the use of the universal ring-dial. worgan, john, surveyor. 1696 approx. 2 kb of xml-encoded text transcribed from 1 1-bit group-iv tiff page image. text creation partnership, ann arbor, mi ; oxford (uk) : 2008-09 (eebo-tcp phase 1). a96930 wing w3581a estc r186866 47683577 ocm 47683577 173032 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. a96930) transcribed from: (early english books online ; image set 173032) images scanned from microfilm: (early english books, 1641-1700 ; 2661:37) the use of the universal ring-dial. worgan, john, surveyor. 1 sheet ([1] p.). s.n.], [london : 1696. attributed to john worgan by wing (2nd ed.). "this instrument, or any other useful for the mathematicks, are made and sold by tho. walpoole at the sign of the mariner and compass in the minories. 1696." the words "mariner and compass" have been lined through and replaaced in ms. by "unicorn." reproduction of original in: william andrews clark memorial library, university of california, los angeles. 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 sundials -early works to 1800. advertising fliers -early works to 1800. broadsides -england -london -17th century. 2007-06 tcp assigned for keying and markup 2007-06 apex covantage keyed and coded from proquest page images 2007-07 pip willcox sampled and proofread 2007-07 pip willcox text and markup reviewed and edited 2008-02 pfs batch review (qc) and xml conversion the use of the universal ring-dial . to find the hour of the day . set the stroke in the sliding piece , to the latitude of the place , and set the stroke crossing the little hole in the bridge or middle piece , to the day of the month ; then open the rings , and hold them up by the little ring , and move the bridge , or middle piece , so towards the sun , according as you may think to be near the hour , and move it gently this way and that , till the sun shining thorow the little hole in the bridge , you can discern a little ray or speck of light , to fall upon the aequinoctial , within side , among the hours ; and the point in the middle line , whereon the ray or speck falleth , is the true hour of the day . to find the altitude of the sun. set the stroke in the sliding piece , to the beginning of degrees , and put a pin thorow the hole on the back-side of the ring , and the shadow of the pin shall shew the altitude on the large quadrant on the back-side . to find the suns declination . set the stroke crossing the little hole in the bridge , to the day of the month , and against it , on the other side , is the sun's declination . to find the latitude of the place . first find the sun's meridian , or greatest altitude for that day observ'd ; then find the sun's declination for that day . if that you make your observation in the summer half year , viz. from the 10th of march to the 12th of september , then you must substract the declination from the altitude ; but if you observe in the winter half year , viz. from the 12th of september to the 10th of march , then you must add the declination of the altitude , and either of those numbers is the height of the aequinoctial , which substract from 90 degrees , is the latitude of the place . this instrument , or any other useful for the mathematicks , are made and sold by tho. walpoole at the sign of the mariner and compass in the minories 1696. a table of the equation of days, shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial, every day of the year. tompion, thomas, 1639-1713. 1683 approx. 17 kb of xml-encoded text transcribed from 1 1-bit group-iv tiff page image. text creation partnership, ann arbor, mi ; oxford (uk) : 2009-10 (eebo-tcp phase 1). b06166 wing t1862a estc r185376 53299330 ocm 53299330 180055 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. b06166) transcribed from: (early english books online ; image set 180055) images scanned from microfilm: (early english books, 1641-1700 ; 2811:22) a table of the equation of days, shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial, every day of the year. tompion, thomas, 1639-1713. 1 sheet ([1] p.) printed for tho. tompion, clockmaker ..., london : 1683. caption title. reproduction of original in: national library of scotland. 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 clocks and watches -calibration -early works to 1800. time, equation of -early works to 1800. sundials -early works to 1800. broadsides -england -17th century. 2008-01 tcp assigned for keying and markup 2008-07 spi global keyed and coded from proquest page images 2008-09 megan marion sampled and proofread 2008-09 megan marion text and markup reviewed and edited 2009-02 pfs batch review (qc) and xml conversion a table of the equation of days , shewing how much a good pendulum watch ought to be faster or slower than a true sun-dial , every day in the year . days . januar. februa . march april . may. june . july . aug. sept. octob. nov. dec. mi.   sec. mi.   sec. m.   sec. m.   sec. m.   sec. m.   sec. m.   sec. m.   sec. mi.   sec. mi.   sec. mi.   sec. m.   sec. 1 8 watch too fast . 52 14 watch too fast . 46 10 watch too fast . 08 0 w. too f. * 46 4 watch too slow . 12 1 w. too slow . * 02 4 watch too fast . 52 4 watch too fast . * 42 3 watch too slow . 40 13 watch too slow . 15 15 watch too slow . 29 5 watch too slow . * 53 2 9 14 14 45 9 51 0 30 4 14 0 49 4 59 4 32 4 00 13 28 15 21 5 25 3 9 36 14 43 9 34 0 14 4 14 0 36 5 06 4 21 4 21 13 42 15 12 4 57 4 9 58 14 40 9 17 0 * w. too slow . 01 4 14 0 24 5 13 4 11 4 42 13 55 15 02 4 27 5 10 19 14 36 9 00 0 17 4 14 0 12 5 20 4 00 5 03 14 08 14 51 3 57 6 10 38 14 32 8 42 0 32 4 13 0 * watch too fast . 01 5 27 3 48 5 24 14 20 14 40 3 28 7 10 58 14 27 8 24 0 46 4 12 0 14 5 33 3 36 5 45 14 32 14 27 2 59 8 11 17 14 21 8 06 1 00 4 10 0 27 5 37 3 23 6 06 14 43 14 14 2 30 9 11 35 14 14 7 47 1 14 4 08 0 40 5 41 3 10 6 26 14 53 14 00 2 00 10 11 52 14 07 7 28 1 28 4 05 0 53 5 44 2 56 6 47 15 03 13 46 1 29 11 12 09 14 00 7 09 1 40 4 02 1 07 5 48 2 42 7 08 15 12 13 30 0 59 12 12 26 13 52 6 50 1 52 3 58 1 20 5 51 2 27 7 28 15 21 13 13 0 28 13 12 40 13 43 6 32 2 04 3 54 1 33 5 54 2 12 7 49 15 29 12 56 0 * watch too fast . 02 14 12 53 13 33 6 13 2 16 3 48 1 46 5 55 1 56 8 09 15 36 12 38 0 32 15 13 06 13 23 5 54 2 27 3 43 1 58 5 56 1 40 8 29 15 42 12 18 1 02 16 13 18 13 12 5 36 2 37 3 37 2 11 5 56 1 23 8 49 15 48 11 59 1 32 17 13 30 13 01 5 17 2 47 3 30 2 23 5 56 1 07 9 09 15 53 11 39 2 01 18 13 42 12 49 4 58 2 57 3 23 2 36 5 55 0 50 9 29 15 57 11 18 2 31 19 13 51 12 36 4 38 3 06 3 15 2 49 5 54 0 33 9 49 16 00 10 56 3 00 20 13 59 12 23 4 19 3 15 3 07 3 01 5 52 0 15 10 08 16 02 10 34 3 29 21 14 08 12 10 4 01 3 23 2 59 3 12 5 50 0 * watch too slow . 03 10 26 16 04 10 11 3 57 22 14 16 11 56 3 42 3 30 2 51 3 23 5 47 0 22 10 44 16 05 9 48 4 25 23 14 23 11 42 3 23 3 37 2 43 3 34 5 43 0 41 11 02 16 05 9 24 4 53 24 14 29 11 28 3 05 3 43 2 33 3 45 5 39 1 00 11 20 16 05 8 59 5 20 25 14 33 11 13 2 46 3 49 2 22 3 55 5 34 1 19 11 37 16 04 8 34 5 48 26 14 37 10 57 2 28 3 54 2 10 4 06 5 28 1 38 11 54 16 01 8 08 6 15 27 14 41 10 41 2 11 3 58 1 58 4 16 5 22 1 58 12 11 15 58 7 42 6 41 28 14 44 10 25 1 53 4 02 1 46 4 25 5 15 2 18 12 28 15 54 7 14 7 07 29 14 45     1 36 4 06 1 34 4 34 5 07 2 39 12 44 15 49 6 47 7 33 30 14 46     1 19 4 08 1 24 4 43 4 59 2 59 13 00 15 43 6 20 7 58 31 14 46     1 02     1 13     4 51 3 19     15 36     8 22 set the watch so much faster or slower than the time by the sun , according to the table for the day of the month , when you set it ; and if the watch go true , the difference of it from the sun any day afterward will be the same with the table . london , printed for tho. tompion , clockmaker , at the three crowns in fleet-street , at water-lane end . 1683. the description and uses of the general horological-ring: or universal ring-dyal being the invention of the late reverend mr. w. oughtred, as it is usually made of a portable pocket size. with a large and correct table of the latitudes of the principal places in every shire throughout england and wales, &c. and several ways to find a meridian-line for the setting a horizontal dyal. by henry wynne, maker of mathematical instruments near the sugar-loaf in chancery-lane. wynn, henry, d. 1709. 1682 approx. 