of 2 in the parallel of 16 hours long , and so successively for the rest , for so many hours from the last sun-setting : for , from 6 the last night in the equinoctial , to 12 this noon , is 18 hours ; but in ♑ , from 47′ after 3 at sun-set , to the next noon , is 20 hours and 13′ , as in the figure foregoing . but for the babylonish-hours , who reckon by equal hours from the sun rising , as before , count 2 hours and 13 minuts after 6 in ♑ ; and 2 hours and 13′ before 6 in ♋ ; and just 6 in ♈ , and that shall draw the line of the suns rising ; then count 3 hours and 13′ after 6 in ♑ , and 7 in ♈ , and 1 hour 13′ before 6 in ♋ ; and that shall be the first hour after sun rising , and so successively till night . but if you use the parallel of the length of the day , the work is easier ; for then 5 , 7 , and 09 , in the parallels of 16 , 12 , and 8 hours , shall be points for the first from sun-rising ; and 6 , 8 , and 10 , shall shew the second hour from sun-rising , and so forwards , as in the table following . v. to draw the azimuth-lines . for all erect-dials , both direct or decliners , deal with the declination of the plain , as you did with the inclination of meridians ; and at the meridian , or 12 , set the plains declination ; and then for rumbs , take 11 deg . 15′ as often as you can ; and what the last number wants of 11-15 , set on the other side of the substile , and to that add 11-15 till you have enough , as in the table annexed for a dial , whose declination was 35 degrees westwards . then make the perpendiculer height of the stile radius , or tangent of 45 , and on the horizontal-line lay off the = tangents of the rumbs last made ▪ in the table , from the foot of the stile their right way , and draw lines through those points , all parallel to 12 , for the rumbs , or virtical circles required ; on the meridian write south , and the rest in their due order . to draw the azimuth or virtical-circles on reclining , or inclining-plains . points . d. m s. e. 80 00 s. e. by s. 68 45 s. s. e. 57 30 s. by e. 46 15 south . 35 00 s. by w. 23 45 s. s. w. 12 30 s. w. by s. 1 15 substile   s. w. 10 00 s.w. b. w. 21 15 w. s. w. 32 30 w. by s. 43 45 west . 55 00 w. by n. 66 15 w. n. w. 77 30 n. w. b. w. 88 45 n. w.   in all reclining , or inclining plains , these azimuths , virtical circles , or rumbs , do meet in a point ( called the vertical point ) found in the meridian , or 12 a clock line , right over ( in incliners ) or under ( in recliners ) the apex or top of the stile , that is to give the shadow , when set in its right place , right over the substile-line ; and as far off the foot of the stile ( being a point in the substile , square , or perpendiculer to the apex or top of the stile ) in a vertical line drawn through the foot of the stile , = to the perpendiculer line of the plain ) , as the co-tangent of the reclination , making the perpendiculer height of the stile to be radius or tangent of 45 degrees . also , the co-tangent of the reclination of the plain , to the same radius , laid from the foot of the stile , in the same virtical-line , shall give the point in the vertical-line , to draw the horizontal-line by ; for a rule laid to this point , and the crossing the equinoctial-line and hour of 6 , shall draw the true horizontal-line . then make the distance between this point , and the meeting of the equinoctial and 6 , a = tangent of the west or east azimuth in the table , and then the sector is set , to lay off all the rest , by taking the = tangents of the numbers in the table , and laying them from the vertical-point in the horizontal-line , both wayes on the horizontal-line . for , from hence you may note , that the sun , being in the equinoctial , doth rise and set near 6 ; and also doth rise near the east-point , and set near the west ; therefore the same point in the dial , must be for the hour 6 in the morning ; and the east aximuth , or the hour 6 at night , and the west azimuth , according as the plain declines eastwards or westwards . then right lines drawn from the vertical-point in the meridian , and to all these points in the horizontal-line , shall be the azimuth-lines required . a thus for example in the figure annexed , being the third sort of south-recliners before-going ; declining 35 degrees southwest , and reclining 60 degrees , ch is the substile , cg the stile , h the foot of the stile , ik the vertical-line drawn through the foot of the stile hi the vertical-point in the crossing of 12 ( and the vertical-line ) and yet right under g the apex ( considering the reclination ) and the raising of g the apex , square , or perpendiculer to h the foot of the stile ; then , i say , a plumb-line let fall from g , will rest in i , the vertical-point ; the dial being set in its due place . then gh , the perpendiculer height , made a = tangent of 45 ; hi is the co-tangent of the reclination , viz. 30 ; and hk the tangent o● the reclination 60 , being the vertical-point in the horizontal-line , from whence to lay the = tangents , of the rumbs in the table last made , into the horizontal-line . then lines drawn from the vertical-point i , to those points in the horizontal-line , shall be the rumbs or points of the compass , vertical circles , or azimuths required . otherwise , when you have made the tables of the angles at the zenith , as before , you may by this canon make tables of angles at the vertical-point , between the vertical-line and the rumb , to be drawn on the plain . as the sine of 90 , to the co-sine of reclination , or inclination ; so the tangent of the angle at zenith , to the tangent at the vertical . this table being made , you may set one point in the vertical-point , and describe a circle to any radius , and therein prick off from the vertical-line , the several chords of the rumbs , as in the table you shall make by the last canon . a table , shewing at what hour and minute the sun is in , in an even azimuth , or point of the compass in ♑ , ♈ , ♋ , for 51-32 . degr. rumbs . alt ♋ h. m. alt. ♈ h. m. alt ♑ h. m. 00-00 south● 62-00 12-00 38-28 12-00 15-00 12-0 11 15 s. by e. 61 39 11 38 37 58 11 24 14 17 11 16 22 30 s. s. e. 60 33 11 15 36 19 10 48 12 05 10 27 33 45 s.e.b. s. 58 31 10 46 33 29 10 10 8 17 9 33 45 00 s. e. 55 40 10 17 29 22 9 28 3 00 8 39 56 15 s.e.b. e 51 35 9 42 23 51 8 42     67 30 e. s. e. 46 06 9 02 16 56 7 51     78 45 e. by s. 39 03 8 15 8 49 6 57     90 00 east . 30 38 7 21 0 0 6 00     78 45 e. by n. 21 21 6 24         67 30 e. n. e. 11 10 5 19         56 15 n. e.b e 3 48 4 18         45 00 n.e.             33 45 n.e.b. n             22 30 n. n. e             11 15 n. by e             00 00 north.             lastly , by help of this table , being general for all dials in the latitude 51-32 , it is done thus ; then you shall see that a rule laid to the vertical-point , and any one of those three points shall cut the other two , if the former lines be true , and you estimate the minuts well . note , that this last table in the equinoctial , is thus readily made , by sines and tangents . as the sine of 90 , to the sine of the latitude ; so is the tangents of the azimuths from the meridian , being the first column in the table , to the tangent of the angle between the meridian , and azimuth line on the equator , which are the numbers in the 6th column , reduced into hours and minuts . so that you see the azimuth of 45 , or rumb of s.e. will cross the equinoctial at ●8 minuts past 9 , as in the table ; which table is easily made by the trianguler-qua●rant , by the rules in chap. xv. vi. to describe the almicanters , or the parallels of the suns altitude above the horizon . first , on the equinoctial , these lines shewing the suns altitude , cannot be expressed . on the horizontal dial they are circles , making the perpendiculer height of the stile radius , or tangent of 45 ; prick off on the hour-line 12 , from the foot of the stile , the = tangent of 10 , 20 , 30 , 40 , 50 , 60 , &c. then one point of a pair of compasses set in the foot of the stile , and the other opened to 10 , 20 , 30 , &c. draw those circles for the parallels of the altitude required . for all erect dials , whether direct or declining , they are best done thus ; if the stile be in , and right set , then the distance from the nodus , to the crossing of the horizontal-line , and azimuth line , on which you would prick down the altitudes shall be the = tangent of 45 ; then the sector so set , the = tangents of 10 , 20 , 30 , &c. laid from the horizontal-line on the respective azimuths , shall be points to draw the parallels of altitude by ( or by applying the rule to the nodus and plain , and the thred to the almicanter ) as afterward is plainly shewed . but if the stile is not in , then the secants to the same numbers and radius , that pricked down the azimuth lines , shall be the several radiusses to use as before ; where you may note , that the suns meridian altitude in the whole even signs , will help to prove the truth of your work . the east and west erect dials , are fitted with parallels of altitude in the same manner ; for the perpendiculer height of the stile , is a tangent of 45 , and the = tangents of 11-15 , 22-30 , 33-45 , &c. laid from the foot of the stile in the horizontal-line , draws down-right lines for the azimuths ; and the secant of 11-15 , 22-30 , 33-45 , &c. shall be the several radiusses to prick off the = tangents of 10 , 20 , 30 , 40 , 50 , 60 , ( or what you will ) on those perpendiculer azimuth lines , for the almicanters , or parallels of altitude required . but for declining reclining plains , you must first draw the azimuth lines , as before is shewed , and then find also the length of the axis of the horizon , as mr. gunter calls it , which is thus done ; make the length of the perpendiculer-stile a = tangent of 45 , viz. gh fig. ii. then hi is the co-tangent of the reclination , and hk the tangent of the reclination ; and then , as the sector stands , the secant of the complement of the reclination , shall be the length of the axis of the horizon required , viz. gi , or by the sines and tangents artificial . as the sine of the reclination , to the sine of 90 ; so is the length of the stile on the line of numbers , being taken in inches and 100 parts , to the length of the axis in the same parts . which is an imaginary diagonal line , reaching from the apex to the vertical-point . this being found , you must find the angles between this axis and the horizontal-line , on every particular azimuth ; and lastly , the distance between the vertical-point , and the parallels of altitude , on every particuler azimuth last drawn . for the doing whereof , you must work as you did before , to lay off the signs , or the parallels of the length of the day , for these almicanters , bear the same respect or proportion to the horizon , as the parallels of the length of the day have to the equator , and are described in the same manner , as followeth . see figure iii. first , draw the line ab , and make ab a chord of 60 , and sweep the arch of a circle , and lay off 10 , 20 , 30 , &c. and draw the lines from a the center , and mark them with 10 , 20 , 30 , 40 , 50 , 60 , the even 10th degrees ; or , 45 for equal , 26-34 for double , 11-19 for 5 times , the length of the shadow and object , or what you please . then , draw ac perpendiculer to ab ▪ and lay off the length of the axis of the horizon from a to c ; then make ac the co-sine of the reclination , and as the sector stands , take out the sine of the reclination , and lay it from a to d , and this will be the distance from the apex to the horizon ; also , the sine of 90 shall reach from c to d , the distance between the vertical-point and the horizon ; also the nearest distance from a to cd , is the perpendiculer height of the stile ah . then , take the distance from i , the vertical-point point on the plain , to the horizon on every particular azimuth line , and lay them in the trygon , or iii figure , from c to the horizontal-line ad , produced if need be ; and draw those obscure lines , as in the figure , and mark them with the names of the rumbs , to avoid confusion ; then is your trygon made ready for use . then , take the distance from c in the trigon , to every crossing of the azimuth-line and almicanter , and lay it on the plain from the vertical point i , on its proper azimuth , finishing one almicanter before you meddle with another , and the work with patience and diligence will be performed ; the line● are to be drawn from point to point , with a steady hand , or a bending thin ruler , being conical sections . note , that when the vertical-line of the plain falls on an even azimuth , then half the number of rumbs will serve , being laid each way on both sides at once . or , having a table of the angles at the zenith , the same as you made to draw the azimuth-lines , draw a line at any convenient distance , parallel to ac ; the further from ac , the larger and better , as def in the figure ; and note , where cd crosses the last line ef , as at d ; make de a parallel sine of 90 , and lay off the sine complements of the angles at the zenith in the table , from e towards d , and draw and mark the lines , as in the figure . otherwise , the stile being fixed , and the dial set in its place where it must be , or at least set to the same reclination , and declination that it must be ; then if you apply the side of the trianguler quadrant to the nodus , and the corner at the end of the same edge that toucheth the middle of the nodus to the plain ; and at the same time , the thred and plummet playing neatly on the almicanter you would draw , you may find as many points , and mark them as you please , without all the former trouble , and it may be every whit as true ; if the under-side be in●onvenient , you may use the upper ; only be sure , that the side you apply , and the thred and plummet play at the angle of the almicanter required . vii . to draw the circles of position , or houses . the circles of position , or 12 houses , meet and cross one another in the crossing of the meridian and horizon ; therefore the horizon is the begining of the 1st and 7th houses , beginning at the east , and reckoning under the earth , by imum coeli , to the descendant , or 7th house , at the west-part of the horizon ; and so to medium coeli , the beginning of the 10th house , to the ascendant , or horoscope , the beginning of the 1st house . to draw these on the horizontal-dial , where they are parallel lines to the hour 12 , do thus ; take the distance from the apex to the equinoctial-line , and make it a = tangent of 45 ; then the = tangent of 30 degrees laid both wayes on the equinoctial , shall give points to draw lines by , = to 12 , for the houses required . for east and west dials , take the radius as before , viz. from the apex to the equinoctial-line on the plain , which here is the meridian ( and but the length of the stile ) a tangent of 45 ; then the = tangents of 30 , 60 , and laid from 6 on the equinoctial-line , gives points to draw lines parallel to the horizon , for the houses required . for east and west recliners , the perpendiculer height of the stile made a secant of 0 ; then the secant of the stiles elevation , shall be radius to prick off the = tangents of 30 , 60 , on the equinoctial-line from the foot of the stile , whereby to draw lines parallel to the horizon for the circles of position required . all these lines may most elegantly and ●asily be drawn and expressed , on a large ceiling , with competent exactness in this manner following . first provide a quadrant of brass , or thin wood , of about a foot radius , or 14 , 15 , or 16 inches ; also , a semi-circle of brass , of about half an inch broad , and about an inch less radius than the quadrant : the semi-circle must have at each end , somewhat more than to make up 180 degrees , to nail to the transum , or stroke of the window , where your glass is to lie . also , to one ray of the quadrant must be fastened two strong wyres , to fasten the quadrant to play after the manner of a casement , one point in the ray of the quadrant next the center , sticking in the hole where you intend the glass shall lie ; and the other end fastened to a piece of wood nailed on the two upright posts of the window , so that howsoever you turn the quadrant , fixed on those two points , it may be precisely perpendiculer , the semi-circle playing all the while through a hole in the other ray of the quadrant , that lies horizontally ; having a skrew to stay the quadrant at any azimuth , as in figure iv , is plainly expressed to your view . then having degrees on the semi-circle , and also on the quadrant , and having fitted the quadrant on his points to play precisely perpendiculer , which the plummet in the quadrant will shew , by turning it round about , and put in the semi-circle through the hole in the horizontal ray of the quadrant , and nailed it so to the stoole or transum of the window , by putting two little bits of wood under the ends , that the quadrant may play evenly and smoothly on the semi-circle-to almost the half-round , for quite the half-round will not be necessary , or useful . then is the instrument set fit for its operation . then first , to find the declination , or rather the true meridian-line . turn the quadrant till the edge be just against the sun , and at the same instant get the suns azimuth ; then if you count so much as the suns azimuth is , on the brass semi-circle , from the place the quadrant stands at , the right way , a line drawn from the center of the semi-circle , or quadrant , to that place , is the true meridian line ; which place you must carefully find by two or three tryals , and then mark it with ink or otherwise , on the brass semi-circle to count from thence , in setting the quadrant to the suns azimuth , at every hour and quarter in those points you intend to draw on the ceiling ; which a crooked rule set to 00 , on the semi-circle ▪ to pass to and fro with the quadrant , will make easie . then , having a table of the suns altitude , and azimuth , at every hour in that latitude you draw the dial for ; first , set the quadrant to the azimuth at the hour , counted the right way from the marked meridian-line on the semi-circle , and there skrew it fast ; then extend the thred fastened in the center of the quadrant , till it cut the altitude of the sun at the same hour and azimuth , on the degrees of the quadrant , and extending the thred to the ceiling , make a mark for that hour and altitude ; that point at that time , gives the true place where the reflected spot will fall , at that hour , azimuth , and altitude on the ceiling of the room . this work repeated as many times as there be hours and quarters in the summer , and winter tropicks , for about 5 hours , ( and in the equinoctial , and any where between , if you please ) shall give points enough to draw the dial , and also the tropicks , and azimuths , and altitudes also , if it were convenient to mark it ; or , to any other altitude you mind to have at that azimuth , all at once , or at most with two slips of the thred ; the italian , babylonish , or iewish-hours , as easily drawn by points found in the other lines . also , on the meridian-line , you may add the day of the month , or any thing that depends on the suns meridian altitude ; which work being well done , and drawn with smoth lines , and well ornamented , would be a comely & pleasant ornament to a ceiling , and far cheaper then some fret ceilings are done , and more useful . lastly , when all is done , to put the glass in right , the foile being first rubbed off , to to cause it to give but one spot , let the superficies of the glass lie just so high as the center of the quadrant was , in the drawing the lines , and put some putty under it , and the sun shining , make it to play right on the true hour , altitude , and azimuth ; or , if it be just at noon , then bend it on the putty with your finger , till it fall just on the meridian , and day of the month also in the meridian-line . also note , that look what altitude the sun hath at any time , the same will the reflected altitude be , at the same time , if the glass lie true , which two observators at the same time may carfully prove . the making of the tables of the suns altitude and azimuth , is very largely shewed in the 15th chapter , vse the 37th and 38th , where you have wayes both general and particular , for any one or more latitudes . the figure explained . a , the place on the transum for the glass to lie on , and in the middest thereof one point , in the ray of the quadrant , is to play : ih , a piece of wood to be nailed fast at h and i , for the other point to play in at g : l , the hole for the plummet to play in , being cut through the thin quadrant : b and c , the ends of the semi-circle , nailed on the transum or stoole : k , the hole in the quadrant for the semi-circular-ring to pass through : fe , the posts of the window : d , the beginning of the degrees on the semi-circle : am , a thred extended from the center of the quadrant to the ceiling . thus you have the usual wayes of dialing in a competent measure , plainly , and practically handled , which may be useful to many a learner ; and i hope will be as well accepted , as with free-will ( though with little ability , and less leisure ) readily imparted . a table of the suns azimuth from the south , at every hour and quarter , 51-32 .   ♋ ♊ ♌ ♉ ♍ ♈ ♎ ♓ ♏ ♒ ♐ ♑   d. m. d. m. d. m d. m d. m d. m d. m 12 00 00 00 00 00 00 00 00 00 00 00 00 00 00   07 10 05 53 5 37 4 55 03 54 03 53 3 52   14 22 13 24 11 10 9 40 07 57 07 40 7 10   21 27 19 53 16 44 14 13 12 11 11 10 10 37 11 1 27 54 26 00 22 13 18 52 16 22 14 45 14 13   34 14 32 01 27 32 23 22 20 15 18 17 17 38   40 12 37 40 32 41 27 41 24 06 21 53 20 59   45 39 43 04 37 32 32 08 29 44 26 30 24 25 10 2 50 51 47 57 42 05 36 24 31 50 28 52 27 49   55 31 52 41 47 04 40 30 35 35 32 18 31 00   61 43 57 11 50 51 44 26 39 10 35 39 34 17   64 11 61 18 54 33 48 13 42 36 38 50 37 34 9 3 68 10 65 16 58 47 51 56 46 02 42 04 40 36   71 52 69 56 62 33 55 33 49 25 45 16 43 40   75 26 72 37 66 10 58 59 52 43 48 21 46 42   78 48 76 05 69 38 62 22 55 55 51 22 49 40 8 4 82 00 79 20 72 57 65 40 59 00 54 22     85 08 82 42 76 12 68 53 62 08       88 10 85 38 79 23 72 06 65 12       91 09 87 50 82 28 75 03 68 12     7 5 94 05 91 34 85 30 78 06 71 10       96 51 94 25 88 27 81 09         99 38 97 16 91 25 84 06         102 25 100 07 94 22 87 04       6 6 105 08 102 54 97 12 90 00         107 52 104 58 100 02           110 36 108 26 103 00           113 18 111 14 105 54         5 7 116 03 114 03             118 50 116 54             121 41 119 47             124 29 122 37           4 8 127 24               130 28             a table of the suns altitude , at every hour and quarter in each sign , for 51-32 .   ♋ ♊ ♌ ♉ ♍ ♈ ♎ ♓ ♏ ♒ ♐ ♑   d. m d. m d. m d. m d. m d. m d. m 12 62 90 58 42 50 00 38 28 27 00 18 18 14 59   61 50 58 34 49 52 38 24 26 54 18 13 14 55   61 25 58 09 49 31 38 07 26 40 18 00 14 38   60 41 57 29 49 00 37 38 26 15 17 37 14 20 11 59 43 56 34 48 12 36 58 25 40 17 06 13 52   58 32 55 26 47 10 36 07 24 02 16 26 13 10   57 04 54 06 46 02 35 06 23 59 15 38 12 35   55 29 52 36 44 38 33 56 22 58 14 49 11 30 10 53 45 50 55 43 12 32 37 21 51 13 38 10 36   51 52 49 07 41 33 31 10 20 33 12 27 9 20   49 54 47 12 39 46 29 36 19 10 11 09 8 9   47 50 45 12 37 56 27 54 17 35 9 44 6 41 9 45 42 43 06 36 00 26 07 15 58 8 13 5 16   43 30 40 57 33 55 24 14 14 14 9 49 3 41   41 16 38 45 31 50 22 16 12 27 4 54 2 04   39 00 36 30 29 40 20 14 10 32 3 07 0 17 8 36 41 34 13 27 31 18 08 8 35 1 15     34 22 31 55 25 09 15 59 6 30       32 02 29 13 22 56 13 46 4 27       29 42 27 16 20 37 11 33 2 17     7 27 23 24 56 18 18 09 17 0 06       25 03 22 36 15 58 06 58         22 45 20 16 13 38 04 40         20 28 17 58 11 18 02 20       6 18 11 15 41 09 00 00 00         15 56 13 25 06 41           13 44 11 11 04 23           11 35 8 59 02 08         5 9 32 6 50 00 06           7 23 4 44             5 26 2 41             3 36 0 41           4 1 32             the description and some uses of the sphear for dialling , and for the better understanding of the general and particular scheams . next the foot and semi-circle frame for supporting of it , you may consider ; 1. the fixed horizon , to which the foot is fastened with 4 skrews , numbred and divided into 360 degrees , or four 90 deg . whose count begins at the dividees side of the meridian-circle . 2. the meridian circle , whose fore-side at the nadir-point stands in the center of the foot ; this is also divided into 4 90s s , and begins to be numbered at the south and north part of the horizon , upwards toward the zenith , and downwards toward the nadir ; which circle is alwayes fixed as the horizon is . 3. the equinoctial circle , made fast at the east and west points of the horizon , moving up and down upon the meridian-circle , according to the elevation of the equinoctial in any latitude ; this is divided ●●kewise into four 90s s , & numbred from the meridian each wayes to the east and west points of the horizon . 4. on the meridian circle , is set 2 moveable poles , to be elevated or depressed fit to the latitude of any place ; on the fiducial-edge of which , is fastened the thred , representing the axis of the world , at any elevation of the pole. 5. on the 2 pole points , is fastened the hour circle , which delineates or represents the motion of the sun , or any fixed star , moving in its supposed diurnal motion about the poles of the world , and may not improperly be called the moveable meridian circle , or hour circle , divided as before . 6. the moveable horizon , that moveth about to any azimuth , and slideth or moveth in the fixed horizon . 7. the plain , fixed in 2 opposite points to the moving horizon ; being set , either horizontal , when it lies parallel to the fixed horizon ; or erect , when perpendiculer thereunto ; or set to any reclination or inclination , by help of the semi-circle of reclination , fastened to the backside of the plain in the 2 poles thereof . 8. you have the upper moving semi-circle , in turning about of which , whateve● degree the fore-side of the semi-circle cuts the perpendiculer-point cuts the comple●ment thereof , and to be called the upper semi-circle , or circle alwayes perpendicu●ler to the plain . 9. there ought to be a thred fastened in the center of the plain , to be extended to any altitude or azimuth required . thus much for description , repeated again in short thus ; the horizon ; the meridian ; the equinoctial circles ; the 2 pole points , and axis ; the hour circle , or moveable meridian ; the moveable horizon ; the plain ; the semi-circle of reclination ; the upper semi-circle , and , the thred . note also , every circle is divided into 4 times 90 , and numbred the most useful way . also , on the plain is set the 12 months , and every single day ; on which every respective day , if you extend the thred , then in the degrees , is the suns right ascention in degrees ( on the innermost circle , the same in hours and quarters ) from the next equinoctial-point , on the line of declination , his mean declination ; on the line of ●he suns place , his mean true place , sufficiently true for any illustration in mathematical practice . the uses whereof in some part follow . 1. to rectifie the sphear to any latitude , count the elevation of the pole on the meridian circle , from the horizon upwards , and downwards from the north and south parts of the horizon ; and there make fast , with the help of the small skrew , the fiducial-edge of the poles points , carrying the hour circle fixed upon them , then the pole is rightly elevated . 2. count the complement of the poles elevation on the meridian , from the south part of the horizon , and to it set the divided side of the equinoctial circle , then is that rectified also ( in the northern hemisphere , or in the southern , if you call the north pole the south pole ) . 3. extend the thred or axis passing through the center to the south pole , and there make it fast , and then the sphear is rectified for many uses in that latitude . use i. the day of the month being given , to find the suns true place . lay the thred in the center of the plain on the day of the month , and in the line of the suns place , you have his place . example . on the 5th of november , it is 23 degrees in ♐ ; or if the suns place be given , look for that , and just against it , in the months , is the day required . example . the suns place being 15 degrees ♌ , i look for it in the line of his place , and just against it i find iuly 28 day . use ii. to find his declination any day . look for the day given , and right against it in the line of declination , is his due declination required . example . august the 5th ; the declination is 14 degrees 5 minuts from the next equinoctial-point , viz. ♎ . note , in the northern sines , or summer-time , the sun hath north declination ; or in southern sines , or winter-months , the sun hath south declination . or if you have the suns declination , find that in the line declination , and right against it in the months is the day required . example . 21 degrees south declination , beginning from the equinoctial towards the winter solstice , i find novemb. 15. the like work had been , if the suns place had been given , to find his declination . use iii. the day given , to find the suns right-ascention . this is usually reckoned from ♈ to ♈ , round , in 24 hours ; but twice 12 is as useful , and then it is thus ; find the day amongst the months and dayes , and just against it , in the time of hours , is the suns right ascention ; ( but note , it is not right figured for this use ) counting onwards from ♈ , or the 10th of march , to the 13th of septemb. and from thence to aries again ; likewise the degrees are to be reckoned from ♈ onwards , as the months proceed . example . on the 12 of may , what is the suns right ascention ? lay the thred on the 12th of may , and in the line of hours it cuts 9-57′ counting from aries onwards ; or in degrees 59-15 , counting as before . thus , if any one of these 4 general things be given , the other may be found . use iv. the suns declination and latitude being given , to find the suns meridian altitude . the sphear being rectified , count the declination on the meridian , from the equinoctial , that way the declination is , either north or south ; and where the count ends ; there is the meridian altitude required for that day , or declination . example . iune 11. declination 23-30′ ; count 23-30 , from 38-30 , the place where the equinoctial stands , for 51-30 latitude , and the account will end at 62 degrees , the suns meridian altitude at that declination northwards : but , if it had been 23-30 south declination ; then count as much from the equinoctial downwards , and the count will end at 15 degrees , for the suns meridian altitude , at 23-30 south declination . use v. the suns declination and latitude being given , to find the suns rising or setting , and amplitude , east or west . count the suns declination on the hour-circle towards his proper pole , that is south-declination toward the south-pole , and north-declination towards the north-pole ; and thereunto lay the thred that is fastned in the center ; then bring the hour-circle and thred both together , till the thred touch the horizon ; then the thred on the horizon shews the amplitude , and the divided-side of the hour-circle , shews the suns rising and setting on the equinoctial , counting the meridian alwayes 12 , and the 2 east and west-points 6 , and 15 degr . for an hour , and every deg . 4 min. example . iune 11. declination 23-30 , the sun riseth at 13′ before 4 , and the amplitude is near 40 deg . again , april 10. declination 11-30 , the amplitude is 18-30 from the east to the north , and riseth at 5 , the hour-circle cutting 15 degrees on the equinoctial . use vi. the declination & latitude , & suns altitude given , to find both hour & azimuth . rectifie the sphear , and set the plain horizontal ; that is , level or parallel to the horizon ; then apply the thred to the declination , counted the right way on the hour-circle ; then turn the hour-circle and upper semi-circle about , till the thred cuts the degrees of altitude in the upper semi-circle , and the hour-circle , shews the hour in the equinoctial , and the semi-circle cuts the suns azimuth in the deg . on the horizon or plain . example . declination 10 , latitude 51-30 , and the suns altitude 30 ; the hour will be 8-27 , and the azimuth 66 , from south eastwards if in the morning , or the contrary if in the afternoon . use vii . the hour , or azimuth , and the suns declination given , to find the altitude . the sphear rectified , as before , and the hour being given , set the hour-circle to the hour on the equinoctial ; then bring the thred to the declination , counted on the hour-circle ; then bring the upper semi-circle , till the fore-side do just touch the thred , and the thred on the semi-circle , shall shew the altitude required ; and on the horizon , the azimuth at that hour , and altitude . but if the azimuth be first given , then set the upper semi-circle thereunto , counted on the fixed horizon ; then the thred laid to the declination , on the hour-circle , and turned about till it touch the upper semi-circle , there it shews the altitude ; and the hour-circle on the equinoctial , shews the hour . use viii . to find the suns height in the vertical-circle . set the upper semi-circle to the east or west-point , or 90 degrees of azimuth ; then lay the thred to the declination on the hour-circle , and then bring it and the thred together , till it just touch the upper semi-circle , and it shall there shew the altitude at east or west required . example . at 10 degrees declination north , it will be east at 16 degrees of altitude . use ix . to find the suns altitude at 6. set the hour-circle to 6 on the equinoctial , and the thred to the declination ; then bring the semi-circle to the thred , and it shewes the altitude at 6 required . example . at 23-30 declination , the altitude 18-15 above the horizon in north-declination ; and as much under in south-declination ; for , you must observe that the surest working is from the upper or divided-sides of the rings , on every occasion to use it . use x. to find the hour of the day , when the sun shineth . rectifie the sphear , and set the plain parallel to the equinoctial-circle ; then set the meridian-circle due north and south , and the shadow of the axis shall on the plain , shew the true hour . or , otherwise thus ; at the true place of the suns declination , on the hour-circle make a mark , or stick the point of a pin , then turn the sphear about , till the shadow of that mark , fall on the center ; ( the sphear standing horizontal , as near as may be ) then the hour-circle shall , on the equinoctial , shew the hour of the day required . note , a small bead , or knot on the thred , will do the business as well as may be . thus any the like questions may be wrought for the stars ; or the manner of raising the canon for any spherical triangle whatever , to work the same exactly by the logarithms . as thus ; suppose i would make the canon , or proposition , to find the suns height in the vertical circle at any declination . first , the sphear being rectified , and the plate set horizontal , bring the upper semi-circle to the east-point , and laying the thred to the declination on the hour-circle , bring it and the thred together , till it just touch the upper semi-circle . the rings or circles so standing , and being great circles of the sphear , there is constituted a spherical-triangle in this form ; wherein you have , zae , 51-30 the latitude , the angle at the equinoctial ; and ze 90 , the upper semi-circle ; and ab 23-30 , the declination , part of the hour-circle ; to find be , part of the upper semi-circle : now this being a right-angled spherical-triangle , and the parts which are given , being one right angle , viz. the angle at a , and the side ab , the suns declination ; and the angle at e 51-30 , to find the side be ; now the sines of the sides of spherical-triangles are proportional to the sines of their opposite angles , and the contrary . therefore , as the sine of the triangle bea 51-30 is to the sine of the side ab 23-30 so is the sine of the angle bae 90-00 to the sine of be 30-39 and the like for any other , as by comparing the rules in mr. norwood's trigonometry , and the circles of the sphear together , the use and convenience thereof will evidently appear unto you ▪ only note this plain observation . that the side of a right-angled triangle , which subtends the right angle , is most properly called the hypothenusa ; the other which you make or suppose radius , the base . the other , the perpendiculer . or more short , the hypothenusa and leggs : therefore if the hypothenusa and one leg be given , the proportion is wrought by sines alone ; but if the two legs be given , and first and second in the question , then the proportion is wrought by sines and tangents together . as for example . as the sine of ♈ ♋ 90-00 to the sine of ♋ ae 23-31 so is the sine of ♈ ♉ 30-00 to the sine of ♉ r 11-31 the suns declination in ♉ . again secondly , as the sine of ♈ ae 90-00 to the tangent of ae ♋ 23-31 so is the sine of ♈ r 27-54 to the tangent of r ♉ 11-31 the declination as before .   but if the one acute angle , and his opposite leg or side be given , then the proportion is made by sines only , as in the foregoing example . again , in vertical triangles that have the same acute angle at the base , as the triangle p ♉ ♋ , and ♈ ♉ r , being equal angled at ♉ ; the sines of the bases are proportional to the tangents of the perpendiculer , and the contrary . likewise , the sines of the perpendiculers , as proportional to the sines of the hypothenusaes , and the contrary . as for example . thus for perpendiculers and bases . as the sine of the base ♉ ♋ 60-00 to the tangent of perpend . ♋ p 66-29 so the sine of the base ♉ r 11-31 to the tangent of perpend . ♈ r 27-54 or , as the tang. p ♋ the perpend . 66-29 to the sine ♉ ♋ the base 60-00 so the tang. ♈ p perpend . 27-54 to the sine r ♉ the base 11-31 also for the second , viz. hypothenusaes and perpendiculers . as the sine of hypothen . p ♉ 78-29 to the sine of perpend . p ♋ 66-29 so the sine of hypothen . ♈ ♉ 30-00 to the sine of perpend . ♈ r 27-54 or the contrary thus ; as sine of perpend . p ♋ 66-29 to sine of hypothen . p ♉ 78-29 so sine of perpend . ♈ r 27-54 to sine of hypothen . ♈ ♉ 30-00 this being premised , when to use sines alone , and when to use sines and tangents together , you may rectifie the scheam to your present purpose , and see there how the triangle lies in its natural parts , very plain and demonstratively to be apprehended . the uses of the sphear in dyalling . to this purpose , you must take notice , that the sphear is very excellent to demonstrate that art ; especially all those dials whose stiles have any competent elevation . therefore , first to explain the terms . the sphear being rectified to the latitude ; then , first the plain , or broad-plate , is to represent any plain howsoever scituate , either horizontal , or erect direct , or direct reclining or inclining , or east and west erect , or reclining , or inclining , or erect and declining , or south declining , or reclining or inclining , less or more than to the pole or north declining ; or re-inclining less , to , or beyond the equinoctial . of which in their order . 1. by horizontal i mean , when the plain is set even with the fixed horizon , and the notch which the semi-circle of reclination passeth in just against the meridian ; then if you stretch the axis streight , and bring the upper semi-circle just to touch the axis ; then the axis , on the semi-circle , sheweth the stiles height ; and the edge of the semi-circle on the plain , shews the substile to be in the meridian . for all the hour-arks on the plain , do thus ; set the hour-circle to every hour and quarter on the equinoctial ; and then if you bring the loose thred , fastened in the center of the plain , along the plain till it just touch the hour-circle , then on the plain it shall shew the angle from 12 , for that respective hour and quarter the hour-circle stands at on the equinoctial , accounting 3-45 for a quarter , and 7-30 for half an hour , and 15 deg . for every whole hour , as was hinted before . 