44 kb of xml-encoded text transcribed from 23 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-10 (eebo-tcp phase 1). a67225 wing w3778b estc r221060 99832437 99832437 36910 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. a67225) transcribed from: (early english books online ; image set 36910) images scanned from microfilm: (early english books, 1641-1700 ; 2143:12) the description and uses of the general horological-ring: or universal ring-dyal being the invention of the late reverend mr. w. oughtred, as it is usually made of a portable pocket size. with a large and correct table of the latitudes of the principal places in every shire throughout england and wales, &c. and several ways to find a meridian-line for the setting a horizontal dyal. by henry wynne, maker of mathematical instruments near the sugar-loaf in chancery-lane. wynn, henry, d. 1709. [4], 30, [2] p., [1] folded leaf :bill. (engraved) printed by a. godbid and j. playford, for the author, london : 1682. with a final leaf of advertisements. 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 oughtred, william, 1575-1660 -early works to 1800. scientific recreations -early works to 1800. sundials -early works to 1800. mathematical instruments -early works to 1800. 2005-02 tcp assigned for keying and markup 2005-02 aptara keyed and coded from proquest page images 2005-03 mona logarbo sampled and proofread 2005-03 mona logarbo text and markup reviewed and edited 2005-04 pfs batch review (qc) and xml conversion the description and uses of the general horological-ring : or universal ring-dyal . being the invention of the late reverend mr. w. oughtred , as it is usually made of a portable pocket size . with a large and correct table of the latitudes of the principal places in every shire throughout england and wales , &c. and several ways to find a meridian-line for the setting a horizontal dyal . by henry wynne , maker of mathematical instruments near the sugar-loaf in chancery-lane . london , printed by a. godbid and j. playford , s for the author , 1682. to the reader . i formerly published half a sheet on this subject , and having disposed of all i printed , i found my self necessitated to print more , to gratify those who bought the instruments of me , but considering with my self the scantiness of that paper , i took the pains to write a larger which should be more effectual , and as i hope will give better satisfaction . 't is confest that there is very little new in this ( as in most other books written now a days ) but what may be found among former authors . my chiefest care herein hath been to collect and alter so that it might serve my present purpose . as for the instrument it self , being carefully made and graduated as is here described , i know of none for portableness , universality , and exactness , that doth exceed it , i mean with respect to its finding the hour , whereby it becomes absolutely useful for any gentleman to carry in his pocket , or to rectify his watch or pendulum by it , &c. i have endeavoured to be as plain as possible for the sake of young beginners , that the reading of this might create in some a farther inclination to the mathematicks , which i heartily wish may flourish not only as they are my trade , and consequently it is my interest to promote them , but because they are of so great and general use and advantage to the kingdom . h. w. the description and uses of the general horological ring : or universal ring-dyal . 1. of the name . this instrument serveth as a dyal to find the hour of the day , not in one place only ( as most sorts of dyals do ) but generally in all countries whether northern or southern ; and therefore it is called the general horological ring , or vniversal ring-dyal . 2. the parts . it consists of these parts , viz. 1. a little ring and its slider to hang it by . 2. two circles which fold one within the other . 3. a diameter a cross in the middle . 4. to this diameter there is another slider . 3. the name of each part . the names given to the parts are : 1. the little ring and its slider is called the cursor of the meridian , and is represented ( figl . ) by the letter z. 2. of the two circles , the outermost m m m m , is called the meridian , and the innermost ae ae ae ae , is called the aequinoctial . 3. that which crosseth the middle noted with a a is called the bridge , or more properly the axis . 4 the slider within it noted c is called the cursor of the bridge or axis . 4. the divisions on each part . one side of this instrument according to fig. i. is thus divided . 1. the cursor of the meridian hath but one division or notch as at o. 2. one half of the meridian is divided into twice 90 degrees , which are again subdivided into halfs , and these halfs are distinguished from the degrees , by a shorter line , these degrees are numbered at every ten , from their middle o both wayes , by 10 , 20 , 30 , &c. to 90 , and in these degrees are the latitudes of places reckoned when you would find the hour of the day . 3. the aequinoctial is divided into 24 hours , and each hour is subdivided into eight parts , viz. halfs , quarters , and half quarters , and some of them have the hours divided into 12 parts , and then every division stands for five minutes of time , whereof 60 make one hour , these hours are numbred with i. ii. iii. &c. to twice xii . from the two opposite points in the meridian where this circle is fastned . 4. on this side the axis is divided into months and dayes , every division expressing 2 days , except in june and december , at which time the alteration of the suns course is almost insensible for several days together , these months are known on one side the slit by these letters , i. f. m. a. m. i. signifying , january , february , march , april , may , june , on the other side by these , i. a. s. o. n. d. for july , august , september , october , november , december . 5. the cursor of the axis hath a little hole through it and a line a cross the hole , which line when it is used is to be set to the day of the month. the other side according to figure ii. hath only the meridian and the axis divided 1. the meridian hath a quadrant or 90 degrees divided on it , whose center is at h. these degrees are again subdivided into halfs , and this i call the quadrant of altitudes , it serving to give the altitude of the sun , by the shadow of a pin , or such like wire , which shall be stuck upwright in the center or hole h. 2. the axis on that side the slit d is divided into twice 23 ½ and numbered both ways from the middle o by 10 , 20 , &c and this is called the line of declination , its use being to give the declination of the ⊙ , &c. on the other side the slit r , are divided four hours and a half , which are again subdivided , numbred by iiii / 8 v / 7 vi / 6 vii / 5 viii / 4 and this line is to shew the sun 's rising and setting at london , but because it is particular this line is left out in most dyals . the cursor on this side as on the other hath the little hole and a line a cross it . besides these divisions on each side , on the inside the aequinoctial , in the middle , is a line upon which is graduated the 24 hours , and parts agreeable to those on the side described in fig i. note that the instrument thus made is general , and will serve wheresoever you are , and therefore most proper for seamen and those that travel far . but for such as shall use them about these his majesties dominions , it will be sufficient to have but one quadrant of latitudes graduated , and no more than 18 hours or thereabouts , viz. from 3 in the morning to 9 at night , and then the instrument may be afforded so much the cheaper . vses of the instrument . the principal uses of this instrument ( although larger may be made to perform many more ) are as followeth . 1. knowing the day of the month to find the suns declination . 2. to find the altitude of the sun at the meridian and all hours . 3. by knowing the suns declination and meridian altitude , to find the latitude of any place . 4. to find the hour of the day . 5. to find at what time the sun rises and sets on any day at london or any other place lying under the same latitude . 