2. for an erect direct south or north-dial . just as the plain stood before , that is to say , the notches of the moving horizon against the meridian ; turn the fixed semi-circle , till the divided side of the horizon cuts no deg . on the fixed semi-circle , then the upper-edge of the plain respects the zenith , and the lower the nadir ; and the two notches in the moveing horizon ( being alwayes the poles of every plain ) are just in the meridian ; therefore it is a direct plain , and erect , because upright without any reclination , as the fixed semi-circle sheweth . then being so fixed , and made fast there , pull the axis streight , and bring the upper moving semi-circle just to touch the thred or axis ; then on the upper and lower semi-circles , the axis sheweth the stiles elevation ; and on the plain the semi-circle cuts the substiles distance from 12 , viz. 00 , because a direct plain . and for all the hour-arks on the plain , set the hour-circle to every hour , quarter , and half hour on the equinoctial ; and bring the thred easily along the plain , till it just touch the hour-circle ; then on the plain it sheweth the ark from 12 required . also note , the several triangles made on the meridian , equinoctial , and hour-circle , at every hour it is set unto . as thus ; suppose at the pole , i set p ; at the cutting of the equinoctial , and meridian , ae ; at the upper-end , or zenith , set z ; on the meridian , and where the hour-circle cuts the equinoctial , at 1 & 11 , set 15 ; at 2 & 10 , 30 ; at 3 & 9 , 45 ; at 4 & 8 , 60 ; at 5 & 7 , 75 ; and at 6 & 6 , 90. then the triangle runs thus ; as the whole sine pae 90-00 to the tang. of ae 15 15-00 one hour on the equinoctial . so the sine of pz 38-30 to the tang. of z 11 11-28 the measure on the plain for 11-1 . the like work serves for all the rest . but note , because the hour-circle cannot pass by 12 , you must turn the other-side , or half , for the afternoon hours . also note , that if the back-side of the plain do not well represent the south-side , being the more useful dial ; then if you hold the sphear with the foot upward , the zenith becomes the nadir , and the north plain a south plain , to appear more plain to the apprehension . 3. for a direct reclining dial. for these dials , set the plain direct , as before , and let the upper part of the horizon cut the semi-circle of reclination , according to the plains reclination , and there make it fast ; then the axis drawn streight , and the upper semi-circle brought to it , sheweth the stile and substile ; and the thred and hour-circle , laid as before , giveth the hour-arks on the plain , and sheweth also how the proportion runs . to find any requisite also you may observe for all north-recliners and south-incliners , that the complement of latitude and reclination put together , doth give the poles elevation , or stiles height , for all those plains , which sometime will be above 90 from the south part of the meridian ; and then the complement to 180 , is to be set from the north part of the meridian : but if it be a south-recliner , then substract the reclination out of the comp. lat. and the remainder is the stiles elevation : but if the reclination be more than the complement latitude , then substract the complement latitude out of the reclination or inclination , and the remainder is the stiles elevation . note also , that the upper-face of the plain , that beholds the zenith , is the recliner ; and the under-face that beholds the nadir , is the inclining-plain . and note , that both plains , viz. both incliners and recliners have the same requisites in each of them . but , the hours proper to the recliner , are not to be put on the incliner ; for when the sun shines on the one , it can't shine on the other . therefore to know what hours are fit for these or any plains whatever , do thus ; the sphear rectified , and the plain set to his true scituation , lay the thred on the suns declination , on the hour-circle ( according to what time of year you would know when the sun begins and ceases to shine on any plain ) and turn the hour-circle , with the thred so laid , till the thred do but just touch the plain , and the hour-circle doth on the equinoctial , cut the hour and minuit required ; when the sun comes on the east-side , and when it goes off from the west-side of the plain . example . suppose you have a direct north-plain that reclines from the zenith towards the equinoctial 25 degrees , you shall find the stiles elevation to be 63-30 , the substile from 12. the north-pole to be elevated on the recliner , and the south-pole on the incliner ; and that the sun shines on the north-recliner in the longest dayes , viz. 23-31 , declination , from the rising 13′ before 4 , till 10 ; and then it begins to shine on the south-incliner , and shines till 2 afternoon ; then it comes on the north-recliner again , and continues till it sets . but in the shortest dayes , when the declination is 23-30 towards south , then on the north-recliner it shines not at all , but only on the south-incliner , from rising to setting ; and so doth it all the time the sun hath south-declination . this rule serves for all sorts of dials whatsoever . note , that the circles of the sphear shews the canon to work this question exactly , whereof you have a large discourse in wells his art of shadows , from pag. 391 , to 408 , in 35 chap. 4. for a direct east or west erect-dial . the sphear being rectified to the latitude , bring the notch in the moveing horizon , to the east or west-points on the fixed horizon , viz. to 90 degrees ; then set the plain erect , and make it fast there ; then you shall perceive the axis lie close to the plain , it shews the stile to have no elevation , but must be set parallel to the plain , at any quantity you please , which is to be the radius of a tangent-line , whereby to pr●●k down the hours ; and that the s●bstile or place where the cock or stile must stand is in 6 , being the hour-circle , till it be ●ust against the upper semi-circle , touch●ng the thred , and in the equinoctial it cuts 6 , the true place where the stile must stand . also , by the fore-going rule you shall find the sun shine all the year from the rising , till 12 on the east-side ; and on the west-side from 12 , till his setting . 5. for an east or west-recliner . turn the moving horizon to 90 degrees in the fixed , as before ; then set the plain to his due reclination , and make it fast there , and pull the axis streight , and bring the upper semi-circle just to touch it , and straitway you have the stile , and substile , and 12 , the inclination , meridian , and hour-arks on the plain . as for example . an east-plain reclining from the zenith towards the horizon 45 degrees , hath his meridian , or 12 a clock line in the horizon ; for if you extend the thred from the center to the fore-side of the meridian , just there the 12 a clock line must alwayes be , which in this plain lies in the horizon . the substile doth lie 41-40 from thence upward , as the upper semi-circle doth shew ; the inclination meridian is thus found ; bring the hour-circle , till it stand even and parallel to the upper semi-circle ; then on the equinoctial it cuts 58-7′ , the inclination of the meridian , with which you must make a table of hours , or arks at the pole , to calculate the arks on the plain , if you work arithmetically . but by the sphear , set the hour-circle to the hours on the equinoctial , and the thred being brought along the plain till it touch the hour-circle , shall shew on the plain the angle from the horizon or perpendiculer ; or with some more trouble , from substile or 12. also , it shews , that the north-pole is elevated on the west-reclining ; and the south , on the east-inclining opposite thereunto ; and that the recliner in ♋ , shews from 9 in the forenoon , till 8 at night ; and the east incliner from the rising , till 9 forenoon in summer ; and in winter , till a 11 in the forenoon . now to make these plains , as erect decliners , let the complement latitude become a new latitude ; and the complement declination a new declination ; then they may become erect decliners , as in the next sort following . 6. of erect decliners east or west . by declination , i mean the quantity of the angle that the meridian or pole of place makes between the meridian , or pole of the plain ; therefore to set the sphear to any declination , do thus ; the sphear being set to the latitude , turn the sphear as well as you can guess , to the scituation of the place ; that is , put the north part of the meridian towards the north ; and the south part towards the south ; then turn the notch of the movable horizon , alwayes to the degrees of the plains declination , from north or south , towards either east or west , and upright also as in erect dials : then is the plain set to his declination , viz. the distance of the horizon between the meridian , or pole-place , which is alwayes 12 a clock , and the meridian , and pole of the plain , being alwayes just where the notch is in the moving horizon . now according to these rules , a plain that declines 30 degrees from south to west , the stiles elevation is 32-35 . the substile from 12 , 21-40 . the inclination of the meridian 36-24 . the south pole is elevated on the south-side , and the north pole on the north-side : and the sun shines on the north-side from rising , to 8 ; and on the south-side , from 8 to 7 at night ; and on the north again , till sun-setting , by working as in the former directions is expressed . note , in those erect decliners , whose declinations is above 60 degrees , you shall find the stiles elevation to be very small ; therefore to make it exact , you must use arithmetical calculation ; for the doing of which , the sphear , with due consideration , gives the best directions , with these proportions or canons . as sine 90 zn 90-00 to sine declination nc 30-00 so co-tangent latitude pz 38-30 to tang. subst . from 12. zh 21-40 as sine 90 zn 90-00 to co-sine declination na 60-00 so co-sine latitude zp 38-30 to sine stiles elevation ph 32-25 as sine latitude pn 51-30 to sine 90 pae 90-00 so tangent declination nc 30-00 to tangent elevation merid. aei 36-24 as co-tangent latitude zp 38-30 to sine of 90 zpq 90-00 so sine declination zia 30-00 to co-tang . 6 from 12 aq 57-50 note , if you set p , at the pole. z , at the zenith . n , at the north-end of the horizon , at the declination , or pole-plain . h , on the plain , just against the moving semi-circle , or substile . a , at the plain on the horizon . ae , on the equinoctial . i , at the hour-circle , cutting the equinoctial , set just against the upper semi-circle . note , q is to be set on the plain , right against the hour-circle , being set to the hour . having , i say , by these rules , and the like , made and found the requisites , then proceed to draw the dial thus ; by help of a sector with sines and tangents , to 7-5 ; such as are usually made ▪ but for very far decliners , use that help as directed in chap. 4. the like work serves to help all sorts of dials with low stiles , polar , and meridian . dials also . the other 6 sorts , yet behind , i shall demonstrate only in two of them , which do properly enough comprehend them all ; and the work of one , is as easie as the work of the other , especially by the help of the sphear , where the hardest is as plain as the horizontal . therefore , 7. of declining , reclining-dials . 1. for south recliners , they may recline short of , to , or beyond the pole , at any declination , as the putting up and down the plain , doth plainly demonstrate . therefore , first , of one that declines south-west 35 , and reclines 20 from the zenith . set the notch , or pole of the plain to the declination , and the reclining circle to its reclination , and there make it fast ; then extend the axis streight , and bring the upper semi-circle just to touch it , and the hour-circle exactly even with the moving semi-circle . then , first , the axis shews the stiles height on the semi-circle to be 12-13 . the thred brought along the plain while it touches the meridian , and that shews the meridians elevation above the horizon , on the north recliner to be 76-32 ; or its depression below the horizon in south-recliners , and that from the east-end , as the sphear sheweth . then , 3. the substile from the perpendiculer line of the plain , is 21-6 , as the upper semi-circle sheweth ; but from the hour 12 , or meridian 7-58 , and stands on the east-side of the meridian . the inclination of the meridian is 33-29 , as the degrees on the equinoctial , between the meridian and hour-circle , shew . all the hour-arks are easily found from the plains perpendiculer eastwards and westwards , by applying the thred to the hour-circle and plain , being set to the hours on the equinoctial . the south pole is elevated in the south-recliner , and the north , on the north incliner . if you set letters to the sides and angles , according to the former discourse , you will see how all the canons in the arithmetical calculation lie , as i shewed you before in the declining dials . and as again thus ; on the pole set p. on the zenith z. at the west-end of the plain , set a. at the east-end b. at the south pole of the plain c. at the north pole d. at the east-end of the horizon e. at the west-end w. at the north-end of the meridian , set n. at the south-end s. where the hour-circle cuts the plain f. where the meridian cuts the plain g. where the fixed semi-circle cuts the plain , set e. as in the figure before . then these canons in short run thus ; as sine base zd 90-00 to tang. perpend . nd 35-00 so sine of base ze 20-00 to tang. perpend . ge 13-28 whose complement ag 70-32 , is the meridians elevation . as sine of the side ge 13-28 to sine of the angle cze 35-00 so sine of the angle gfz 90-00 to sine of the side gz 23-57 which taken from zp 38-28 , leaves 14-33 , the distance of the meridians place from the pole to the plain , viz. gf . as sine of hypothen . gz 23-57 to sine of perpend . ze 20-00 so sine of hypothen . pg 14-33 to sine of perpend . pf 12-13   the stile . as tangent of perpend . zf 20-00 to sine of base ge 13-28 so tangent of perpend . pf 12-13 to sine of base fg 7-58   the substile to 12. as the sine of the side ze 20-00 to the sine of the side ge 13-28 so is the sine of the angle pfg 90-00 to the sine of the angle fpg 33-28   inclin . merid. for the hours in all dials , say thus ; as sine of 90 , to sine of stiles height ; so tangent of the angle at the pole , to tangent of the angle on the plain . 8. for north declining reclining-dials . for these plains also , you must rectifie the sphear to the latitude , and set the plain to his declination , and inclination , which is given , and for which you are to make a north declining reclining dial. as you did in the south-recliner , so work in all respects , as you shall bring forth the quesita's , either by the sphear or arithmetical-calculation , as is largely shewn . and for a plain that declines 55 degrees from the north towards the east , and relines 20 from the zenith , you shall find the requisites to be as followeth . 1. the meridians elevation above the horizon , is found to be 63 deg . 58 min. but yet observe , you must make use of that part of it which is below the horizon , because the sun being elevated high on the south-part of the meridian , must needs cast a shadow on the north-part thereof ; therefore in drawing the dial-part , part is only to be made use of for the sun to shine on . 2. the stiles elevation is 37 degrees 00 minuts . 3. the substile from 12 , 65-24 ; or from the plains perpendiculer 39-22 . the north pole is elevated ; and in regard the plain declines to the east , the stile must be set towards the west , and it shines on the plain in summer-time , from the rising unto 12 : but in the winter-time , but a few hours . note also , that these declining reclining-plains , may be referred to a new latitude and declination , wherein they shall become upright decliners , as before is hinted . the poor-mans dial-sphear ; or another way to demonstrate the mystery of dyalling , both for declining and inclining plains , in a very plain , easie way , for one 6th part of the cost of the other brass-sphear . first , as to the description , and afterward for the vse . as to the description , the figure annexed , and a few words shall suffice ; wherein consider , first , the plain flat-board , representing the horizon , as abcd. secondly , the two upright pieces , as east and west-points , as ae , and bf , to support the moving plain . thirdly , the moving-plain , moving to any inclination , on the two points e and f , with 180 degrees upon the plain , and noted by abef . fourthly , also a brass-circle as g , fastened to the plain , to set it to any degree of inclination ; and a skrew , as at h , that may stay it steady , when set to any reclination . fiftly . on the middle of the horizontal-board , is fastened at the point m , a true horizontal-dial , drawn fit for your latitude , and to turn round on the point m , as imkl . sixtly , a thred fastened in l , the center of the horizontal-dial ; and in n , the center of the plain ; to be both a stile for the horizontal-dial , and to represent the axis of the world ; also a small woodden-quadrant will be useful , such a one as half the plain is , to draw perpendiculers , and measure angles , as afterwards in the uses . the uses follow . use i. to find the declination of a plain by the sun-shining . apply the side ab to the wall , and hold the instrument level , as by help of a point plummer , fastened at n , and the point playing right on m , it is easie to do ; then by the trianguler-quadrant , having first observed the true hour , turn the horizontal-dial about on the point m , till the shadow of the thred ( or axis ) shew the same hour ; then the point on the north-end of the horizontal-dial , shall shew the true declination of the plain . for any south decliner , the use is obvious . but for north-decliners , you must turn the plain out of the way of the thred , still keeping the same side , ab , to the wall ; and if the horizontal-dial hinder , put a parallel-piece between , as your rule , or any other thing , and you shall have the point give the declination on the southern semi-circle on the fixed horizon . use ii. the declination of any erect decliner given , to find the substile , and stile , inclination of meridians , and every hour and quarters distance from 12 , being the perpendiculer line on the plain . first , set the point at 12 on the horizontal-dial , to the declination of the plain , toward the east or west , and set the plain upright . then first for the substile . apply the side of the quadrant to the plain , and cause the shadow of the thred to play parallel to the perpendicular ray of the quadrant , and at the same time it shall shew on the degrees on the plain , the true substiles distance from 12. example . suppose the plain decline 20 degrees south-west , you shall find the substile to be 15 deg . and 12′ from 12 , and to stand on the east-side of 12 , in a south declining west 20 degrees , latitude 51-30 . again , for the stiles elevation . apply the quadrant to the flat of the plain , on the substile line , so as the thred may cut the center of the quadrant ; and then the thred shall cut on the quadrant 35-46 for the stiles height . again , for the inclination of meridians . the shadow of the thred when it cuts the substile 15 deg . 12′ on the plain , shall on the horizontal-dial cut 1 hour 36 min. which reduced to degrees , is 24 deg . 