6. to find what days and nights throughout the year are equal . use i. to get the suns declination by knowing first the day of the month . explanation . the sun moves not alwayes in the aequinoctial , but declines from it sometimes toward the north , and sometimes towards the south , every day , either moving in it or in circle parallel to it , this diversity of motion is called the suns declination , now about the 10 day of march and 13 of september the suns course is in the aequinoctial , and then he is said to have no declination , and from the 10 of march to the 13 of september , the sun moves on the north side the aequinoctial , and it called his northern declination , also from the 13 of september to the 10 of march his motion being on the south side , is called southern declination . by this variety of the suns motion , is caused the diversity of seasons and inequalities of day and night . note also , that the greatest declination on either side exceeds not 23 degrees and ½ . now to find it , the rule is : slide the cursor of the axis to the day the month , and then turn it on the other side , and the division crossing the same hole will shew the suns declination in the line d. note that the axis may be turned without turning the whole dyal . example 1. march the 10 , i slide the cursor to the day of the month , and turning the other side , the division stands at o , which shews the sun hath no declination that day , but moves in the aequinoctial . example 2. april the 8 , i slide the cursor to the day of the month , and turning the otherside , the division shews 11 degrees to be the suns declination on that day northward . example 3. october the 20 , the cursor being set to the day , on the other side it will shew 14 deg. for the suns declination on that day to the southward . use ii. to find the suns altitude on the meridian and all hours . explanation . the altitude or height of the sun is the the number of deg . contained between the middle or center of the sun , and the horizon or circle which bounds our sight , and the meridian altitude is its height every day just at 12 a clock , the sun at that time coming to touch the meridian . to find it , the rule is : when the sun shines slide the division on the cursor of the meridian to the beginning of the degrees in fig. i. marked with o , then turn the dyal and stick a wire or pin upright in the hole h , fig. ii. and holding it by the little ring turn it gently towards the sun , so that the shadow of the pin may fall among the degrees in the quadrant of the altitudes , now the deg. whereon the shadow falleth is the suns altitude at that time , but to know the meridian altitude you must observe the suns height just it 12 , now that you may be sure to have it right make several observations just about 12 , and the greatest is the truest , for as the sun all the morning from its rising grows higher and higher untill it comes to the meridian where it is highest , so having past the meridian , all the afternoon it grows lower and lower until it sets : wherefore the suns greatest altitude on any day is the meridian altitude for that day . examples .     deg . m. march the 10 th . the suns meridianaltitude , at londonwill be found by the foregoing rule to be 38 28 april the 8 th . 49 28 october the 20 th . 24 28 june the 11 th . 61 58 now before i proceed further to shew the uses , it will be necessary to explain some terms in astronomy , such as i shall here make use of , that the young practitioner may with more ease understand what follows . 1. degrees and minutes . and first what is meant by degrees and minutes . all circles according to astronomy are conceived to be divided into 360 parts , which are called degrees , every degree is subdivided into 60 minutes , every minute into 60 seconds , &c. so that one degree is the three hundred and sixtieth part of a circle , and one minute the 60th part of a degree , &c. now the whole circle containing 360 degrees , the half must contain 180 deg . the quadrant , or quarter part of a circle , contains 90 deg . so likewise one deg . containing 60 minutes , 45 min. are 3 quarters , 30 min. are one half , 20 min. one third part , 15 min. are one quarter , 12 min. are one 5 part , 10 min. are one 6 part , 5 min. are one 12 part , &c. on the meridian of the dyal fig. i. there are two quadrants , or twice 90 deg. graduated , one of which next n p is called the northern quadrant of latitudes , and serves for those places whose latitudes are on the north side the aequinoctial , the other is the southern quadrant , and serves in south latitudes . 2. meridian . it is a great circle imagined in the heavens , lying directly north and south , dividing them into two equal parts , the eastern and western , passing through both poles , and the zenith and nadir ; to this circle when the sun cometh at all times it is noon or midnight , and note that every place hath a several meridian , except such as ly directly north and south one from the other . 3. poles . the poles are two imagined points in the heavens opposite to each other , one north the other south . 4. axis . a right line imagined to run from one pole to the other , is called the axis . 5. zenith . the zenith or vertex is the point in the heavens directly over our heads . 6. nadir . the nadir is the opposite point to the zenith , it being directly under our feet . 7. equinoctial . the equinoctial is a great circle imagined to run directly east and west , it exactly crosseth the meridian , and lyeth in the middle between the poles , and divideth the heavens into two equal parts , the northern and southern , when the sun moves in this circle , which is twice a year , the days and nights are of an equal length throughout the world . 8. tropicks . the tropicks are two lesser circles dividing the heavens into two unequal parts , they are parallel to the equinoctial , and distant from it 23 deg . 30 min. one on the north side of it the other on the south , these circles are the utmost bounds of the suns declination . 9. latitude and eleva●●●● of the pole. the latitude of any 〈◊〉 is the number of degrees contained between the zenith of that place and the aequinoctial , which degrees are counted in the meridian , either on the north or south side of the aequinoctial , according as the place is situated . this latitude is always equal to the elevation of the pole , which is the number of degrees in the meridian contained between the pole and the horizon ; thus those that live under the aequinoctial are said to have no latitude , and those that live under the pole , if any such there be , are in 90 deg. of latitude ; hence also it is manifest , that those places which are situate directly east and west one from the other , have one and the same latitude . 10. colatitude . the compliment of the latitude is the number of degrees contained between the zenith and the pole , which is also the same with the distance between the aequinoctial and the horizon , or it is so much as the latitude wants of 90 deg. for subtract the latitude from 90 , the remainder is the colatitude . use . iii. by knowing the suns declination and meridian altitude to find the latitude . the rule . if the suns declination be north , subtract it from the meridian altitude , and the remainder is the colatitude , but if the suns declination be south add it to the meridian altitude , and the sum shall be the colatitude , which subtracted again from 90 deg. the remainder is the latitude . example 1. march the 10. the sun hath no declination , and i find the meridian altitude at london , to be 38 deg . 28 min. therefore 38 deg . 28 min. subtracted from 90 deg . the remainder is 51 d. 32 m. the latitude of london , and by this we see when the sun is in the aequinoctial , its meridian altitude is equal to the compliment of the latitude . example 2. april the 8. the suns declination is 11 deg . north and its meridian altitude 49 deg . 28 m. now subtract 11 deg . from 49. 28. there rests 38 deg . 28 min. which subtracted again from 90 there rests 51 deg . 32. min. the latitude required . example 3. october the 20. the suns declination is 14 d. south , and the meridian altitude is 24 d. 28 m. then add 14 d. to 24 d. 28 m. the sum is 38 d. 28 m. which subtracted from 90 d. there rests 51 d. 32 m. as before . example 4. thus if the declination were 23 d. 30. m. north and the meridian altitude 65 d. 10 m. the latitude would be found to be 48 d. 20. m. example 5. let the declination be 12 d. 15 m. south , and the meridian altitude 39 d. 40 m. the lat. would be 38d . 5 m. note that these rules hold good only for finding the latitudes of such places as ly to the north of the aequinoctial , for south lat. the contrary are true , for there if the declination be north , you must add it as you do now when it is south , and if the suns declination be south , you must subtract it as you do here when it is north. and least it be thought troublesome to find the lat. there is added at the end of this book a table of the latitudes of the principal places in england , scotland , and ireland . so that being near any of those places you may make use of the lat. of that place , for 10 or 20 miles in this case will make a very insensible or no alteration . use iv. to find the hour of the day . note that although the equinoctial fold up within the meridian to render the instrument the more portable , yet when you would find the hour , the aequinoctial must be drawn forth according to fig. iii. and 't is a little ray or speck of light that coming through the hole of the cursor of the axis falleth upon the line in the middle of the aequinoctial and sheweth the hour . the rule . first the latitude being got by the foregoing rules , or by the table at the end of this book , slide the division on the cursor of the meridian to it , either in the north or south quadrants , according as the place is situated . secondly slide the cursor of the axis to the day of the month . thirdly open the equinox as far as 't will go , which is just to cross the meridian , then guess as near as you can at the hour , and turn the axis towards the hour you guess , that the sun may the better shine through the hole , and holding the instrument by the little ring so that it may hang freely , move it gently this way and that , till the sun shining through the hole you can discern a little ray or speck of light to fall upon the aequinoctial within side among the hours and parts , now the point in the middle line whereon the ray falleth is the true hour . a little practice will make it very easie . fi. iii. representeth the dyal as it is when you would find the hour , where the cursor z is set to the lat. of london , 51 32. the cursor of the axis is set to the day being april the 8 , and the aequinox is drawn open to cross the meridian . now when the dyal is thus set , and shews the true hour , the meridian of it hangeth directly north and south , according to that imagined in the hea ens , the point n p represents the north pole , s p represents the south , the cursor z represents the zenith , and its oposite point n represents the nadir ; the axis lyeth according to that of the world passing from pole to pole , the points of vi and vi in the aequinoctially directly east and west , and the middle line within lyeth according to the true aequinoctial in the heavens . use v. to find the suns rising and setting . note this line of rising and setting is particularly for the latitude of london , or any other place , situated directly east or west from it , but it may indifferently serve the whole kingdom . note also that the great figures stand for the rising and the other for the setting . the rule . slide the cursor of the axis to the day of the month , then turn the other side , and the division crossing the hole , shews the suns rising , and setting in the line r. example 1. i slide the cursor to march the 10 , and on the other side it shews vi. and 6 , for then the sun rises at 6 and sets at 6. example 2. april the 8 , i set the cursor to the day , and on the other side it shews v. and 7 , which is 5 for the suns rising and 7 for its setting . example 3. october the 20 , the cursor being set to the day , on the other side it will shew the rising to be at a quarter after vii , and the setting three quarters after 4. now having found the suns rising and setting , you may likewife from thence find the length of the day and night , for double the time of the suns rising , and you have the length of the night , and double the time of its setting , gives you the length of the day , as will appear by the three following examples . example 1. march the 10 , the sun rises at 6 and sets at 6 , now twice 6 is twelve for the length both of day and night . example 2. april the 8 , the sun rises at 5 and sets at 7 , now twice 5 is 10 the length of the night , and twice 7 is 14 the length of the day . example 3. october the 20 , the sun rises at a quarter after 7 and sets at 3 quarters after 4 , now twice 7 and a quarter is 14 and a half for the length of the night , and twice 4 and 3 quarters is 9 and an half for the lenghth of the day ; in all which examples it appears that both the sums of the length of the day and night being added together will make 24 , the hours contained in a natural day . use vi. to find what days and nights throughout the year are equal . the rule . the days on one side the slit are equal to the days on the other . example . slide the cursor to march the 10 , and the day equal to it will be found on the other side sept. the 13 , so equal to april the 8 is august the 14. and the day equal to the 20 of october is february the second . now these days are said to be equal each to the other , in these respects ; 1. in respect of the suns declination , it being on both the same . 2. of the suns altitude , for what altitude the sun has on any hour on one , the same will be its altitude on the same hour on the other . 3. the time of the suns rising and setting is on both the same . 4. they are equal in length both of day and night . a table shewing the latitudes of most of the principal places in every shire throughout england and wales . shires . places names . d. m. anglesey , beaumaris , 53 27 holy-head , 53 33 berkshire , abington , 51 42 newbery , 51 25 reading . 51 28 bedfordshire , bedford , 52 09 dunstable . 51 53 brecknockshire , bealt , 52 12 brecknock . 52 04 buckinghamshire , alesbury , 51 45 buckingham . 52 00 cambridg●hire , cambridge , 52 06 ely. 52 30 cardiganshire , aberistwith . 52 35 cardigan . 52 20 carmarthenshire , carmarthen , 51 58 kidwelley . 51 50 carnarvonshire , arberconway , 53 30 bangor , 53 21 carnarvon , 53 18 cheshire , chester , 53 15 nantwich . 53 03 clamorganshire , cardiff , 51 30 landaff . 51 34 cornwall , fallmouth , 50 20 the lizard , 50 10 truro . 50 25 cumberland , carlisle , 55 00 cockermouth . 54 45 derbyshire , chesterfield , 53 20 derby . 53 00 denbighshire , denbigh , 53 18 ruthyn . 53 12 devonshire , dartmouth , 50 20 exeter , 50 41 plymouth . 50 30 dorset shire , dorchester , 50 40 shaftsbury , 50 58 weymouth , 50 32 durham , aukland , 54 45 durham . 54 50 essex , colchester , 52 00 harwich . 52 05 flintshire , st. asaph , 53 25 flint . 53 20 gloucestershire , gloucester , 51 56 tewxbury . 52 15 hampshire , portsmouth , 50 45 southampton , 50 54 winchester . 51 03 hertfordshire , hertford , 51 50 ware. 51 48 herefordshire , hereford , 52 12 lemster . 52 24 huntingtonshire , huntington , 52 15 st. ives . 52 20 isles of gernsey , 49 38 jersey , 49 28 man , douglas , 54 25 wight , newport . 50 45 kent , canterbury , 51 15 dover , 51 25 rochester . 51 30 lancashire , lancaster . 54 15 manchester , 53 39 preston . 53 55 leicestershire , harborough , 52 33 leicester , 52 40 lincolnshire , boston , 53 16 lincoln , 53 16 stamford . 52 48 merionethshire , bala , 52 57 harlech . 53 00 middlesex , london , 51 32 stanes , 51 30 uxbridge 51 35 monmouthshire . chepstow , 51 42 monmouth . 51 54 montgomery , 52 40 montgomeryshire . montgomery , 52 40 welchpool . 52 50 norfolk , linn , 52 52 norwich , 52 44 yarmouth . 52 40 northamptonshire . northampton , 52 15 peterborough . 52 38 northumberland , barwick , 55 50 newcastle . 55 03 nottinghamshire , nottingham , 53 00 workensope . 53 25 oxfordshire , banbury , 51 57 oxford . 51 45 pembrookshire , st. davids , 52 00 pembrook . 51 48 radnorshire . prestein , 52 30 radnor . 52 25 rutland , okeham , 52 43 uppingham . 52 38 shropshire , ludlow , 52 28 shrewsbury . 52 48 somersetshire , bath , 51 20 bristoll . 51 30 staffordshire . lichfield , 52 48 stafford . 52 52 suffolk , st. edm. bury , 52 22 ipswich . 52 20 surrey , guilford . 51 14 suffex , chichester , 50 49 lewis . 50 46 warwickshire , coventry , 52 32 warwick . 52 28 westmoreland , apleby , 54 40 kendal . 54 24 wiltshire , marlborough , 51 25 malmsbury , 51 35 salisbury , 51 04 worcestershire , kidderminster 52 28 worcester . 52 15 yorkshire , bridlington , 54 50 doncaster , 53 38 hull , 53 48 leeds , 53 50 york . 54 00 the latitudes of the most eminent places in scotland . places names . d. m. aberdeen . 57 06 st. andrews . 56 24 barwick . 55 50 dunblain . 56 20 dunbriton . 56 10 dunbar . 56 03 dundee . 56 31 dunfrees . 55 03 edenburgh . 56 04 fair-head . 58 43 glascow . 56 05 irwin . 55 50 isles of orkney . 58 50 kaithness . 57 48 larnack . 55 51 montross . 56 44 nairn . 57 30 perth or st. johns town , 56 32 sterlin . 56 15 withern . 54 57 the latitudes of the most eminent places in ireland . places names . d. m. armagh . 54 23 athloon . 53 21 bantry . 51 30 belfast . 54 41 cashell . 52 24 casherlash . 52 46 clare . 52 44 corke . 51 43 craven . 54 01 droughdagh . 53 44 dublin . 53 20 dundalk . 54 02 dungarvan . 51 57 dunnagall . 54 40 galloway . 53 12 james town . 53 53 kildare . 53 08 kilkenny . 52 34 kingsail . 51 30 knockfergus . 54 50 limrick . 52 33 londonderry . 55 04 longford . 53 42 slego . 