50 min. the inclination of meridians . again , for every hours distance , in degrees and minuts from 12. turn the whole instrument about , ( as it is then first set ) till the shadow of the thred shall fall on every hour and quarter , and then the shadow shall cut on the degrees on the plain , the distance of every hour and quarter from 12 , for that declination , in degrees and minuts ; which you may draw into a table , for your use and purpose ; or hereby examine your more exact calculation , and prevent all gross mistakes in your former work . use iii. any declining north-east , or north-west-dial being given , to find the former requisites for those dials . in the true proper using the sphear for north-dials , the stile should look upwards , which will appear so to do , if you turn the instrument the bottom upwards , for the further help to your fancy ; but observe that the hour-arks , and angles , are the same for the north , as for the south , only the difference is in the scituation , as to the contrary-side , and looking upward instead of the south decliner , looking downward , as by turning the instrument appears ; so that if you draw the dial as a south-west , when you would make a north-west ; and set right figures , and the right way , and then your work is effected to your mind , to the right intent and purpose . example of a north-east , 30 degrees , latitude 51-30 . set the point at 12 , to 30 degrees westward , and apply the square to the plain , till it just touch the thred ; and on the degrees on the plain , it cuts 21-40 for the substile ; and at the same time almost half an hour past 2 for inclination of the meridians ; and applying the quadrant to the substile-line on the plain , and to the thred ; it cuts 32-35 for the stiles height , being the same , and the same way found as for the south decliner east . but observe , that for the hour-arks , you must note , that the north-dial cannot shew 12 at noon , nor any hours very near noon , which will be seen on the south decliner east ; therefore 4 in the morning , is here called 8 ; and 5 is called 7 ; and 6 is 6 : 7 in the morning , is called 5 ; and 8 is to be named 4 : and if you turn the instrument , that the shadow of the thred may fall on those hours , it will also cut on the degrees on the plain , the true hour-arks required . as thus ; for 8 , it sheweth it not ; at 7 , it sheweth 77-00 ; at 6 , it sheweth 58 deg . 52 minuts ; at 5 , it cuts on the degrees on the plain 45-38 ; at 4 in the morning , it cuts on the plain 35-27 ; but the shadow falls then on hour of 8 , on the horizontal-dial . also note , that these numbers are not laid from the substile , but from the plains perpendiculer , which in all upright plains is a perpendiculer line ; and in all other plains , a perpendiculer to the horizontal-line , drawn on the plain . and thus proceed with any other ; the affections are best seen when you turn the instrument the upper-part downwards . use iv. to find the requisites , and to draw the hours on a far declining erect-dial , s. w. 80. set the point to 80 , as before s. w. then the thred and the quadrant shall shew 38-2 for the substile ; and 82-8 on the horizontal , for the inclination of meridians ; and 6-12 , for the stiles elevation ; and the shadow of the thred on the horizontal-dial , will shew you how close and inconvenient the hours will be , if not helped by the former directions ; and in like manner will the north-east or west be , and likewise helped . use v. to find the requisites , and hour-arks , from the perpendiculer of a declining inclining plain , with its affections . set the point at 12 to the declination , and move the plain by help of the arch , or circle of brass , to the inclination , and with the skrew make it fast and steady in that place . then for the substile , apply the quadrant to the plain , and also perpendiculer to the axis , as the edge of the quadrant being thick , will neatly shew ; then the thred will shew on the degrees on the plain , the distance of the substile from the perpendiculer , or the complement thereof from the horizon ; which point note with a spot of ink ; for , when the shadow of the thred falls on that spot , on the horizontal dial , it sheweth the inclination of meridians ; that is to say , on what hour and minut , the cock of the dial should stand right over . also , the quadrant , applied to the plain and thred , on the substile-line , sheweth the true stiles elevation above the plain . and lastly , making the shadow of the thred to fall on every hour on the horizontal-dial , it shall at the same time shew how many degrees and minuts on the plain , that hour-line ought to be from the perpendiculer , or from the horizon ; and also which way , either to the right or left , east or west ; or from the substile , or 12 ; if you will trouble your self to count it , from the place found out for the substile , or 12. example of a plain declining 30 s.e. and inclining 20. the substile , by applying the square , you shall find to be 30 degrees on the left-hand of the perpendiculer westward , and the inclination of meridians 48-20 , the stiles height 51-36 , and the meridian on the right-hand of the perpendiculer-line 11-30 eastward ; and the shadow of the thred playing on every hour and quarter , on the horizontal-dial , will shew on the plain the quantity in degrees from the perpendiculer-line . use vi. to find the requisites in a north-east reclining-dial , and the hour-lines . set the instrument as before , and find the substile , stile , and inclination of meridians as before ; but note , as to the affections , which way do thus ; turn the instrument the bottom upward , and as near as you can guess , turn the plain to its scituation ; then you shall first see the stile to look upward in the north-east recliner , which before was downward in the south-east incliner . also , the substile stands on the right-hand of the perpendiculer , 30 degrees westward ( for observe this alwayes , if a plain declines eastward , the substile will stand westward , and the contrary ) . also note , that the meridian-line is to be drawn quite through the center on the other-side ; because , when the sun is in the meridian above , it must needs cast the shadow of the axis , or stile , the contrary way downwards . use vii . to find what are the most hours , that the sun can shine on any plain , whatsoever . first , on all south direct , or declining inclining-dials , the mid-day-meridian is proper to it , unless it incline above 75 degrees , and then it becomes useless in london latitude ; then what hour soever you can make the sun to shine on the plain , and horizontal-dial both together , ( the sun being at that hour above the horizon ) by bending or turning the instrument any way , ( when the point at 12 is first set to the declination ) that , and all those hours are proper to that plain , at one time of the year or other . also note , that several hours that serve for the south-plain , do , at some time of the year , belong to the north-plain also ; as by turning the instrument about , you may plainly see , either by the sun-shine , or by the thred , and your eye cutting the hour-lines and the plain . also observe , that if you would delineate a south reclining plain , you may bring the plain toward the thred , till it becomes a polar-plain . but if it reclines below the pole , then conceive it to become a north reclining-dial , and work as is before directed , and you shall obtain your desire ; for the dials will be the same , the one as the other , as before was hinted at , in the inclining-plains . use viii . the declination of any plain given , to find what reclination will make it a polar-dial , and the contrary . set the north-point to the declination , and bring the plain to touch the thred ; then on the brass circle is cut the reclination required . or contrary ; set the plain to the reclination given , and then bring the thred to the plain , by turning the horizontal-dial , and the point at 12 shall shew the declination required , to make it polar . in like manner you may discover a declining equinoctial , but not so easily , when the substile and meridian are 90 degrees assunder ; the substile being then alwayes in the hour of 6 , as by moving the plain , if the declination be given ; or by moving the thred , if the inclination be given , till the square , touching the thred , it shall shadow or bourn , just upon 6 on the horizontal-dial . note also , that east and west recliners , and incliners , are discovered after the same manner ; so also direct recliners , and incliners , as by moving the plain to and fro , you shall see the plain and true reason , how the stile is elevated or depressed , and how the hour-lines are inlarged or contracted , according to the elevation of the stile . also , in east and west-dials , that the stile hath no elevation , but is parallel to the plain ; and how the meridian lieth in the horizon , in east and west recliners , and incliners . many more uses might be insisted on , which i shall leave to the scruteny of the industrious practitioner , in the art of shadows . chap. ix . how to remedy several inconveniencies in the using of the artificial lines of numbers , sines and tangents , as they are usually made . 1. if the term required happen to be under one degree of sines and tangents , then the line of numbers will supply it , having due respect to the increase of the radius , or caracteristick . as thus ; as the sine of 90 , to the sine of 23-31 , the greatest declination ; so is the sine of 1 deg . 10′ , the suns distance from the equinoctial , to 0-28 , the declination which falls beyond the end of the rule . now to remedy this , the 1 deg . & 10′ , is 70 minuts ; therefore by the numbers say , so is 70 minuts , the suns distance from the equinoctial , to 28 the suns declination on the line of numbers , observing to extend the same way , as from the first to the second term . 2. when you have occasion to use a sine above 90 degrees , then you must count the sine of 80 , for the sine of 100 ; and 70 , for 110 ; and 60 , for 120. so also , the distance from 90 to 60 in the sines , is the secant of 30 degrees ; and the distance from 90 to 50 , is the secant of 40 ; or the point beyond 90 , that represents the secant of 40. 3. if the extent be too large for your compasses , as from 45 or 90 , to 3 or 4 degrees ; then instead of 90 or 45 , make use of a point in the sines or tangents right against the middle 1 in the line of numbers , where you may have two brass center-pins , viz. in the tangent of 5-43 , and the sine of 5-45 ; and the extent from thence backward or forward , shall reach in the numbers , to the 4th proportional number required . example . as tang. 45 , to 1-61 in the numbers ; so is tang. of 15-0 , to 0-43 in the numbers . instead of which , you may say , as the tang. of 5-43 , to 1-61 on the numbers ; so is the tang. of 15 , to 0-43 on the numbers diminishing a radius ; for as tang. 45 to 1-15 , a greater than that ; so is the tang. of 15 , to a greater than 15 also , viz. 0-43 . secondly , in sines & tangents , or sines only , where there is another caution to be observed , as sine 90 , to sine 10 ; so is sine 20 , to sine of 3-24 ⅓ . to work this with small compasses on a large line , do thus ; note , that at 10 on the line of numbers , or sine of 90 , or tang. of 45 , is one compleat radius ; but at the middle 1 , on the line of numbers , is a place , or radius , less ; wherein the logarithm sines , the characteristick is 8. again , at the sine of 0-34 ½ , the characteristick is 7 , ( and at 3 minuts it is 6 , ) which do note the several decreasings of the radiusses ; therefore set the distance from one number given , to the next nearest place against 1 , or next radius , as far from a greater or a less radius , as your occasion serves , and note the place . as thus for example . in this operation , the extent from the point at 5-45 on the sines , to the sine of 10 degrees , i set the same way from the point at 0-34 ½ and note the place , which will be at near 1 degree ; then the work is thus ; as the place against the middle 1 , instead of 90 , is to the place last found for 10 ; so is the sine of 20 , to sine of 3 deg . 24′ ⅓ , the 4th term required . but in those lines of numbers , sines , and tangents , where the number is double , this is performed by working a-cross only . 4. when the last term in tangents happens to be above 45 , then the remedy is two wayes , as thus ; as sine of 30 , to sine of 90 ; so is the tang. of 30 , to tang. 49-07 . which here happens beyond 45. apply the end of the rule , next 90 , close and even with any thing on which the point of the compasses may stay , till you take from thence to 45 , for that distance laid from 45 , shall reach to 49-07 , reading the tangents as numbred beyond 45. or more neatly thus ; the compasses being set from the sine of 30 , to the sine of 90 ; set one point in the tangent of 45 , and turn the other on the tangents , and keep it there fixed ; then remove the other from 45 , and close it to the third term , being here the tangent of 30 ; then this last extent laid from 45 , shall reach to 49-07 , the tangent required . 5. when the first term is a tangent above 45 , and the second under 45. take the excess of the first number above 45 , and set it the same way from the second number ; then the extent from the second number to 45 , shall be the true distance between the first and second terms . example . as the tangent of 51-30 , to the tangent of 30 ; so is the tangent of 40 , to tangent 21-04′ . for the extent from 45 , to 51-30 on the tangents , set the same way from 30 , does reach to about 24-30 ; then the extent from thence to 45 , shall reach from 40 to 21-04 on the tangents , the 4th number required . or , if it had been from a tangent above 45 , to a sine , the same way would have remedied the defect . 6. when the third term exceeds 45 of tangents , then thus ; example . as sine 90 , to sine 30 ; so is the tang. of 50 , to tang. of 30-48 . the compasses set from the first term sine 90 , to sine of 30 the second , a less ; then set one point in the tangent of 45 , and extend the other backwards in the tangents , and note the place , keeping one point there close , the other to 50 the third term ( being above 45 , by counting backwards ) then , i say , that extent laid from tangent 45 , shall reach to tangent 30-48 , the 4th proportional tangent required . if the proportion had been increasing , then there had been no trouble at all . also note , that working a-cross , or changing the terms , is a good remedy also . as thus ; as sine 90 , to tang. 50 ; which is properly increasing , for the tang. of 50 being more than the sine of 90 , yet taken on the rule from 90 to 40 , the complement thereof , as if it were decreasing ; so is sine 30 , to tang. 30-48 , the contrary way : therefore , as from the first term , properly counting to the second . 7. lastly , when one or two radiusses ( or alterations of the characteristick ) falls between the first and second term . as thus for example . first , by the line of numbers only ; as 8000 is to 10 , so is 5000 to 6 ¼ , or 25. to work this properly , and naturally , the unite on the numbers should be four times repeated , which is seldom more done than twice , as here : but this , and any other , by the line of numbers is not interrupted , having a due respect to the number of places . for to work this , the best way , is changing of terms thus ; as 8000 , to 5000 in the same radius ; so is 10 , to 6-25 in the same radius also . or , without changing ; as 8000 , to the next 1 ; so is 5000 turning the compasses the same way , to 6-25 . but to call it so , and not 625 , your reason must guide you more than precepts . but in using sines and tangents , the way in the third remedy will fit you . example . as sine 90 , to 1 degree ( or under ) ; so is sine 30 degrees , to sine 30 minuts . this being too wide an extent for the compasses , the third rule is a remedy for it ; which on a large radius several times repeated , as in mr. oughtred's circles of proportion , is as easie as may be ; being sure to remember the number of radiusses between the first and second term , that you may have so many between the third and fourth term also . much more might have been said as to this ; but this observation being alwayes kept , that as the extent from the first term to the second , is either increasing or decreasing ; so alwayes must the extent be from the third to the fourth , increasing or decreasing , in like manner , when you use sines and tangents ; and numbers also , except , as before , in a few particuler rules ; then you will be truly resolved . the end of the book of dyalling . an appendix to the use of the trianguler-quadrant in navigation . where it performs the uses of the davis-quadrant , the cross-staff , bow , sinical-quadrant , and sector , with as much ease and exactness as any , or all of them , will do in observation or operation , naturally or artificially . being first thus contrived , and made by iohn brown , dwelling at the sign of the sphear and sun-dial in the minories , near to aldgate , london . london , printed by iohn darby , for iohn wingfield , and are to be sold at his house in crutched fryers ; and by iohn brown at the sphear and sun-dial in the minories ; and by iohn selle at the hermitage-stairs in wapping . 1671. chap. i. the description thereof for sea-uses . the description of the instrument , is largely and plainly set down in the first part , and first chapter . but , in regard that is the general description of all the lines that can conveniently be put on , and those necessary for this use being far less , i shall repeat the description again , as far as concerns the use thereof for sea-observations . 1. first for length , it ought to be two foot-long at least , when shut together , and not above 3 foot at any time for sea-uses ; ( but for land-uses it may be 6 , 8 , 10 , or 12 foot in length , to find altitudes or distances to seconds of a degree certainly ) . 2. the form of it is the same , as before , viz. an opening joynt of about an inch and quarter , or half quarter broad each leg ; and 6 tenth parts of an inch in thickness , with a loose-piece of the same length , breadth , and thickness , to make it an equilateral-triangle . as the figure sheweth . 3. the lines necessary for sea-uses are , first , the 180 degrees upon the moving-leg and loose-piece , numbred as before is shewed . also , 60 degrees on the innermost-edge of the loose-piece . the kalendar of months and dayes , and degrees of the suns place , and right ascention , on the moveable-leg . for the speedy and ready finding the suns place , and declination , which you may do to a minut at all times , by help of the rectifying table , and astronomical cautions of time and longitude . also , on the head-leg , is the general scale of sines and lines , to the great and lesser radius , as in the figure . and thus much will serve both for observation and operation , as in the following discourse will fully appear . 4. to this instrument doth chiefly belong the sights for the observations at sea , where the horizon is made use of in the taking the sun or stars altitude . and to this instrument belongs the index and square , that makes it a most compleat sinical-quadrant , for the plain and easie resolving of all plain triangles . also , a weighty plummet and thred , and a pair of large wood or brass compasses for operation . thus much for description , being all put on one side only , unless you shall be pleased to add the artificial numbers , sines , and tangents on the outer-edge , and a meridian-line , and his scale on the inner-edge ; and natural sines , and natural versed-sines on the sector-side : but these as you please . chap. ii. the use of the trianguler-quadrant in observation . that the discourse may be plain , and brief , and general ; there are 10 terms to be named and described , before i come to the vses and examples , which are as followeth . 1. first , the head-leg of the instrument in which the brass-rivit is fixed , and about which the other leg turns , as ab , in the figure ; on which leg , the general scale of sines and lines are usually set . 2. the moveable-leg , on which the months and dayes be , as in the figure , noted by bd ; which leg turns about the head-leg . 3. the loose-piece that is joyned to the head , and moving-leg , by two tennons at each end thereof , noted by da in the figure . 4. the head-center , or center-pin on the round-part of the head-leg , being center to the 60 degrees on the in-side of the loose-piece ; which point is known by b , in the figure . 5. the leg-center , being near the end of the head-leg , which is the center to the degrees on the moving-leg , and out-side of the loose-piece , being in all 180 degrees ; and noted in the figure by the letter c. 6. the great radius , or greater line of sines , issuing from the leg-center toward the head , having the tangents on the moveable-leg to the same radius ; and the measure from the leg-center to the tangent on the moving-leg , a secant to the same radius ; as ce in the figure . 7. the little radius that issues from the leg-center toward the end , having the tangents , on the out-side of the loose-piece to the same radius , and the measure from the center to those tangents for secants to the same radius ; as cf. 8. the turning sight alwayes to be skrewed to the head , or leg-center , known by his shape and skrew-hole , as 9. the sliding horizon-sight to slide on the moving-leg and loose-piece , noted with its bigness and hole to look through , as 10. the shadow sight , and 2 others , to pin the instrument together , which you may call the object-sights , alwayes fixed in the two holes at the ends of the moving-leg , and the head-leg ; and the shadow-sight is to set to and fro to any place required ; noted in the figure with 〈◊〉 and the other two with 〈◊〉 and thus you have their name and description at large , which in brief take thus for easie remembring . 1. the head-leg . 2. the moveable-leg . 3. the loose-piece . 4. the head-center . 5. the leg-center . 6. the great radius . 7. the less radius . 8. the turning-sight . 9. the horizon sliding-sight . 10. the shadow-sight , and the two objest-sights ; the open-part in one is next to , and the other remoter from the rule , to answer to the upper or lower-hole in the turning-sight , according as you please to use them in observation . thus much for the terms , the vses follow . use i. to find the suns , or a stars altitude , by a forward observation , as by a fore-staff . skrew the turning-sight to the head-center , and put the object-sight into the hole at the end of the head-leg , and put the sliding horizon-sight on the in-side of the loose-piece ; then setting the turning-sight to your eye , and holding the loose-piece in your right-hand , and the moveable-leg toward your body , then with your thumb on the right-hand , thrust upwards , or pull downwards the horizon-sight , till you see the sun through the object-sight , and the horizon through the horizon-sight ; then the degrees cut by the line on the middle of the horizon-sight , shall shew the true altitude required . also observe , that if you like to use the upper or lower-edge of the horizon-sight , instead of the small bar a-cross the open-hole , after the manner of the ends of a fore-staff , that then the degrees and minuts cut by the edge of the brass , is the altitude required , to be counted as it is figured from the object-sight , toward the horizon-sight ; the degrees between them being the angle required . note also , that if the altitude of the sun , or star , be above 30 degrees , you will find it a hard matter , to behold the horizon and sun with a bare roling the ball of the eye only , and a stirring of the head , will easily cause a stirring of the hand , which will spoil the exactness of observation , unless the instrument shall be fixed to a ball-socket and three-legged-staff , which is not usual at sea. therefore to remedy this , you may observe with the open oval-hole in the turning-sight set to the eye , or taking the turning-sight quite away ; observe just as you do with a fore-staffe , setting the round part of the head , to the hollow-part beside your eye , so as the head-center-pin may be as near the very sight of your eye as possibly as you can ; which center is the center to the degrees now used in a forward way of observation . or , rather use this way when the weather will suffer , by a thred and plummet , which i shall add as a second use. use ii. to observe the sun or a stars altitude , by a forward observation , using the thred and plummet . skrew the turning-sight to the head-center , as before , and put the two object-sights into the two holes at the two ends of the rule ; and on the leg-center-pin hang the thred with a weighty plummet of two pound , or above a pound at least . then hold up the trianguler-quadrant , setting the small-hole on the turning-sight close to your eye ; and if the sun , or star , be under 25 degrees high , then look to the sun or star through the turning-sight , and that object-sight , which stands in the end of the moveable-leg , letting the thred and plummet play between your thumb and fore-finger , as a brick-layers plummet in his plum-rule doth in a bendid hole , that you may keep it in order whilst you look at the sun or star , and the weighty plummet will pull the thred streight , and let you know by feeling which way it is playing , till it playeth evenly and truly , whilst you have the object precisely in the midst thereof , whether it be sun , moon , or any star , or other object , whose altitude you would observe ; then , i say , when the plummet playes well , and you behold the object right , bend back the quadrant , and see what the thred cuts on the degrees on the moveable-leg , which shall be the true altitude required ; and in my opinion , must needs be more exact than any other way of a forward observation , because you are not troubled to mind the horizon and sun both at at once . an objection may be , the boisterous winds , and the rouling of the ship , will hinder such an observation . answ. so it will any other way , though happily not so much . again , i answer , one object is better and more certainly seen , than two at any time together ; and though the wind blow hard , if you can stand to observe at all , the heavy plummet will be sure to draw the thred perpendiculer ; and for ought i know , you may come as near this way as any other ; however this , at most times , may confirm and prove the other , and may be useful in rivers , and harbours , and misty-dayes , when you may see the sun well enough , but not the horizon at all . use iii. to find the suns altitude by a backward observation , as with a back-staff , or davis-quadrant . skrew the turning-sight to the leg-center ( or center to the degrees on the moveable-leg ) ; and set the object-sight to the long stroke by 00-60 on the out-side of the loose-piece , and put the sliding horizon-sight on the out-side of the moveable-leg ; then hold the object-sight upwards , and the small-hole in the piece turning on the ●dge ( or to the small-hole in the middle ) of the horizon-sight ( which you please ) close to your eye ; and looking through that hole , and the middle-hole of the turning-sight , to the true horizon , turning your self about , and lifting up , or pressing down the horizon-sight , close to the moveable-leg , till the shadow of the upper-edge of the shadow-sight , being next to the sun , fall at the same time just on the middle of the turning-sight ; then , i say , the edge or middle of the horizon-sight , that you looked through , shall cut the true altitude of the sun required . being the same way as you do observe with a davis-quadrant , or back-staff . use iv. to find the suns distance from the zenith , by the trianguler-quadrant . skrew the turning-sight to the leg-center , and put the object-sight , whose oval-hole is remotest from the quadrant , in the hole in the end of the head-leg , or rather in a hole on the general scale , between the turning-sight , and the sun ; and put the horizon-sight on the out-side of the moveable-leg ; then hold the turning-sight toward the sun , and the small-hole in the edge of the horizon-sight to your eye ; then look through that hole and the turning-sight , till you see the shadow , the object-sight , to fall just on the turning-sight , or the shadow of the turning-sight to fall just on the object-sight , which is all one , though the first be more easie , because you shall see the horizon through the turning-sight , and that , both at once ; then , i say , the degrees cut by the horizon-sight , shall be the suns distance from the zenith required ; being the very same work , and done in the same manner , and producing the same answer , viz. the suns distance from the zenith , that the davis-quadrant doth . note , that this way you may observe very conveniently , till the sun be 20 degrees distance from the zenith ; and by the adding of a 60 arch , as in davis quadrant , or to 45 will be enough , it will do as well as any davis quadrant , being then the same thing . but i conceive , the complement of the altitude being the same , will do as well ; which altitude is better found by this instrument , than the distance from the zenith by a davis quadrant is , as in the next use will be seen . use v. to find the suns altitude when near the zenith , or above 90 degrees above some part of the horizon . in small latitudes , or in places near the equinoctial , or under it ; the sun will be found to be in , or near the zenith : and if you count from some part of the horizon , above 90 degrees distant from it ; then instead of setting the sliding object-sight , to the long stroke at 00 on the loose-piece , you must set it 30 degrees more towards the head-leg ; then observe , as you did before , and whatsoever the horizon-sight cuts , you must add 30 degrees more to it , and the sum shall be the true altitude required . example . suppose that in the latitude of 10 deg . north , on the 10th of iune , when the suns declination is 23 degrees and 31 min. northward ; suppose that at noon , i observe the suns meridian altitude , skrewing the turning-sight to the leg-center , and setting the object-sight to the 30 degrees on the loose-piece , near the end of the head-leg , and the horizon-sight on the movable-leg ; then hold up the quadrant , with the shadow-sight toward the sun , and the small-hole in the horizon-sight toward your eye , and look to the horizon through that , and the turning-sight , the shadow of the right-edge of the shadow-sight , that cuts the degree of 30 , at the same time falling on the middle of the turning-sight , you shall find the horizon-sight to cut on 46-29 minuts ; to which if you add 30 , the degrees , the shadow-sight is set forwards , it makes up 76-29 , the suns true altitude on that day in that latitude ; 76-29 the meridian altitude , and 23-31 the declination , added together , make 100 deg . 00 ; from which taking 90 , there remains 10 , the latitude of the place . 1. in this observation , first you may note this , that if you had stood with your back toward the south , you would have had 103 degrees and 31 minuts , for the sliding horizon-sight would have stayed at 73 degrees 30 ; to which if you add 30 , it makes 103-31 ; which a davis quadrant will not do . 2. in the holding it , you may lean the head of the rule to your breast , and command it the better , as to steady holding . 3. you may turn the turning-sight about , to any convenient angle , to make it fit to look through to the horizon , and also to receive the shadow of the shadow-sight . if the brightness of the sun offend the eyes , you may easily apply a red or a blue glass , to darken the sun beams , and the sights may be painted white , to make a shadow be seen better . use vi. to find the latitude at sea , by a forward meridian observation of the altitude , according to mr gunter's bow. skrew the turning-sight to the leg-center , and set the shadow-sight to the suns-declination , and the horizon-sight to the moving leg ( or loose-piece ) , and the turning-sight to your eye ; then let the shadow-sight cut the horizon , and the horizon-sight the sun , moving it higher or lower till it fits ; then whatsoever the sight sheweth , adding 30 degrees to it , is the latitude of that place required . example . suppose on the 10th of march , when the declination is only 10′ to the northward , as in the first after leap-year it is ; then set the edge , or stroke on the middle of the shadow-sight to 10′ of declination toward the head , and the horizon-sight , on the same leg toward the end , and slide only the horizon-sight till it cuts the sun , and the other the horizon ; then suppose ●t shall stay at 21-30 : then if you count the degrees between the two sights , it will amount to the suns meridian altitude ; but if you add 30 degrees to what the sight cuts , it shall give the latitude of the place where the observation was made for 21 and 30 , to which if you add 30 , it makes 51-30 , the latitude of london , the place where the observation was made . note here , that in small latitudes the sun will be very high , in summer time especially , and then the sliding-sight , must be set on the loose peice . as thus for example . suppose on the 10th of may 1670 , when the declination is 20-7 in the latitude of 30 degrees , i observe at a meridian altitude , i shall find the sliding-sight to stay at 00. on the loose peice ; then it is apparent that 30 added to 0 , makes but 30 degrees for the latitude ●equired . but if the sliding-sight shall happen to pass beyond 00 on the loose peice , then whatsoever it is you must take it out of 30 , and the remainder is the latitude required . example . suppose on the 11th of iune 1670 you were in the latitude of 10 degres to the northward , and standing with your back to the north , as you must needs do in all forward observations in more northern latitudes , you shall find the sight to pass just 20 deg . beyond 00 on the loose peice ; therefore 20 taken from 30 the residue is 10 , the latitude required . again . suppose that in the same place you had obsereved on the 11th of december , when the sun is most southwards , if you set the one sight to 23-31 southwards against the 11th of december ; then if you observe forwards with your face toward the south , as before , you shall find the moving-sight to stay at 20 degrees beyond 00 on the loose-piece ; then , i say , 20 taken from 30 , rests 10 , the latitude required , because the sight passed beyond 30 on the loose-piece . lastly , if the moving-sight shall happen to pass above 30 degrees beyond 00 on the loose-piece , when the other sight is set to the suns declination , and you observe with your face toward the south ( part of the meridian ) ; then , i say , the latitude is southwards as many degrees as the moving-sight stands beyond 30 , on the loose-piece toward the head-leg . so that the general rule is alwayes , in north latitudes ( observing the suns meridian altitude , to find the latitude , by a forward observation , according to mr. gunters bow ) , your face must be toward the south ; although that thereby in some latitudes , the altitude may seem to be ( as indeed it is above the south-part of the horizon ) above 90 degrees . then , if the sliding-sight stay any where on the moving-leg , or loose-piece , short of 00 , add it alwayes to 30 , and the sum shall be the true latitude north ; if it pass beyond 00 , then so much as it doth , take out of 30 , and the remainder is the latitude north ; but if it shall stay just at 30 on the loose-piece , then the latitude is 00 ; but if it pass beyond 30 , then so much as it is , the latitude is southward . the same rule serves , if you were in south latitude , then you must in forward observations , to find the latitude , as with a gunters bow , stand with your face to the north , and in setting the fixed-sight to the declination , you must count south declination toward the head ; because those that have southern-latitude , have their longest dayes , when those that live in northern-latitude have their shortest dayes . the same rule serves for the stars also , for being in north-latitudes ; and observing a latitude forwards , have your face alwayes toward the south , and set one sight to the declination , counting the stars northern or southern declination ▪ the same way as the suns , ( and the contrary in south-latitude ) ; then holding your eye close to the great-hole of the turning-sight , slide the moving-sight till its middle-bar or edge ( as is most convenient ) cuts the star , and the other the horizon ; then whatsoever the edge of the moving-sight cuts short of 0 , added to 30 , or beyond 00 , taken out of 30 , shall be the latitude required . example . suppose the middle-star of orions-girdle , whose declination is 1-28 south , being in the meridian , i set one sight to 1-28 of south-declination ; and slipping the other-sight till it cuts the star , the fixed-sight being set to the horizon , you shall find it stay in the latitude of 51-30 , at 21-30 on the moving-leg , which added to 30 , makes 51-30 . note , that if the corner of the instrument be inconvenient for the sight to slide on , as for about 5 degrees it will , then you may remedy it by slipping the sight set to the declination 10 degrees more , any way that is convenient , increasing or decreasing ; but then note , that instead of adding even 30 , to what the moving-sight stayeth at , you must add 10 degrees more , viz. 40 , when you slip it towards the end of the moving-leg ; or 10 degrees less , viz. 20 degrees , when you slip it 10 degrees more toward the head , as is easie to conceive of . use vii . to find the latitude at sea , by a backward meridian observation , according to the way of mr. gunter's bow. skrew the turning-sight to the leg-center , and set the sliding-horizon-sight to the suns declination ( the middle or edge of it , as you can best like of ) and the shadow-sight on the loose-piece , or moving-leg , with your face alwayes to the north , in north latitudes ; or supposing your self to be so , though it may be you are not . then looking through the hole in the horizon-sight ( standing at the declination ) and the turning-sight to the horizon , with your hand gently slide the shadow-sight till the shadow fall just on the middle of the turning-sight , as you do in observing the altitude with a davis quadrant ; then , i say , whatsoever the shadow-sight shall stay at under 0 , add to 30 ; or over 0 , take out of 30 , and the sum or remainder , shall be the latitude north : but if it happen to stay at just 30 , the latitude is 00 ; if beyond , it is so much to the southwards : this is only the converse of the former , and needs no example , but a few words to demonstrate it ; which may be thus , in the way of an example . suppose that on the 11th of december , in sayling toward the east-indies , about the isle of st. matthews , supposing our ship to be in north-latitude , i set the horizon-sight to 23-31 , south declination ; and the shadow-sight on the loose-piece , then standing with my face to the north , as another then would do , as at other times , and looking through the horizon , and turning-sight to the north-part of the horizon , i find the shadow-sight when it playes well over the turning-sight , to stay at 33 degrees on the loose-piece . then , consider that the distance between the wo sights , is the altitude of the sun above the south-part of the horizon ; which if you do count on the trianguler-quadrant , you will find to be 36-29 , and 33 , which put together , make 69-29 , for the suns meridian altitude ; to which if you add 23-31 , his declination , it makes 93-00 the distance of the north-pole and zenith , or 3 degrees of south latitude ; for had you been just under the equinoctial , the altitude would be 66-30 ; or had you been more northward , it would have been less ; therefore by considering , you may soon see the reason of the operation . also , if the shadow-sight be too near the corner , or too far from the turning-sight to cast a clear shadow ; then , set the horizon-sight that stands at the declination 10 , 20 , or 30 degrees more toward the end of the moving-leg , and you shall see the inconvenience removed ; but then you must take 10 , 20 , or 30 degrees less than the shadow-sight sheweth , for the reason abovesaid . the reason why even 30 is added , is because that 0 degrees of declination , stands at 60 on the moving-leg , instead of 90 , or 00. note , if you had rather move the lower-sight than the upper , then count like latitudes and declinations from 00 on the loose-piece toward the head-leg , and unlike the contrary , and then set the shadow-sight fixed there ; then observing , as in a back-observation , the horizon-sight shall shew the complement of the latitude required , without any adding of 30. thus you see , that the trianguler-quadrant , containing 180 degrees in a triangle , brings the shadow-sight near the center , and with one manner of figuring , gives the suns altitude above the horizon , backwards or forwards , and his distance form the zenith , and the latitude of the place south or north , or north or south , backwards or forwards , by the sun or stars , by one side only , as conveniently and with fewer cautions , and as exactly , if well used , as any other instrument whatsoever ; so that by this time you see it is a fore-staff , quadrant , and bow. the other uses follow . use viii . to find the latitude by a meridian observation , by the thred and plummet , by the sun or stars . this way of observing without a horizon , must be done by an astrolabe , which is a plummet it self , or else with a plummet fitted to another instrument , and at some times may do better service than the horizon , and for an altitude barely , is shewed already . for the latitude thus ; count the declination , which is the same with the latitude , from 00 on the loo●epiece toward the moveable-leg ; and contrary declinations , both of sun or stars , count the other-way toward the head-leg , and thereunto set the edge of the horizon-sight , that hath the small-hole on it . then let the sun-beams shine through the small-hole on the turning-sight , to the small-hole on the horizon-sight , the thred and plummet duly playing , shall shew the latitude of the place required . but if you look at a star , having the same declination , then set your eye to the horizon-sight , and behold the star through the turning-sight , and the thred shall fall on the latitude required , when you look toward the south , being in northern latitudes . so also , when you turn your face toward the north , in observing those stars , it is best done when they come to the meridian below the pole ; but for their coming to the meridian above the pole , then their declination is increased by the quantity of their distance from the pole , or the complement of their declination . as thus ; the declination of the pole-star , when in the meridian below the pole , is 87-20 from the equinoctial ; but when the same star is on the meridian above the pole , then it is 92 deg . 