54 17 waterford . 52 09 wexford . 52 17 how to place an horizontal dyal upon a levell plane , and to find the meridian several wayes . 1. prepare a smooth board or stone , and place it truly horizontal or levell , which may be done with such an instrument as the artificers call a plumb-rule , or otherwise , then find the hour of the day by such an instrument as is before described , or by some other as true , or having a good watch go to some sun-dyal that you know to go true , and set the watch by it , afterwards turn the dial ( which you are to place ) about , untill it shews the same hour with your instrument or watch , and there fasten it . 2. or having prepared your plain as before , near the middle of it set up a wire which shall stand exactly perpendicular or upright , and the sun shining clear , observe a little before noon when the shadow of the wire is at the shortest , and there make a point , and through that point and the center where the wire stood draw a line , upon which place the 12 a clock line of your dyal , and fix it . 3. and which is better , near the middle of your plain choose a point as a center , and thereon describe a circle of a convenient bigness , and erect a wire at right angles to your plain as before , then observe in the forenoon when the shadow of the top of the pin just toucheth the circle , and there make a mark , and again in the afternoon watch when the shadow of the top of the pin just toucheth the circle , and there make another mark , then with a pair of compasses divide the space between those two markes into two equal parts , and there make a third mark , through this last point and the center of the circle where the wire stood , draw a line and it shall be a true meridian-line . this last conclusion may be done with more ease , if there be several circles described one within another on the same center , also then you may make several observations for the doing it with more certainty . 4. the meridian may be found by the help of a good magnetical needle , well made and fitted to a square box , if in the useing of it there be an allowance made for the variation , the use of which is so plain , even to those that have but seen them , that i think it needless here to treat of . i shall set down only two ways more , which will require more knowledge in the mathematicks than any of the former , and so conclude . the first is in dary's misscellanies , page 22 , thus . 1. let a piece of mettal or wood be made a true plain , then in some convenient point thereof ( taken as a center , ) erect a gnomon of sufficient length at right angles to the plain , this done , fix the plain truly horizontal ; secondly if you take the suns co-altitude ( that is his distance from the zenith ) 3 several times in one day , and according to the stereographick projection having a line of tangents by you set off from the center of your plain or foot of the gnomon , the tangent of half each arch upon his respective azimuth or shadow ( continued if need be ) made by the gnomon , at that instant when the co-altitude is taken , so shall you insert three points upon the plain . thirdly if you find out the center to those 3 inserted points , then a right line infinitely extended by this center found and the foot of the gnomon or the center of the plain , is the true meridian line . 2. the other way , is by the help of the suns azimuth , and it is hinted in most books of dyalling , thus , 1. your plain being prepared as before , hold up a string and plummet , so that the shadow of the string may fall a cross an assigned point in the plain , and in the same line of shadow make another point at a convenient distance from the first , then through these two points draw a right line , secondly at the same instant get the suns azimuth or horizontal distance from the south part of the meridian , and having a line of chords by you , set off the angle of the azimuth from the assigned point , either on the west side of the line drawn , if your observation be made in the morning , or on the east side if your observation be in the afternoon , and draw the line . thirdly this last line so drawn shall be in the true meridian . finis . errata . page 4 line 9 read subivided and numbered . page 8 line 15 dele the. page 13 and 14 read complement . page 17 line 14 read heavens . page 19 line 14 read length . all sorts of mathematical books are sold and instruments made relating to arithmetick . trigonometry . surveying . stereometry . gauging . astronomy . geography . navigation . opticks . dyalling . geometry . architecture . fortification . gunnery . mechanicks , &c. at reasonable rates : by henry wynne , near the sugar-loaf in chancery-lane . the art of painting wherein is included the whole art of vulgar painting, according to the best and most approved rules for preparing an [sic] laying on of oyl colours : the whole treatise being so full, compleat, and so exactly fitted to the meanest capacity, that all persons whatsoever may by the directions contained therein be sufficiently able to paint in oyl colours, not only sun-dials, but also all manner of timber work ... / composed by john smith, philomath. smith, john, b. 1648? 1676 approx. 64 kb of xml-encoded text transcribed from 50 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-12 (eebo-tcp phase 1). a60467 wing s4099 estc r37566 16974146 ocm 16974146 105578 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. a60467) transcribed from: (early english books online ; image set 105578) images scanned from microfilm: (early english books, 1641-1700 ; 1159:17) the art of painting wherein is included the whole art of vulgar painting, according to the best and most approved rules for preparing an [sic] laying on of oyl colours : the whole treatise being so full, compleat, and so exactly fitted to the meanest capacity, that all persons whatsoever may by the directions contained therein be sufficiently able to paint in oyl colours, not only sun-dials, but also all manner of timber work ... / composed by john smith, philomath. smith, john, b. 1648? [15], 82, [2] p. printed for samuel crouch ..., london : 1676. "licensed may 10, 1676. roger l'estrange": p. [1] running title: the art of painting sun-dials. reproduction of original in the bodleian 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 painting, industrial -early works to 1800. sundials. decoration and ornament -early works to 1800. 2005-04 tcp assigned for keying and markup 2005-07 spi global keyed and coded from proquest page images 2005-08 john cords sampled and proofread 2005-08 john cords text and markup reviewed and edited 2005-10 pfs batch review (qc) and xml conversion licensed , may 10. 1676. roger l'estrange . the art of painting . wherein is included the whole art of vulgar painting , according to the best and most approved rules for preparing , an laying on of oyl colours . the whole treatise being so full , compleat , and so exactly fitted to the meanest capacity , that all persons whatsover may by the directions contained therein be sufficiently able to paint in oyl colours not only sun-dials , but also all manner of timber work , whether posts , pales , pallisadoes , gates , doors , windows , wainscotting , border boards for gardens , or what ever else requires either use , beauty , or preservation from the violence or injury of weather . composed by john smith , philomath . london , printed for samuel crouch , at the corner shop of pope's head ally , on the right hand next cornhill , 1676. to the reader . it is well known to all persons , that understand the mathematicks , that dialling ( one excellent part thereof ) hath with much care and industry been improved of late years , and abundance of treatises have been written thereof , wherein several wayes have been delivered , for delineating the hour lines on dial playns , some more speedy , as the instrumental way ; some for more exactness , as the arithmetical , and the geometrical way good in it self , and in some cases serving , where the other two cannot conveniently be made use of . dialling being thus plainly and familiarly communicated to the world , it 's easie for an industrious and ingenious spirit to attain the knowledge of it , so far as to be able to draw his draught ; but then wanting the knowledg of painting it on the playn , he is fain to sit down and satisfie himself with having taken much pains , to attain that , which in the end will not profit him , by reason that he hath learnt but half his art. just like a surveyor , that can hardly draw the draught , and give the