40′ distant from the same northern part of the equinoctial . so that if you make 60 on the moveable-leg , to represent the north-pole ; then you may count or observe any star that is 25 degrees distant from the pole , both above or below the pole ; then adding 30 degrees to what the thred falls on , shall be the true latitudes complement required ; because you have removed the pole from 90 to 60 , 30 degrees backward . example . the declination of the uppermost star in the great bears back , is 63-45 ; that is , 26-15 below the pole ; or , 25 degrees 17 minuts above the horizon , when on the meridian below the pole ; but the same star , when on the south-part of the meridian , is 77-47 above the horizon , or 26-15 above the pole. therefore , the star being below the pole , you may set the hole in the middle of the horizon-sight , to the declination , counting 90 the pole , and looking up to the star , as usually , the plummet will fall on 38-28 , the latitudes complement required . again , the same star being on the south-part of the meridian , above the pole , i count 60 on the moving-leg for the pole , and 26-15 beyond that pole further , viz. to 86-15 , which is as far as you can well go , counting 60 the pole ; then observing , as you did before , you shall find the thred to play on 08-28 , the latitudes complement required , for the distance between 08-28 , and 86-15 is 77-47 , adding 30 degrees , because of 60 instead of 90 , for the pole-point . note , that the thred playing near the corner , may prove somewhat troublesome to observe , without help of another person ; but if you will be exact in this or any other observation , a staff and a ball-socket , should be applied to this , as well as to other instruments , to stand steady and sure in the time of observation . these wayes are ready and easie , without taking notice of those regulations and cautions , which are to be observed in finding the altitude , barely ▪ as in the seamans kalender , and mr. wrights errors in navigation , is plainly seen . but if you know them all , and had rather use those rules in those books ; then , i say , a thred and plummet by this instrument , will do as conveniently as any other , or the three sights and horizon , as before is shewed , to find the altitude . chap. iii. to rectifie the table of the suns declination . thus much as for the way of observation ; now , that your operation may be true also , it is necessary that you have a table of the suns declination , for the first , second , and third year , after the leap-year . but in consideration , that the second after the leap-year , is a mean between the other three ; i have made a table for that , and the months on the trianguler-quadrant are agreeable thereunto ; and for the first , third , and leap-year , have added a rectifying table to bring it to a minut at least to the real truth , wherein i have followed the suns place , according to mr. streets table of the suns place , for 1666. in which table , you have degrees and minuts ; and a prick after , notes a quarter of a minut ; and two pricks , half a minut ; and three pricks , three quarters of a minut more . now , by the rule , you may count to a minut , and the rectifying table tells you how many minuts more you must add to , or substract from the degrees and minuts the table or rule shall shew it is , in the second year . a table of the suns declination every day at noon for london in the year 1666 , the second year after the leap-year , according to mr. street's tables of longitude . calculated by iohn brown , 1668. month dayes . ianu. febr. march april . may. iune . d.m. d.m. d.m. d.m. d.m. d.m. 1 21 45 ... 13 50. . 03 29. 08 31. 18 02 23 11. 2 21 36 13 30. . 03 05. . 08 53. 18 17. 23 14. . 3 21 25. . 13 10. 02 42 09 15. 18 32 23 18. 4 21 14 ... 12 49 ... 02 18. 09 36 ... 19 46 ... 23 21 5 21 03. . 12 29 01 54 ... 09 58 19 01 23 23 ... 6 20 51 12 08. 01 31 10 19. 19 14 ... 23 26 7 20 40 11 47. 01 07 10 40 ... 19 28. 23 27 ... 8 20 27. . 11 25 ... 00 43. 11 01 19 41. 23 29. 9 20 15 11 04. s. 19 ... 11 22 19 54. . 23 30. 10 20 01 ... 10 43 n. 04. . 11 42 ... 20 07 23 30 ... 11 19 48. 10 21 00 27 ... 12 03 20 19 23 31 12 19 34. . 09 59. 00 51. . 12 23 20 31 23 30. . 13 19 20. 09 37 01 15 12 43. 20 42. . 23 30 14 19 06 01 15 01 38. . 13 03 20 54 23 29 15 18 51 08 52 02 02. . 13 22. 21 04. 23 37. . 16 18 35 ... 08 29 ... 02 26 13 42 21 15. 23 15. . 17 18 20. 08 07. . 04 49. 14 01 21 25. 23 23 18 18 04 07 44 ... 03 13 14 20 21 35 23 20. . 19 17 48 07 22 03 36 14 38. 21 44. 23 17. 20 17 31. 06 59 03 59. . 14 57 21 53. 23 14 21 17 14. . 06 36 04 22. . 15 15 22 01 ... 23 10 22 16 57. 06 13 04 45 ... 15 33 22 10. 23 05. . 23 16 40 05 50 05 0● ... 15 50 ... 22 17 ... 23 01 24 16 22. 05 26. . 05 32 16 08 22 25. 22 55 ... 25 16 03 ... 05 03. 05 34. . 16 25. 22 32. . 22 50. 26 15 45 04 39 ... 06 17. 16 42 22 39 22 44. 27 15 27 04 16. 06 40 16 58. . 22 45. . 22 37 ... 28 15 08 03 52 ... 07 02. . 17 14 ... 22 51. . 22 31 29 14 49   07 25 17 30 ... 22 57 22 23 ... 30 14 29. .   07 47 17 47 23 02 22 16. 31 14 10.   08 ●9 .   23 06. .   a table of the suns declination every day at noon , &c. month dayes . iuly . augu. septem octob. novem decem d.m. d.m. d.m. d.m. d.m. d.m. 1 22 09 15 14. 04 26. . 07 12. . 17 37 ... 23 07. 2 22 00. 14 56 04 03 ... 07 35. 17 35 ... 23 12 3 21 51. 14 37. . 03 40 ... 07 58 ... 18 10. 23 16 4 21 42. . 14 19. 03 17. 08 20. 18 25. . 23 19. . 5 21 33 14 00 ... 02 54. 08 42. . 18 41. 23 22. 6 21 23. 13 41. . 02 31 09 04 ... 18 56. 23 25 7 21 13. 13 22. 02 08. . 09 27 19 10 ... 23 27. 8 21 02. . 13 03 01 44. 09 49 19 25. 23 28 ... 9 20 52 12 43. 01 20 ... 10 11 19 39. 23 30 10 20 41 12 23 00 57. 10 32. . 19 53 23 30 ... 11 20 29 12 02 ... n 34 10 53 ... 20 06. 23 31 12 20 17. 11 43 n 10. . 11 15. 20 19 23 30. . 13 20 05 11 21. . s 13. 11 36. 20 31 ... 23 30 14 19 52 ... 11 02 00 36 ... 11 57 20 44 23 29 15 19 39. . 10 41. . 01 00 ... 12 18. 20 55 ... 23 27 16 19 26 ... 10 20. . 01 24 12 38 ... 21 07 23 25. 17 19 13. 09 59. 01 47. . 12 59. 21 18. 23 22. 18 18 59. 09 38 02 10 ... 13 20 21 28 ... 23 19. . 19 18 44 ... 09 16 ... 02 34. . 13 39 ... 21 38 ... 23 16 20 18 30. 08 55. 02 58 13 59. 21 48. ▪ 23 11 ... 21 18 15 ... 08 33. . 03 21. 14 19 21 58 23 07. . 22 18 00. . 08 11. . 03 44. . 14 38 22 07 23 02. . 23 17 45. 07 49 ... 04 08 14 57. . 22 15. . 22 57 24 17 29. 07 27. . 04 31. . 15 16 22 23. . 22 51. . 25 17 31. . 07 05. . 04 54. . 15 35 22 31 22 44 ... 26 16 57 06 43 05 18. 15 53. 22 38. 22 38 27 16 40. . 06 20. . 05 41 16 11. 22 45 22 30. . 28 16 24 05 57. . 06 04 16 29. 22 51. . 22 23 29 16 06. . 05 35. 06 27 16 46 ... 22 57. 22 15 30 15 49. 05 12. . 06 49. . 17 03 ... 23 02 ... 22 06. 31 15 32     17 20.   21 57. a table of the suns declination for every 5th and 10th day of every month of the four years ; calculated from mr. street's tables of the suns place , made for the years 1665 , 1666 , 1667 , and 1668 ; the nearest extant . m. d 1. 1665 2. 1666 3. 1667 l.y. 68 ianuary . 05 21 01 21 03. . 21 06. 21 09. 10 19 58. . 20 01 ... 20 05 20 08. . 15 18 47. 18 51 18 54. . 18 58. 20 17 27. 17 31. 17 35 17 39. . 25 15 59 ... 16 03 ... 16 08 16 12 ... 30 14 25 14 29 ... 14 34. . 14 39 february . 05 12 24 12 29 12 34 12 39 10 10 39. . 10 43 10 48 10 53. . 15 08 47 08 52. . 08 57 ... 09 03. 20 06 53. . 06 59 07 04 ... 07 10. 25 04 57. . 05 03. 05 08 ... 05 14. . 28 03 47. s 03 52 ... 03 58. . 04 04. march. 05 01 48 ... 01 54 ... 02 00. 01 42. s 10 00 10 n 00 04. n 00 01 ... s 00 16. n 15 02 08 02 02. . 01 56. . n 02 14. . 20 04 05 03 59. . 03 54 .. 04 11. . 25 06 00. 05 54. . 05 49. 06 06. . 30 07 52 ... 07 47 07 42. 07 58. april . 05 10 03. 09 58 09 53 10 09 10 11 47 ... 11 42 ... 11 37 ... 11 35. 15 13 27. 13 22. 13 17. . 13 32. 20 15 01. . 14 57 15 52. . 15 06. . 25 16 29. 16 29. 16 21. 16 34 30 17 50. . 17 47 17 42 ... 17 54. . may. 05 19 04. 19 01 18 57. . 19 08 10 20 10 20 07 20 04 20 13. . 15 21 07 21 04. 21 02 21 10 20 21 55. . 21 53. 21 51 ... 21 57 ... 25 22 34 22 32. . 22 30 ... 22 36 30 23 03 23 02 23 01 23 04. . iune . 05 23 24 23 23 ... 23 23 23 25. . 10 23 30 ... 23 30 ... 23 30. . 23 31 15 23 27 23 27. . 23 28 23 26 ... 20 23 13 23 14 23 15 ... 23 12 25 22 48 ... 22 50. 22 51 ... 22 47 30 22 19 ... 22 16. 22 18 22 12. a table of the suns declination , &c. m. d 1. 1665 2. 1666 3. 1667 l.y. 68. iuly . 05 21 31 21 33 21 35. 21 28 10 20 38 20 41 20 43. 20 35 15 19 36 19 39. . 19 42. . 19 32 ... 20 18 25. . 18 30. 18 34 18 23 25 17 09 ... 17 13. . 17 17. . 17 05 30 15 45. . 15 49. 15 53. . 15 40. . august . 05 13 55. . 14 00 ... 14 05 13 50 10 12 18 ... 12 23 12 28 12 13 15 10 36. . 10 41. . 10 46. . 10 31 20 08 50 08 55. 09 00. . 08 44 25 07 00 07 05. . 07 10. . 06 53 ... 30 05 07 n 05 12. . 05 18 05 00 ... september . 05 02 48. . 02 54. 02 59 ... 02 42. 10 00 51. . n 00 57. n 01 03 n 00 45. n 15 01 07 s 01 00 ... s 00 54 s 01 12. s 20 03 05. . 02 58 02 52. 03 09 ... 25 05 00. 04 54. . 04 49 05 06. . 30 06 55. 06 49 06 44. 07 01. october . 05 08 48 08 42. . 08 38. . 08 54 10 10 37 ... 10 32. . 10 27. 10 43. . 15 12 23 12 18. 12 13. 12 28 ... 20 14 04 13 59. 13 54. . 14 09. 25 15 39. 15 35 15 30. . 15 44 30 16 08. 17 03 ... 16 59 ... 17 12 ... novemb. 05 18 45 18 41. 18 37. 18 49. 10 19 56 19 53 19 49. . 19 59 ... 15 20 58. 20 55 ... 20 53 21 01. . 20 21 51 ... 21 48. . 21 46. . 21 53. 25 22 32 ... 22 31 22 29. . 22 35 30 23 04 23 02 ... 22 01. . 23 05 ... december . 05 23 20. 23 22. 23 21 23 23 ... 10 23 30 ... 23 30 ... 23 30 ... 23 31 15 23 27 23 27 23 27. . 23 26 20 23 11 23 11 ... 23 13 23 09 25 22 43 22 44 ... 22 46. 22 41 30 22 05 ... 22 06 ... 22 08 ... 22 01 ... a rectifying table of the minuts and quarters that are to be added or substracted from the fore-going table of the suns declination , made for the second year after leap-year , for every day at noon in the meridian of london . m. d 1 year 3 year l. year . ianuary . 05 s. 2. . a. 2 ... a. 5 ... 10 s. 3 a. 3. a. 6. . 15 s. 3. . a. 3. . a. 7 20 s. 4 a. 4 a. 8. 25 s. 4. . a. 4. a. 9. . 30 s. 4 ... a. 5. a. 9 ... february . 05 s. 5 a. 5. . a. 10 10 s. 5. a. 5. . a. 10. 15 s. 5. . a. 5 ... a. 10. . 20 s. 5 ... a. 5 ... a. 10 ... 25 s. 5 ... a. 5 ... a. 11 28 s. 6 a. 5 ... a. 11. . march. 05 s. 6. a. 5. . s. 12 10 a. 6. . s 5. . a. 12. . 15 a. 6. s 5. a. 12 20 a. 5 ... s. 5. a. 11 ... 25 a. 5 ... s. 5 a. 11. . 30 a. 5 ... s. 4 ... a. 11. april . 05 a. 5. . s. 4 ... a. 11 10 a. 5. . s. 4. . a. 10. . 15 a. 5 ... s. 4. . a. 10 20 a. 5 s. 4. . a. 09 25 a. 4 ... s. 4 ... a. 08 30 a. 4. . s. 4 a. 07 ... may. 05 a. 3. . s. 3. . a. 7 10 a. 3 s. 3 a. 6 15 a. 2 ... s. 2 a. 5. . 20 a. 2 s. 1 ... a. 4 25 a. 1. . s. 1. . a. 3 30 a. 1. s. 1 a. 2 iune . 05 a. 0. . s. 0 ... a. 1 10 a. 0 s. 0. . a. 0 15 s. 0. . a. 0 s. 0. . 20 s. 1 a. 1 s. 2 25 s. 1. . a. 1. s. 3 30 s. 1 ... a. 1. . s. 4 iuly . 05 s. 2. . a. 2. . s. 05 10 s. 3 a. 2 ... s. 06 15 s. 3. a. 3 s. 06 ... 20 s. 3. . a. 3. . s. 07. 25 s. 3 ... a. 4 s. 08. . 30 s. 4 a. 4. s. 09. august . 05 s. 4. a. 4. . s. 09 ... 10 s. 4. . a. 4 ... s. 10 15 s. 5 a. 5 s. 10. . 20 s. 5. a. 5. s. 11 25 s. 5. . a. 5. s. 11. 30 s. 5 ... a. 5. . s. 11. . september . 05 s. 5 ... a. 5 ... s. 11 ... 10 s. 5 ... a. 5 ... s. 12 15 a. 6. . s. 6. . a. 12. 20 a. 6. s. 5 ... a. 12. . 25 a. 5 ... s. 5 a. 11 30 a. 5. . s. 4 ... a. 11. . october . 05 a. 5. . s. 4. . a. 11. . 10 a. 5. s· 4 ... a. 11 15 a. 4 ... s. 5 a. 10. . 20 a. 4 ... s. 5 ... a. 10 25 a. 4 ... s. 5. a. 09. . 30 a. 4. . s. 4. . a. 09 november . 05 a. 4 s. 4 a. 08 10 a. 3 s. 3. . a. 07 15 a. 2. . s. 3. a. 06 20 a. 2 s. 2. . a. 05 25 a. 1 ... s. 2 a. 03. . 30 a. 1. . s. 1. . a. 02 december . 05 a. 1 s. 0 ... a. 01. . 10 a. 0 s. 0 a. 00. 15 a. 0. ● . 0. s. 01 20 s. 0. . ● . 1. s. 02 25 s. 0 ... a. 1. . s. 03 30 s. 1 a. 2 s. 04 ... 1665 1667 1668   1665 1667 1668 1669 1671 1672 for these years . 1669 1671 1672 1673 1675 1676 1673 1675 1676 a table of the magnitudes , right ascention in hours and minuts , and degrees and minuts , and the declination north or south of 33 fixed stars . n. names of the stars . m. r. asc. r. asc decli . n.   d. m. h. m. d. m. s. 01 pole-star , or last in little bear. 2 7 53 0 32 87 33 n. 02 andromedas girdles 2 12 31 0 50 33 50 n. 03 medusaes head 3 41 27 2 46 39 35 n. 04 perseus right side 2 44 30 2 58 48 33 n. 05 middle of the pleides 5 51 22 3 26 23 06 n. 06 bulls eye 1 64 0 4 16 15 48 n. 07 hircus or goat 1 72 44 4 51 45 36 n. 08 orions left foot 1 74 30 4 58 8 38 s. 09 mid-star in orions girdle 2 79 45 5 19 1 28 s. 10 orions right shoulder 2 84 5 5 36 7 18 n. 11 auriga , or waggoner 2 84 45 5 39 44 56 n. 12 great dog 1 97 24 6 30 16 13 n. 13 castor , or apollo 2 108 00 7 12 32 30 n. 14 little dog 1 110 20 7 21 6 6 n. 15 pollux , or hercules 2 110 25 7 22 28 48 n. 16 hydraes heart 1 137 36 9 10 7 10 s. 17 lyons heart 1 147 30 9 50 13 39 n· 18 great bears fore-guard 2 160 48 10 43 63 32 n. 19 lyons tayl 1 172 45 11 31 16 32 n· 20 virgins spike 1 196 43 13 07 9 11 n· 21 last in great bears tayl 2 203 36 13 34 51 5 n· 22 arcturius 1 209 56 14 00 21 4 n. 23 little bears fore-guard 2 222 46 14 52 75 36 n. 24 brightest in the crown 3 231 00 15 24 27 43 n. 25 scorpions heart 1 242 23 16 09 25 37 s. 26 hercules head 3 254 40 16 59 14 51 n. 27 lyra , or harp 1 276 17 18 25 38 30 n. 28 eagle , or vulture 1 293 28 19 35 8 1 n. 29 swans tayl 2 307 30 20 30 44 5 n 30 dolphins head 3 307 53 20 32 15 0 n. 31 pegassus mouth 1 321 50 21 27 8 19 n 32 pomahant 3 339 30 22 38 31 17 s. 33 pegassus lower wing 2 358 50 23 55 13 22 n. as for example . to find the suns declination for the year 1670 , on the 12th day of may : first , if you divide 70 ( being the tens only of the year of our lord by 4 , rejecting the 100s s ) you shall find 2 , as a remainder , which notes it to be the second after leap-year ; and if 0 remain , then it is leap-year . then , look in the table of declination for 1666 , the second after leap-year , as the year 1670 is , and find the month in the head of the table , and the day on one side , and in the meeting-point you shall find 20 deg . 31 min. for the declination on that day at noon required . or , if you use the trianguler quadrant , extend the thred from the center over the 12th of may , and you shall find it to cut in the degrees just 20 deg . 31 min. the true declination for that year and day . note , that if you have occasion to use the declination before noon , then observe that the difference between stroke and stroke , is the difference of declination for one day ; and by consequence , one half of that space for half a day ; and a quarter for a quarter of a day , &c. as thus for example . suppose i would have the suns declination the 18th of august 1666 , at 6 in the morning ; here you must note , that the 18th stroke from the beginning of august , represents the 18th day at noon just . now the time required being 6 hours before noon , lay the thred one fourth part of the distance for one day , toward the 17th day , and then in the degrees , the thred shall cut on 9-43′ , whereas at noon just , it will be but 9-38 ; and the next , or 19th day at noon , it is 9 degrees 16 min. and 3 quarters of a min. as the three pricks thus ... in the table doth plainly shew ; but by the rule , a minut is as much as can be seen , and so near with care may you come . note also farther , that if you shall use it in places that be 4 hours , 6 or 8 , 10 or 12 hours more eastward , or westward in longitude , the same rule will tell you , the minuts to be added in western-longitudes , or to be substracted in eastern-longitudes , as reason and experience will dictate unto you with due consideration . for if being eastwards , the sun comes to the meridian of that place before it comes to the meridian of london ; then lay the thred as in morning hours : but if the place be to the westwards where it comes later , then lay the thred so many hours beyond the noon-stroke for london , as the place hath hours of western-longitude more than london , counting 15 degrees for an hour , and 4 minuts for every degree ; and then shall you have the declination to one minut of the very truth . but if it happens to be the leap-year , or the first or third year after the leap-year , then thus ; suppose for the 5th of october 1671 , being the third after leap-year , i would have the declination . first , if you lay the thred over the 5th of october , in the degrees , it gives 08 deg . 42 minuts .. , for the declination in the second year after leap-year ; then , because this is the third year , look in the rectifying-table for the 5th of october , and there you find s. 4 .. , for substract 4 minuts and a half from 8-42 .. rests 8-38 , the true declination required for the 5th of october 1671. the like work serves for any other day or year ; but for every 5th and 10th day , you have the declination set down in a table for all 4 years , to prove and try the truth of your operations ; and by that , and the line of numbers , or the rule of three , you may continue it to every day by this proportion . as 5 dayes , or 120 hours , to the difference of declination in the table , between one 5th day and another ; so is any part of 5 dayes , or 120 hours , to the difference in declination to be added or substracted to the 5 dayes declination immediately fore-going the day required . example . suppose for the 18th of february 1669 , the first after leap-year , i would know the declination by the table made to every 5th day only ; on the 20th of february , i find 6-53 ½ ; on the 15th day , 8-47 ; the difference between them is 1-53 ½ ; then the extent of the compasses from 5 , the number of dayes , to 1-53 ▪ the minuts difference ( counted properly every 10th for 6 minuts ) shall reach from 3 , the dayes from 15 toward 18 , to 1 degree 7 minuts and a half , which taken from 8-47′ , the declination for the 15th day , leaves 7 degrees 38 minuts and a half , the true declination for the 18th day of february , in the first after leap-year . or , by the line of numbers thus ; the extent from 5 , the difference in dayes , to 113 ½ , the difference in min. for 5 dayes , shall reach from 3 , the difference in dayes , to 68 , the difference in minuts for 3 dayes , to be added or substracted , according to the increasing or decreasing of the declination at that time of the year . proved thus ; if you substract 5′ ½ from 7 deg . 44 ... the declination in the second year , there remains 7 deg . 38′ ½ , the declination for the 18th of february , 1669. these tables may serve very well for 30 years , and not differ 6 minuts in declination about the equinoctial , where the difference is most ; and in iune and december not at all to be perceived . thus you may by the rule and rectifying table , find the suns declination to a minut at any time , without the trouble of calculation . chap. iv. the use of the trianguler-quadrant in the operative part of navigation . use i. to find how many leagues , or miles , answer to one degree of longitude , in any latitude between the equinoctial and pole. first , it is convenient to be resolved how many leagues or miles are in one degree in the meridian or equinoctial , which mr. norwood and mr. collins hath stated about 24 leagues , or 72 miles . or. if you keep the old number , making the miles greater , viz. 60 miles , or 20 leagues ; then the proportion , by the numbers , sines and tangents , runs thus ; as sine 90 , to 20 on the numbers for leagues ; so is co-sine of the latitude , to the leagues , on the numbers , contained in one degree of longitude in that latitude . but in miles , to have the answer , work thus ; as sine 90 , to 60 on numbers ; so co-sine latitude , to the number of miles . example , latitude 51° 32′ . as sine 90 , to 60 ; so sine 38-28 , to 37 miles 30 / 100. but by the trianguler-quadrant , or sector , work thus ; take the latteral 20 for leagues ( or 60 for miles ) from the line of lines from the center downwards ; and make it a parallel in the sine of 90 , laying the thred to the nearest distance . then , the nearest distance from the co-sine of the latitude , to the thred , measured latterally from the center , shall shew the true number of leagues required . example , latitude 51° 32′ . as — 60 , to = sine of 90 ; so = sine of 38-28 , to — 37-30 , on the lines . as — 20 , to = sine of 90 ; so is = sine of 38-28 , to — 12-40 for leagues . or , as — 24 , to = sine of 90 ; so is = sine of co-lat . to — 15 , the number of leagues , after the experiment made by mr. norwood , of which true measure you may read more in the second part of the plain scale , by mr. collins . or , if you multiply the natural sine of the co-lat . by 2 , it gives the leagues ; or by 6 , it gives the miles in one degree , cutting off the radius from the product . note also , that if you take the natural-number of the secant of the course or rumb , and multiply it by 2 , cutting off the radius from the product , it shall give the leagues required , to raise one degree , at the rate of 20 leagues to one degree of a great circle . use ii. to find how many leagues , or miles , answer to raise , or to depress the pole one degree on any rumb from the meridian . first , by the artificial sines , tangents , and numbers . as the co-sine of the rumb from the meridian , to 20 leagues ( or 24 leagues ) on the numbers ; so is the sine of 90 , to the number of leagues required . which , when you have sayled on that rumb , you shall raise or depress the pole one degree . but by the trianguler-quadrant , thus ; as — 20 , taken from the line of lines , or any equal parts , to the = co-sine of the rumb , laying the thred to the nearest distance . so is the = sine of 90 , or nearest distance from sine 90 , to the thred , to — number of leagues required , to sayl on that rumb , and to raise the pole one degree . use iii. to find how many miles or leagues answer to any number of degrees in any parallel of latitude . suppose you sayling in the latitude of 48 degrees , have altered your longitude 30 degrees , and would then thereby know how many leagues you had sailed . first , bring ( or reduce ) the 30 degrees to leagues , by multiplying them by 20 , or 24 , ( the leagues resolved to be in one degree ) which makes 600 , ( or 720 ) . then by the numbers and sines . the extent from the sine of 90 , to 42 the co-sine of the latitude , shall reach the same way , from 600 on the numbers , to 400 the leagues required ; or from 720 , to 480 , according to mr. norwood . by the trianguler-quadrant . take — 600 from the line of lines , or any equal parts , and make it a = in the sine of 90 , laying the thred to the nearest distance . then , the nearest distance from the sine of 42 , ( the co-sine of the latitude to the thred ) and it shall give 400 on the lines , or equal parts , the leagues required . which is thus more briefly ; as — 600 , to = 90 ; so is = sine 42 , to — 400 , as frequently before . use iv. to work the six problems of plain sayling by gunter's lines on the edge , or the trianguler-quadrant . note , that in this art of navigation , or plain sayling , that the angle that any degree of the quadrant , or point of the compass makes with the meridian , or north and south-line , that is called the rumb or course . but the angle that it maketh with the east and west-line , or parallel , is called the complement of the rumb or course . note , that in plain triangles , the sines and tangents give angles , and the numbers give sides . note also , that in plain sayling , the distance run , or course , is the same with the hypothenusa in plain triangles . also note , that the difference of latitude is counted on the meridian , and the difference of longitude or departure from the meridian , is counted on the equinoctial , or on a parallel of latitudes . one of which lines , in plain triangles , is called the base ; and the other , the perpendiculer . the base being a sine , and the perpendiculer a sine complement . note also , that in north latitude , sailing southerly , the latitude doth decrease ; therefore you must substract the difference in latitude , from the latitude you parted from ; but if you sayl northerly , then you must add it to the latitude you parted from : the like in south latitudes . but when one latitude is south , and the other north , then you must add them both together . note also , that the difference in latitude and longitude , ( and departure ) when given in degrees , are to be reduced to leagues , by multiplying by 20 , and counted alwayes on the line of numbers , or equal-parts , when you use the trianguler-quadrant . so then in using the index and square in plain sailing , the distance sayled , is alwayes counted on the index from the center . the course is counted on the degrees from the head toward the loose-piece . the difference of latitude on the head-leg , from the leg-center to the head. the departure or longitude , is counted on the square . the complement of the course or rumb is counted on the degrees , beginning at 00 on the loose-piece . when your number of leagues exceed 100 , you must double the numbers on the index , the square , and head-leg , or count 10 for a 100 , &c. problem i. the course , and distance run on that course , being given , to find the difference in latitude , and departure , or difference in longitude . as sine of 90 , to the distance run ( or leagues sayled ) on the line of numbers ; so is co-sine of the course or rumb , to difference in latitude on the numbers . again , for the longitude or departure . so is the sine of the course , to the departure , or difference in longitude . by the trianguler-quadrant . as — leagues sailed , to = sine 90 , laying the thred to the nearest distance ; so = co-sine of the rumb or course , to — difference in latitude . or , so is = sine of the rumb or course , to = departure , or difference in longitude . by the index and square , after the manner of a synical quadrant , thus ; set the index ( being put over the leg-center-pin ) to the course counted on the degrees from the head , toward the loose-piece . then slide the square perpendiculer to th● head-leg , till the divided edge thereof cuts the distance run on the index ; then shall the index , on the square , give the departure or difference in longitude ; and the square on the head-leg , shall shew the difference in latitude . problem ii. the course and difference of latitude given , to find the distance run , and departure . as co-sine of course , to the difference in latitude ; so is sine 90 , to the distance run . then , as sine of 90 , to the distance run ; so is sine of course , to the departure . by the trianguler-quadrant , without the square . as — difference of the latitude , to = co-sine of the course ; so = sine 90 , to — distance run . so is = sine of the course , to the departure . with the index and square , thus ; set the index to the course , and the square to the difference in latitude ; then on the index , is cut the distance ; and on the square , the departure . problem iii. the course and departure given , to find the distance run , and difference of latitude . as sine course , to the departure on numbers ; so is sine 90 , to the distance . again , as the sine 90 , to distance run ; so is co-sine course , to difference in latitude . by the trianguler-quadrant . as — departure taken from any fit scale , to = co-sine of the course ; so is = sine 90 , to — distance run on the same scale . so is = sine of the course , to the difference in latitude . with the index and square . set the index to the course , and slide the square perpendiculer to the head-leg , till the index cuts the departure on the square ; then the index sheweth the distance , and the square the latitude on the head-leg , counting from the center . problem iv. the distance run , and difference in latitude given , to find the course and departure . as the leagues run , to sine 90 ; so the difference in latitude , to co-sine course . again , as sine 90 , to the distance run ; so is sine of the course , to the departure . by the trianguler-quadrant . as — radius , or a small sine of 90 , to = distance run on the line of lines ; so is = difference in latitude , to co-sine of the course , measured on the small sine . so is — sine of the course , to the = departure , carried = in the lines . by the index and square . set the square to the difference in latitude , and move the index till the square cuts the distance run on the index ; then shall the index shew on the square , the departure ; and on the degrees , the course required . problem v. the distance run , and departure given , to find the course and difference in latitude . as the distance run , to sine 90 ; so is the departure , to sine of the course . then , as sine 90 , to the distance run ; so is co-sine course , to the difference in latitude . by the trianguler-quadrant . as — distance run , to = sine 90 ; so — departure , to = sine of the course . so is = co-sine course , to — difference in latitude . by the square and index . slide the square and index , till the index cuts the departure on the square , and the square cuts the distance run on the index ; then , on the degrees , the index shall shew the course ; and on the head-leg , the square shall shew the difference in latitude . problem vi. the difference of latitude , and the departure given , to find the course and distance run . as the difference in latitude , to 45 degrees ; so the departure , to the tangent of the course . again , as sine course , to the departure ; so is sine 90 , to the distance run . by the trianguler-quadrant . as — radius , or tangent of 45 , to = difference in latitude ; so is = departure , to — tangent of the course on the loose-piece from whence you took 45. then , as — departure , to = sine of the course ; so is = 90 , to — distance run . by the index and square . set the square to the difference in latitude , on the head-leg , counted from the center ; and bring the index to cut the departure on the square ; then the square on the index shews the distance ; and the index , on the degrees , gives the course required . in all these 6 problems , which mr. gunter makes 12 problems , of plain sayling , i have set no example , nor drawn no figure , because the way by the index and square is so plain ; and of it self makes a figure of the work : for the index is alwayes the distance run , the hypothenusa ▪ 〈◊〉 secant : the square sheweth the departure ; and the line of lines on the head-leg , the difference of latitude : and you may not only perform the work , but also see the reason thereof , being a help to the fancy of young learners in these nautical operations : and if your square playes true , you may be more exact than you can by scale and compass , and much more quick and ready ; not only in this , but any thing else in right-angled plain triangles , as in heights and distances , and the like . use v. the use of the meridian line , and his scale . these six problems of plain sayling for short distances , may come very near the matter ; as in making a traverse of the ships way from place to place coasting , as in the streights , and the channel , and the like : but for great distances , it is not so certain as the sayling by mercators chart ; therefore to that purpose the meridian-line was invented , to reduce degrees on the globe , to degrees in planò , as mr. wright hath largely shewed . on the innermost-edge of the rule , or trianguler-quadrant , you may have a meridian-line so large , as to have half an inch for one degree of the equinoctial ; and the inches for measure , to go along by it ; or rather you may have it lie near to the line of lines on the head-leg , as you shall think most convenient , for then it will be the same as mr. gunter's is , and perform his very operations , as near as may be , after his way , by the thred and compasses , or index and compasses . problem i. two places being propounded , one under the equinoctial , the other in any latitude , to find their meridional difference in degrees and minuts , or 100 parts . look for the latitude of the place , scituate out of the equinoctial in the meridian-line , and right against in the equal-parts is the meridional difference of those two places . example . let the river of amazones , under the equinoctial , be one place ; and the lizard , in the latitude of 50 degrees north , another place ; look for 50 on the meridian-line , and right against it , on the equal-parts , is 57-54 , for 57 degrees 54 minuts ; or in decimal parts of a degree 57-90 . problem ii. any two places having both southerly or northerly latitude , to find the meridional difference between them . extend the compasses on the meridian-line , from one of the latitudes to the other ; the same extent laid from the beginning of the scale of equal-parts , by the meridian-line , shall reach to the meridional difference required . or , the measure from the least latitude , to the beginning on the meridional-line , shall reach the same way from the greater , to the difference on the equal-parts . example . if the latitude of one place be 30 degrees , and the other 50 degrees ; extend the compasses from 30 to 50 on the meridian-line , and that extent shall reach on the equal-parts , from the beginning of the line , to 26 degrees 26 minuts . problem iii. when one place hath south latitude , and the other north latitude , to find the meridional difference . extend the compasses from the beginning of the line of meridians , to the lesser latitude ; then that extent applied the same way on the meridian-line from the greater latitude , shall shew on the scale of equal-parts the meridional difference required . example . suppose one latitude be 10 deg . south , and the other 30 deg . north ; the extent from 0 to 10 degrees , shall reach from 30 , to 41° 31′ , the meridional degrees required . problem iv· the latitudes of two places , together with their difference in longitude being given , to find the rumb directing from one to the other . as the meridional difference in latitude , to the difference in longitude ; so is the tangent of 45 , to the tangent of the rumb or course . example . let one place be in the latitude of 50 north , the other in 15 deg . and 30 min. north , as the lizard-point , and st. christophers ; and let the difference in longitude be 68 degrees , 30 minuts ; and let the rumb , leading from the lizard to st. christophers , be required· first , by the meridian-line , and the scale of equal-parts , by problem ii. find the meridional difference in latitudes , which in our example will be 42 degrees , and 12 parts of a 100. then , the extent on the line of numbers , from 42 degrees and 12 minuts , the meridional difference in latitude , to 68 degrees and 30 minuts , the difference in longitude shall reach the same way from the tangent of 45 , to the tangent of 58 degrees and 26 minuts , the rumb from the meridian of the lizard westwards , being two degrees , and better , beyond the 5th rumb from the meridian . by the trianguler-quadrant thus ; as the — tangent of 45 , taken from the loose-piece , is to the = meridional-difference in latitudes on the line of lines ; so is the = difference in longitudes , to the — tangent of the course 58 degrees 25 minuts . but by the index and square , this is wrought very easily and demonstratively thus ; count the meridional difference of latitudes on the head-leg down-wards from the center , as 42 and 12 on the line of lines , and set the square to it . then , count the difference of longitudes on the square , viz. 68-50 , and to that point lay the index ; and then the index on the degrees shall cut the complement of the course , viz. 31-35 , or 58-25 , if you count from the head. having been so large in this , i shall contract the rest . problem v. by the two latitudes and the rumb , to find the distance on the rumb . as the co-sine of the rumb , to the true difference of the latitudes , ( on the numbers ) ; so is the sine of 90 , to the distance on the rumb required , ( on the numbers ) . being given in degrees and decimal parts , and brought to leagues by multiplying by 20 , or 24 , according to mr. norwood , as before . note also , that the true difference of latitudes , is found by substraction , of the less latitude out of the greater . by the quadrant . as — true difference of latitudes , to = co-sine of the course or rumb ; so is = sine of 90 , to — distance on the rumb ( in the same line of lines ) . the index and square is used as in the second problem of plain sayling . problem vi. by the two latitudes , and distance between two places given , to find the rumb . as the distance sayled , in the degrees and 100 parts , counted on the lins of numbers , is to the true difference of latitudes , found as before , by substraction ; so is the sine of 90 , to the co-sine of the rumb required . as — sine of 90 , to = distance sailed ; so is = difference of latitudes , to — co-sine of the course . by the index and square , work as in problem iv. of plain sayling . problem vii . both latitudes and the rumb given , to find the difference of longitude . as the tangent of 45 , to the tangent of the rumb ; so is the meridional difference of latitudes , to the difference of longitude required . as — tangent of 45 , to = tangent of the rumb , ( first laid on the lines from the loose-piece ) ; so is the — meridional difference of latitudes , to the difference of longitudes . by the index and square , work as in the 4th problem last past . problem viii . by one latitude , distance and rumb , to find the other latitude . as sine 90 , to the co-sine of the rumb ; so is the distance , to the true difference of latitude . as — co-sine of the course , to = sine 90 ; so is = distance , in degrees and parts , on the lines , to the — true difference in latitudes , to be added or substracted from the latitude you are in , according as you have increased , or depressed the latitude in the voyage . by the index and square , work as in the 5th problem last past , or 2d or plain sayling . use vi. to find the distance of places on the globe of earth and sea ; or , geography by the trianguler-quadrant . problem i. when two places are scituated under the same meridian ( or longitude ) and on the same side of the equinoctial ; then substract the lesser latitude