proportions , or dimensions for the building of an house , but cannot rear the fabrick thereof , of , without the help of carpenters and masons ; so here , our dialist can only draw the draught , but must be beholding to the painter to finish his dial. the consideration of which hath made me adventure in the ensuing work , to lay down such rules for preparing , mixing , and laying on of oyl-colours , as that the meanest capacity may thereby attain to the knowledg of vulgar painting , and may render our ingenious artist a compleat diallist ; n●t able onely to draw his draught , but , also to finish his work , and make it sit for use ; which i have observed , not one in twenty , that are otherwise knowing in this art , can do . i am not ignorant of the lightness and vanity of the times wherein we live , and therefore expect to be censured by a sort of people so vain , that they are apt to condemn all , before they understand any ; yet let the ingenious know , that the rules herein delivered , are the same , which i my self have alwayes practiced , and that with as good applause as any professor in this kind whatsoever ; and i doubt not , but he that industriously practiseth what he finds here delivered , will soon become my champion to defend my rules , which he finds so true , against all opposers whatsoever . the truth is , the meanness of my style may perhaps not a little detract from the reader 's pleasure ; for which i hope the subject will plead my excuse , which requires to be delivered , rather in demonstrative , than elegant expressions ; and 't is the profit of my reader that i more aym at than his pleasure : however , if he chance to fail of his expectation in both , through sloth or ignorance , yet he cannot accuse me for being prodigal of his time , the discourse being so brief , which i desire may be as kindly accepted , as freely imparted . farewel . a table of the contents of the several chapters . chap. i. the description and use of the several tools used in and about the art of painting . page 1 chap. ii. a catalogue of several colours used in the art of painting ; their nature and use. p. 11 chap. iii. how to order such colours as require to be burnt in the fire , to make them the more sit for some uses . p. 24. chap. iv. how to wash such colours as by their grittiness are not otherwise to be made fine enough for certain uses . pag. 32 chap. v. how to grind colours with oyl . p. 29 chap. vi. how to order colours for working after they are ground . p. 32 chap. vii . how to make gold size to lay gold on when you would guild . p. 36 chap. viii . what colours set off best one with another . p. 40 chap. ix . what colours are sufficient for painting sun-dials . pag. 42 chap. x. some instructions for making of plains and boards to draw dials on . p. 44 chap. xi . how to make the best glew for glewing the joynts of dial-boards . p. 49 chap. xii . a catalogue of such books as are necessary for him that would be a compleat dialist . chap. xiii . the practice of painting sun-dials . pag. 56 chap. xiv . how to guild the figures of sun-dials with gold. pag. 63 chap. xv. how to lay smalt , the only colour that requires strowing . p. 67 chap. xvi . the practice of vulgar painting . pa. 69 chap. xvii . how to scowr , refresh and preserve all manner of oyl paintings . 72 chap. xviii . some improvements in painting to resist weather , and preserve timber or woodden works from rotting . p. 79 the art of painting svn-dials . chap. i. the description and use of the several tools used in and about the art of painting . 2. a grinding stone and mulier ; the stone it self ought to be of a porphyrie , which is the best ; but for want thereof , any stone will serve , whose hardness will not suffer a knife to wear it away , and withal sound and free from small pores : for if your stone be full of small holes , as some are , the colour that you first grind thereon cannot be so cleansed off , but there will remain some of it in those small holes , which will stain and spoyl the next different colour that is ground after it . this grinding-stone ought to be a foot and a half square , and so thick as may make its weight sufficient to keep it firm from moving when you are grinding on it . when at any time you have done using of your stone , you must be sure to cleanse it well from all the colour that shall remain , by scowring it with a cloath and fine dry ashes or other dust ; for if you set up your stone foul with colour , it will put you to great trouble when it is dry to make it clean when you come next to use it . the mulier must be a pebble stone of the fashion of an egg , with one end broken off flat , three inches diameter is sufficient on the flat end , and five inches in height is convenient , that you may with more ease command it in the time of grinding . 2. you must have a piece of lanthorn horn about three inches square , this piece of horn is used to keep the colour together in the grinding , and to take the colour of the stone when it 's ground sufficient . 3. you must have galley-pots , pans or pipkins to put your colours in when you have ground them ; and these pots or pans ought to be proportionable to the colour you grind : for if it be but little , and your vessel great , your colour will be lost and spent in daubing about the sides of it ; therefore for a little quantity of colour , have a small vessel ; and for a larger quantity , a larger vessel : however , let the largest hold not above two quarts , lest it prove too cumbersome and troublesom to you . 4. you must have brushes and pencils of all sorts , some for priming and laying on of colours ; others for drawing lines , figures , letters , and the like ; brushes are made with bristles , and are of several sizes , as from two inches and an half diameter , to a quarter of an inch : their goodness consists in the bristles lying close and even at the ends , and being well bound to the frame . pencils are made of finer hair , as of colaber tayls , &c. they are of several sorts , as of swan , goose , and ducks quill fitht and pointed ; their goodness consists in their being well bound , and that the hair lyes close , and that the pointed ones draw very sharp , being wetted with your tongue , and drawn through your lips . when you have done using of your pencils or brushes , you must wash the colour clean out with sope and warm water ; else the drying of the colour will so mat them , that they will never be fit for use afterwards : however , take notice you need not wash them every time you have done using them , but only when it will be a considerable time before you use them again ; otherwise putting them into a pan of water , or letting them remain covered with colour or oyl , will preserve them sufficiently when you use them often together . 5. you must have an easel for the painting of dials , easie to be made by your self or by a joyner ; it must be almost of the fashion of a ladder about 7 foot high , having the uppermost round moveable with a stay in it on the back-side of the easel to stay it in what posture you please ; the sides of it must be boared full of holes at equal and opposite distances , wherein two pins are to be put on which your dial board is to be set , and may be let lower or set higher at pleasure according as occasion shall require . 6. you must have black-lead pencils , which you may buy at the colour-shops , or at the stationers ; good pencils have of late been very scarce in london , till just upon the writing hereof there came advertisements abroad of good ones to be sold at the prince's arms over against the king on horse-back in the stocks-market . which pencils i have since tryed , and find excellent good ; they are marked thus , r. ♥ f. so that you may easily know them : however , the best way to be sure is to try them on paper ; if they shed their colour freely , and draw a black line oft-times together , they are good ; else not . the use of these pencils are to draw the draught of your dial on paper , and afterwards to draw it on the plain it self , as hereafter is taught . 7. you must have guilding cushion to cut your leaf-gold upon when you guild ; the bottom is a board about 6 inches broad , and about 12 inches long ; one which is fastened to the cover , which should be a piece of fine tanned calves-leather , the flesh side outward : this must be stuffed extraordinary hard , and as flat on the top as may be ; for want of this , the leathern bottom of an ordinary cushion will serve at a pinch , if you have not much to do . 