out of the greater , and the remainder shall be the distance in degrees required , counting 20 ( or 24 ) leagues to a degree on every great circle of the sphear . problem ii. when one place is on one side of the equinoctial , and the other on the other side ; and yet both on one meridian , as was the former ; then the two latitudes ( viz. the north-latitude , and the south-latitude ) added together , shall give the distance in degrees required . problem iii. when the two places differ only in longitude , and are both under the equinoctial , then substract the lesser longitude from the greater , and the residue is the distance in degrees . problem iv. when the two places have both one latitude , or near it , north or south , and differ only in longitude . then work thus ; as sine 90 , to co-sine of the ( middle ) latitude ; so is the sine of half the difference in longitude , to the sine of half the distance . by the trianguler-quadrant , or sector . as — co-sine of the mean latitude , to the = sine of 90 , laying the thred to the nearest distance ; so is = sine of half the difference in longitude , to — sine of half the distance . problem v. when both places have different longitudes and latitudes , as these three wayes following ▪ i way . when one place hath no latitude , and the other north or south , with difference in longitude also ; then , as sine 90 , to co-sine of difference in longitude ; so the co-sine of the latitude , to the co-sine of the distance required . by the trianguler-quadrant , thus ; as — co-sine of difference in longitudes , to the = sine of 90 ; so the = co-sine of the latitude , to the = co-sine of the distance . ii way . when both the places have either north or south latitude , that is , both toward one pole ; then thus , as the sine of 90 , to the co-sine of the difference in longitude ; so the co-tangent of the lesser latitude , to tang. of a 4th ark. which 4th ark , must be taken out of the complement of the greater latitude , when the difference of longitudes is less than a quadrant , or added to it when more , then the sum or difference shall be a 5th ark. then , as the co-sine of the 4th ark , to co-sine of the 5th ark ; so is the sine of the lesser latitude , to the sine of the distance . by the trianguler-quadrant . as — co-sine of difference in longitudes , to = sine of 90 ; so is the = co-tangent of the lesser latitude , taken from the loose-piece , and laid from the center , and from thence taken parallely to the — tangent of a 4th ark , which do with , as before is shewed , to find a 5th ark. and then , as the — co-sine of the 4th ark , to the = co-sine of the 5th ark ; so is the — sine of the lesser latitude , to = co-sine of the distance . iii way . but when one latitude is on one side the equinoctial , and the other on the otherside , viz. one having north-latitude , and the other south . then , as the sine of 90 , to the co-sine of the difference in longitude ; so is the co-tangent of one latitude , to the tangent of a 4th ark. which taken out of the other latitude , and 90 deg . added , when the difference of longitude is less than a quadrant , but added to it if more than a quadrant , and that sum or difference shall be the 5th ark. then , as the co-sine of the 4th ark , to the co-sine of the 5th ark ; so is the sine of the latitude , first taken , to the co-sine of the distance in degrees . by the trianguler-quadrant , or sector ; as the — co-sine of the difference of longitudes , to = sine of 90 ; so the = co-tangent of one latitude ( being first taken from the loose-piece , or moveable-leg , and laid from the center downwards , and from thence taken parallely ) to the — tangent of a 4th ark. which 4th ark you must do with , as before , to obtain a 5th ark. then , as — co-sine of the 4th ark , to = co-sine of the 5th ark ; so — sine of the latitude , first taken , to = co-sine of the distance . that is , when the 4th ark is substracted ; or , to the co-sine of the comp . distance when added . example . suppose i would know how far it is from the lizard , to the cape of good hope ; the lizard having 50 degrees of north-latitude , and the cape of good hope 35 degrees of south-latitude , and the difference in longitude 30 degrees . as the sine of 90 , to the co-sine of the difference in longitude 30 , being best counted from 90 backwards ; so is the co-tangent of 50 , ( viz. at 40 ) to 36 degrees 01 minut , a 4th ark. then 90 degrees , and 35 degrees , the other latitude added , makes 125 ; from which sum , taking the 4th ark , remains 88-59 , for a 5th ark. then say , as the co-sine of the 4th ark 36-1 , to the co-sine of the 5th ark 88-59 ; so is the sine of 50 , the latitude first taken , to the co-sine of the distance 89 deg . 3 min. the nearest distance in the arch of a great circle . note , that here you will have occasion to make use of that help mentioned p. 218 , sect. 3. as thus for instance . the proportion being as the co-sine of 36-1 , to the co-sine of 88-59 ; which is all one , as the sine of 54 and 59 , to the sine of 1 degree and 1 minut , which is too large for ordinary compasses , on ordinary gunters rules ; therefore first lay the distance from the sine of 90 , to the sine of 54 and 59 , the same way from the sine of 5 degrees and 45 minuts , and note the place . also , lay the distance from the sine of 90 , to the sine of 50 , the same way from the sine of 5 degrees and 45 minuts ; and note that place also . then , as the extent first noted for 54-59 , is to 1 degree and 1 minut , the co-sine of 88-59 ; so is the second mark noted for 50 , to 89-3 , the distance in degrees required . which multiplyed by 72 , gives the distance in the arch of a great circle , viz. 6412 miles statute-measure ; or , 5340 miles , whereof 60 make one degree , on a great circle on the superficies of the sea. use vii . to find the distance of places by the natural versed sines in the way of a sector on the trianguler-quadrant , being much more easie than the two former wayes . first , by the pen , find the sum and difference of the complements of the two latitudes , and count that sum and difference on the versed sines latterally , and take the distance between your compasses , and make it a parallel versed sine of 180 degrees . or , by the trianguler-quadrant . if you have not the line set on from the leg-center , then the small line of sines beyond the leg-center , being doubled , will do the work , by taking the distance between the sum and difference , and setting one point in the center-prick at two times the radius of the sines from the leg-center ; and then laying the thred to the nearest distance , or the line of right asceneions under the months , is a fit line . then , take out the = difference of longitude , and that shall reach latterally from the difference to the distance required . example . london and ierusalem , two places in north latitude ; london 51-32 , ierusalem 32-0 , whose two complements 38-28 and 58 added , make 96-28 for a sum , and one taken from the other , leave 19-32 for a difference . now the — distance between the versed sines of 96-28 , and 19-32 , make a = versed sine in 180 , keeping the sector so , or laying the thred to the nearest distance , ( and noting where it cuts in degrees ) . then , the = distance between 47 , the difference of longitude between the two places , shall reach on the versed sines from 19-32 , the difference to 39-14 , the distance required ; which , at 72 miles to a degree , makes 2805 miles . note , this one rule comprehends all the three last wayes , and is not troubled with half so many cautions as the former . use viii . having the latitudes and distance of two places , to find their difference in longitude . find the sum and difference of the two co-latitudes , as before , by addition and substraction ; count them on the versed sines , and take the — distance between , and make it a = versed sine of 180. then , the — distance , between the difference and distance on versed sines , shall stay at the = difference in longitudes required . example . let one place be burmudas isle , and the latitude thereof 32-25 ; let the other place be the lizard-point , and the latitude thereof 50 degrees ; the co-latitudes are 57-35 and 40-0 ; the sum of them is 97-35 ; the difference between them is 17-35 . the distance in the great circle , according to mr. norwood , is 44-30 , or 886 leagues , counting 20 leagues to one degree . then , the — distance between the versed sines of 17-35 , and 97-35 made a = versed sine of 180 , the sector is set . then , the — distance taken between 44-30 , and 17-37 on the versed sines , and carried parallelly , shall stay at 55 , the difference in longitude required between those two places . chap. v. of sayling by the arch of a great circle . in the book called , the geometrical seaman , by mr. phillips , is a very ready figure to shew in a quadrant , or more , by what longitudes and latitudes a ship is to pass in any long-run , which is contained under 90 degrees , or 120 difference of longitude , and the two places having both north latitude . which figure , or quadrant , is neatly and readily performed by the trianguler-quadrant , thus ; upon the back-side of the index , before spoken of , may be graduated from the center , two tangent-lines , one equal to the radius on the loose-piece , the other to the radius on the moving-leg ; then in the use , count the fiducial line in which the leg-center-pin stands , alwayes for the meridian of one place ; and some where in that line , according to the latitude thereof , counting the leg-center the pole of the world ; and the index being hung thereon , by the tangents prick down the latitude ; there , i say , knock in a pin to stay a thred for one place ; then , on the degrees , count the difference of longitude from the head-leg , and lay the index to it , and bring the thred fastened , as before , till on the index it cuts the degree and part of the other latitude , and there make the thred fast with another pin in the loose-piece . then , if you move the index to any degree of longitude between those places , the thred shall cut on the index the degree of latitude that answers unto it ; or if you make the thred cut any degree of latitude , the index gives the longitude required for that latitude . note , if the latitude be small , as between 10 and 30 , the small tangents are most convenient ; but if it be between 40 and 80 , the greater tangent line is best . note , that two threds and a pair of compasses may serve ; but the index is much better and quicker in operation . example ▪ let the two places be the summer-islands and the lizard-point ; the same example that you find in mr. norwood , pag. 126 ; and in mr. phillip's geometrical-seaman , pag. 55. that you may the more readily compare the truth thereof by their operations . the latitude of the lizard point is 50 degrees , the longitude is 10. the latitude of the summer islands is 32-25 , the longitude is 300. the difference of longitudes is 70 , as is computed by their observation . then , hanging or putting the center-hole of the index over the leg-center-pin , and counting the fiducial-line on the head-leg for the meridian of one place , count on the tangent line on the index the co-tangent of one latitude , as suppose the latitude of the lizard-point ( the center alwayes counted as 90 ) and there knock in a pin in a small hole to hang a thred on . then count 70 degrees , the difference in longitude , on the degrees from the head-leg , and there stay it ; then draw the thred put over the first pin , till it cut the complement of the other latitude , and by help of another pin stay it there , which you may conveniently do by one of the sliding-sights ; then the thred being so laid , slide the index to every single degree , or fifth degree of longitude , and then the thred shall shew on the index , the co-tangent of the latitude answerable to that degree of longitude , as in the table annexed . also , if you would have equal degrees of latitude , and would find the longitude according to it ; then slide the index to and fro till the thred cuts on the index an even degree of latitude ; then on the degrees you have the difference of longitude from either place . also note , that the drawing of one line only on the trianguler instrument in the beginning , according to the directions of laying of the thred ; with the thred and compasses , will perform this work also . the table . long d.l. latitude 300 09 32 — 25 305 05 35 — 52 310 10 38 — 51 315 15 41 — 24 320 20 43 — 34 325 25 45 — 24 330 30 46 — 54 335 35 48 — 07 340 40 49 — 04 345 45 49 — 47 350 50 50 — 15 355 55 50 — 31 360 60 50 — 33 05 65 50 — 23 10 70 50 — 00 if this work fit not any case that may happen , there is another way mentioned in page 75 of the geometrical seaman , by the steriographick projection ; and that scheam is drawn the same way , as the horizontal-projection for dyalling was , and somewhat easier ; and any two points given , in a circle , you may draw a great circle to cut them , and the first circle into two equal-parts , by the directions in page 15 ; and the application thereof you have very plainly in mr. phillips his book , to which i refer you , having said more than at first i intended , which was chiefly the use thereof in observation only . so for the present i conclude this discourse , and shall endeavour a further advantage in the next impression , according as time and opportunity shall offer . farewel . the end of the second part. the table of the things contained in this second part. the difinition and kind of dials , page . 7 directions to draw the scheam , 9 to draw lines to represent the several sorts of plains in the scheam , 13 to draw a scheam particularly for one dial , 14 to draw the equinoctial dial , 19 to try when a plain lies equinoctial , 20 to draw a polar-dial , 21 to draw an erect east or west-dial , 24 to draw a horizontal-dial , 27 the d●monstration of the canon for hours , ib. to draw a direct erect south or north-dial , 30 to draw a direct recliner , 33 the use of the figure , 35 to draw a direct east or west recliner , 37 to make the table of arks at the pole , 42 to refer those dials to a new latitude , and a new declination , wherein they may become erect decliners , 46 to find the requisites by the scheam , ibid. to find the declination of a plain by the needle , or by the sun , 49 to take off an angle , or set the sector to any angle required , 53 precepts to find the declination by the sun , and examples also of the same , 58 to draw an erect declining-dial , 62 the proportions for the requisites of erect decliners , 64 to find the requisites three wayes , 66 to draw the erect south decliner , 67 to draw the lines on a north decliner , 70 to draw the hour-lines on a plain , that declines above 60 degrees ▪ 73 of declining reclining plains , 77 the first sort of south recliners , 79 the second sort of south recliners , being polars , 90 the third sort of south-recliners , 98 the first sort of north recliners , 106 the second sort of north recliners , being equinoctial , 114 the third sort of north recliners , 119 of inclining di●ls , 126 to find the useful hours in all plains , 130 to draw the mathematical ornaments on all sorts of dials , 134 to draw the tropicks , or length of the day , 136 to make the trygon , 138 to draw the planetary or iewish hours , 142 to draw the italian hours , 144 to draw the babylonish hours , 145 to draw the azimuth lines , 146 to draw the almicanters , 154 to draw the circles of position , or houses , 160 to draw the hours , and all the rest , on the ceiling of a room , 165 the figure of the instrument , explained , 166 a table of the suns azimuth , at every hour and quarter , in the whole signs , 168 a table of the suns altitude the same time , 169 the description and use of the armilary-sphear for dyalling , several wayes , 172 the description and use of the poor-man's dial-sphear for dyalling , and several uses thereof , 203 how to remedy several inconveniences in the use of the gunter's rule . 220 the use , and a further description of the trianguler-quadrant , for navigation , or observation at sea , 227 for a fore-observation with sights , 233 for a fore-observation with thred and plummet , 235 for a back-observation , as a davis quadrant , 237 to find the suns distance from the zenith , or the co-altitude , 238 to find the altitude , when near the zenith , 239 to find a latitude by a forward observation , as with a gunter's bow , 241 to find the latitude by a back observation , 247 to find the latitude by a meridian observation , with thred and plummet , 252 to find the suns declination , 254 a table of the suns declination for the second after leap-year , 256 , 257 a table of the suns declination for every 5th day the intermediate years , 258 , 259 a rectifying table for the intermediate years , 260 a table of the magnitudes declinations and right ascentions of 33 fixed stars , in degrees , and hours , and minuts , 261 the use of the trianguler-quadrant , in the operative part of navigation , 267 of sayling by the arch of a great circle , 300 finis . errata for the second part. page 6. line 10. for too , read to . p. 18. l. 10. f. h , r. the ends of the arch qp . p. 22. l. 14. f. begins , r. being . p. 15. l. 27. f. latitude , r. co-latitude p. 34. l. 8. f. sun , r. sum . p. 39. l. 24. f. incliner , r. inclination of meridians . p. 61. l. 23. f. place , r. plain . p. 62. l. 20. f. ☉ r. q. p. 66. l. 2. f. i , r. t. p. 69. l. 7. f. 12. r. 7. p. 92 l. 22. r. gives a mark near e , whose measure on the limb from , b. p. 87. l. 8. f. gi●es , r. gives . l. 11. add at r near c. p. 93. l. 25. f. fe . r. pe. p. 100. l. 21. add cd next gives . p. 10● . l. 6. f. 8-5 , r. 8-3 . p. 105. l. 19. f. use , r. have . p. 108. l. 6. f. pole , r. zenith . p. 112. l. 3. f. cuts 12 , r. cuts the substile . p ▪ 113. l. 19. f. df , r. de. l ▪ 19. f. t. r. ct . also in l. ●8 . r. ct . p. 122. l. 6. f. e , r. f. p. 122. l. 13 , 14 , 15 , 16 , add sine . p. 128. l. 26. f. i , r. l. p. 139. l. 21. add , as in this example . p. 140. l. 6. add to . p 170. l. 10. f. divides , r. divided . p. 181. l. 20. f. popsition , r. proportion . p. 119. l. 27. f. from , r. on . p. 193. l. 6. f. being , r. bring . p. 197 l. 4. f. elevation , r. inclination . p. 200 l. 24. f. c , r. g. p. 204. l. 3. f. f , r. e. p 209. add in the last line , or by the upper part of the plain . p. 124. l. 18. add , or remove the thred to turn it further when it reclines beyond the pole. p. 238. l. 7. add of . l. 20 add but. p. 247. l. 13. r. and much better in small latitudes . p. 248. l. 5. f. wo , r. two . p. 251. l. 14. f. 20 , r. 33. also , l. 17. f. 40 , r. 27. advertisements . the use of these , or any other instruments concerning the mathematical practice , or further instructions in any part thereof , is taught by iohn colson near the hermitage-stairs . also by euclide speidwel , dwelling near to white-chappel church in capt. canes rents , or at the custom-house . also , by william northhall mariner , dwelling at the crooked-billet in meeting-house alley on green-bank near wapping . also the instruments may be had at the house of iohn brown , iohn seller , or iohn wingfield , as in the title-page is expressed . there is now extant a large treatise of navigation in folio , describing the sea-coasts , capes , head-lands , the bayes , roads , rivers , harbours and sea-marks in the whole northern-navigation ; shewing the courses and distances from one place to another , the ebbing and flowing of the sea , with many other things belonging to the practick part of navigation . a book ( not heretofore printed in england ) collected from the practice and experience of divers able and experienced navigators of our english nation . published by iohn seller , hydrographer to the kings most excellent majesty , and to be sold by him at the sign of the marine●s compass at the hermitage-stairs in wapping , and by iohn wingfield right against st. olaves church in crouched-fryers . practical navigation ; or , an introduction to that whole art. sold by iohn seller and iohn wingfield aforesaid . notes, typically marginal, from the original text notes for div a29762-e2070 see the general-scheam .