8. you must have a knife of cane to cut your gold on the cushion ; you must form or shape it with a very sharp edged knife , that the edge of your cane may be the sharper and clearer . if you want a cane knife , and know not well how to make it , an ordinary pocket knife will do the business ; provided its edge be very sharp , and free from notches . you must wipe it very dry on your sleeve or some dry cloath ; for if the edge be never so little moist , the leaf-gold will stick to it , and spoyl all . 9. you must have a tuft of cotton , or the hinder foot of a hare or coney to press down your gold after it 's laid on the size , to make it take well and lye smooth . 10. you ought to have several pieces of wood about 3 inches long , some one inch broad , some half an inch broad , and some not above a quarter ; these must have fine pieces of cloath glewed on the bottom , with a small button or handle on the top to hold it by , this is to take your leaf-gold from the cushion when it is cut in shape , and lay it on such figures , letters or mouldings of a sun-dial as you intend to guild ( as hereafter is taught ) if your work be hollow or protuberant that you are to guild , then the gold is commonly taken up on a bunch of cotton , and laid on , pressing it down with the same . 11. you must have brass compasses , which will be useful to you on all occasions ; indeed you cannot be well without them : the best places to buy them at , are the mathematical instrument-makers . 12. you must have rulers of several lengths to draw your lines with ; these must be footed upon one side with little wier pegs to stand from the wood about a quarter of an inch ; this is to keep your ruler up from the board , that when you lay it across lines newly drawn , it may not blot them . 13. fine neat squares are also necessary , they will be useful at every turn to draw perpendicular lines , or what else requires to be true and square . 14. you must have crusiples or melting-pots to burn such colours in that require it : the best places to buy them at , are the iron-mongers in foster-lane . 15. you must have also large earthen vessels to wash such colours in that require washing to be fit for use . 16. you must have also cane-plyers to take your leaf-gold out of the book , and lay it on the guilding cushion to be cut . chap. ii. a catalogue of several colours used in the art of painting ; their nature and use. white-lead and cerus , these two colours are much of a nature , cerus being only white-lead more refined ; which advanceth its price , and renders it something more esteemed among picture-drawers ; but the white-lead is every way as useful : this colour is naturally apt to be ground very fine , and is the onely white colour used in painting with oyl : with this colour the playns of dials are laid for the last colour to draw thereon the hour lines , that they may be the more visible . with this colour posts , payls , palisadoes , gates , doors , windows , divers wainscottings , and other carpentary and joynary work are often coloured both for beauty and preservation . it resists the weather well , but within doors it 's apt to tawnish and grow rusty . this colour dries of it self indifferently well , especially if it be wrought pretty stiff ; however , to make it dry speedily , some put oyl of turpentine to it , in the tempering , which makes it dry much more speedily ; but then without doors it will not resist the weather so well : therefore the best way to make it dry speedily , and yet last long , is to put drying oyl to it , which is made by steeping red-lead in linseed oyl for about a fortnight , stirring it every day once or twice , and afterwards let it settle clear before you use it . lampe-black , is a colour that of it self is very fine , and may be tempered with linseed oyl , and used without grinding , after it hath soaked two or three daies in the oyl ; but when it is thus laid on , it will be a long-time a drying , by reason of a certain greasiness that is inherent to it ; to remedy which , it must be burnt as hereafter is taught , which consumes the fatty substance , and then it dries well . but note , that after it 's burnt , it must be ground on a stone ; otherwise it will not work well by reason of its being hardned or crusted in the fire . this colour is used in the margins of some dials that have their figures guilt ; a little of this colour and much white , make the ash colour ; and according to the quantity of either more or less , it gives several delightful varieties . willow charcoal and sea-coal ; these two are a good black for ordinary uses , only they are something coarse , and require good labour in the grinding to make them fine ; they dry well , especially the charcoal . spanish brown. this colour is a certain earth brought out of spain ; the best is that which is of a deep bright colour , and free from stones ; indeed i think there is little of it free from grittiness more or less : this colour will grind well , notwithstanding its grittiness , if you take pains with it . this is the only colour used in priming of all manner of timber-work , being fittest for that purpose for divers reasons : as first , for its cheapness , it being but of small price . secondly , it dries kindly for that purpose , not so soon but that it gives the oyl sufficient time to pierce into wood ; nor is it so long as to make the time over-tedious . thirdly , it kindly receives all other colours that are laid on it . some are of such a nature , that when they are dry no other will take on them but with great difficulty . for example , white-lead when it 's throughly dry is so greasie , that if you would either draw lines on it , or lay other colours upon it , they will run together just as ink will when you write on greasie paper . this colour of it self is a perfect horse-flesh colour ; it 's the natural shadow for vermillion , and being mixt with white , gives several varieties , according as the quantity of each is predominant . red-lead , is a sandy colour , not to be ground very fine on a stone : the onely way to make it fine is by washing ( which shall be shewed afterwards ) : this colour is an exceeding great dryer and binder , for which purpose it 's many times mixed with other colours ( such as will bear it ) to make them dry speedily : 't is a colour that resists the weather as well as any colour whatsoever , if it have the same advantage in working . it 's of it self an orange colour , and is the onely colour used in making of drying and fat oyls . vermillion . this is a very rich colour , and of a good body ; and if pains and time be bestowed , it will be ground as fine as is possible for a colour to be ground : which it must be , or else it works as bad as any colour whatsoever ; but if it be ground fine even as oyl it self , no colour works better . this colour is used to draw the hour lines on sun-dials , and for divers other not common uses : it drys well if you work it stiff . the best way to buy it is in the stone ; for otherwise it may be sophisticated & spoyled with red-lead if bought in the powder . this colour is a perfect scarlet , mixed with white it gives scarlet carnation in divers varieties , according to the quantity of each colour mixed . the natural shadow for vermillion it self , is spanish brown. lake , is also a rich colour , and may be ground very fine ; it 's often used in ornaments of dials , and a margin of it sets off well with gold figures , especially if a little white be added to it . it 's excellent in divers kinds of flowrages . lake and bice make a purple of divers varieties according to the blew that is mixed with it . lake of it self is an excellent crimson colour . lake and white make an excellent crimson carnation in divers varieties according to the quantity that is mixed of each . lake and white , and a little red lead make a flesh colour . smalt , is a lovely blew at a distance , if strewed on ; if you will work it in oyl , it must be made fine with washing : the truth is , when it 's at the finest , it works but badly in oyl , by reason of its harshness ; it must also have white-lead added to it if wrought in oyl , or else it 's too dark , and shews not its self ; and when all is done , time is apt to turn it black : therefore the best way is to strew it ( as shall be shewed hereafter ) and then there is not a more glorious blew to be made . this is an excellent colour for the margin of a dial , if the figures are guilt , and for several other purposes , as it may by an ingenious spirit be made use of : if you buy it to work in oyl , the finest is best , which they call oyl smalt ; but if it be for strewing , the coarsest you can get is the best , both for colour and continuance . blew bice , is a colour fine enough for almost any use ; it is but a pale colour , and in dial-painting is used for a margin ground to guild figures in small playns or stacks of dials that are near the eye : this colour works well , though it be a little sandy ; bice and pink make a green ; bice and lake make a purple ; bice and white make a light blew , of each several varieties according to the quantity of each . blew verditer , is a colour something sandy ; it 's subject to change and turn greenish , and makes a good green , mixed with yellows , this colour may serve in dial-painting , where bice and smalt are wanting ; but not so good as either of them mixt with whites and yellows ; or both , giveth variety of colours for divers uses . indico , is a very dark blew , and seldom used without a mixture of white , unless to shadow other colours ; it grinds fine , and works well , and is much used in vulgar painting for the last colours of windowes , doors , pales , posts , rails , pallisadoes , or any other timber-work . it resists the weather well , onely it 's something dear , and yet not very chargeable for work , by reason much white must be mixed with it , which makes a little of it go a great way : vulgar painters instead thereof use blew balls , which they buy at the colour-shops which nearly imitates it , but is not so good a colour neither for beauty nor lasting . indico and white make a lead colour ; 't is a pleasant colour to marble white withal , or to shadow it . amber , is a colour that will be ground very fine , but must have labour and time bestowed on it : it 's very apt to furr the mulier , and difficult to be drawn under it , without sleight of hand in the grinding this colour ; dryes and binds exceedingly , and therefore resists weather well : it 's much used in painting , for the many pleasant varieties it giveth . this colour of it self is a perfect hair colour , and being mixed with white , giveth variety of pleasant colours . this colour burnt in a crusipple is the natural shadow for gold ; it likewise shadows divers other colours , and in great varieties . verdigrease , is a good green , something inclining to a blew ; therefore for divers uses it 's willowish colour must be corrected with yellows : this colour is commonly very foul , and requires time and pains to pick and cleanse it : it requires also labour to grind it fine . this colour dryes speedily , and is a green that is used on most occasions that require that colour . it is of it self a perfect willow green ; and being mixed with pink yellow , it makes a pure lively grass-green : and these being mixed with white , gives several varieties of light greens , according to the quantity of each . yellow daker , is of two sorts ; the one gotten in england , the other brought from beyond the seas : the one is a light yellow , much like the colour of wheat straw ; the other is somewhat of a deeper colour . this colour may with labour be ground very fine , but something troublesom by reason of its clamminess : this colour is used to make gold size , and is also much used in vulgar painting . pink yellow , is a colour something inclining to a green ; 't is a good yellow for some uses , and grinds well . it 's chiefly used to mix with other colours to make green. besides these colours , a dial-painter must furnish himself with leaf-gold for guilding , linseed oyl to temper his colours with , and oyl of turpentine to make his colours dry the more speedily , by mixing a little of it among his colours . chap. iii. how to order such colours as require to be burnt in the fire , to make them the more fit for some uses . colours that commonly use to be burnt , are lamp-black , umber , yellow oaker , and spanish brown. lamp-black must alwayes be burnt , otherwise it will never dry kindly . umber works and dryes well enough without burning for many uses : but when you would colour either hair , horse , dogg , or the bodies of some trees , then it must be burnt , which makes it of a deeper and brighter colour ; so likewise for some particular uses the others are burnt , else not : the manner thus , take a crusipple or melting-pot of bigness sufficient to hold the quantity of colour you desire to burn , set it in the midst of a charcoal or other clear fire , and let it continue therein till it be all like a coal ; then take it out , and let it cool of it self ; then grind it and make it fit for use . chap. iv. how to wash such colours as by their grittiness are not otherwise to be made fine enough for certain uses . some colours are of such a gritty sandy nature , that it 's impossible to grind them so fine as some curious works do require ; therefore to get forth the flower and fineness of the colour , you must do thus ; take what quantity of colour you please to wash , and put it into a vessel of fair water , and stirr it about till the water be all coloured therewith ; then if any filth swim on the top of the water , scum it clean off , and when you think the grossest of the colour is settled to the bottom , then pour off that water into a second earthen vessel that is large enough to contain the first vessel full of water four or five times ; then pour more water into the first vessel , and stir the colour that remains till the water be thick ; and after it is a little settled , pour that water also into the second vessel , and fill the first vessel again with water , stirring it as before : do thus so often till you find all the finest of the colour drawn forth , and that none but course gritty stuff remains in the bottom ; then let this water in the second vessel stand to settle till it be perfectly clear , and that all the colour be sunk to the bottom ; which when you perceive , then pour the water clear from it , and reserve the colour in the bottom for use , which must be perfectly dryed before you mix it with oyl to work . the colours thus ordered , are red-lead , blew and green bice , verditor blew and green , smalt , and many times spanish brown , when you would cleanse it well for some fine work , as also yellow oaker , when you intend to make gold size of it . take notice also , that unless you intend to bestow some cost on a piece , you need not be at the trouble to wash your colours , but use them for coarse ordinary work as you buy them at the shops . chap. v. how to grind colours with oyl . when you come to grind colours , let your grinding-stone be placed about the heighth of your middle ; let it stand firm and fast so that it soggle not up and down ; then take a small quantity of the colour you intend to grind ( two spoonfuls is enough ) for the less you grind at a time , the easier and finer will your colour be ground : lay this two spoonfuls of colour on the middest of your stone , and put a little of your linseed oyl to it , ( but be sure you put not too much at first ) then with your mulier mix it together a little , and turn your mulier three or four times about , and if you find there be not oyl enough , put a little more to it , till it come to the consistence of an oyntment ; for then it grinds much better and sooner then when it 's so thin as to run about the stone : you must oftentimes in the grinding bring your colour together with your piece of lanthorn horn , and with the same keep it together in the middle of your stone ; when you find you have ground it fine enough ( by the continual motion of your mulier about the stone , holding it down as hard as your strength will permit , which you must also move with such a sleight , as to gather the colour under it ) and that no knots nor grittiness remains ; then with your horn cleanse it off the stone into a gally-pot , pan , or what ever else you design to put it into : and then lay more colour on your stone , and proceed to grinding as before : do so thus often till you have ground as much of this same colour as shall serve your occasions ; and if you grind other colours after it , let the stone be well cleansed from the first colour with a cloath and fine dry ashes . chap. vi. how to order colours for working after they are ground . when you have ground your colours ( if you observe my directions in grinding ) they will be too thick for use without the addition of more oyl ; therefore when you have ground those colours you desire , and intend to use them either simply by themselves , or compounded with others , according as your fancy or occasions require , you must then add more oyl unto them , till they be so thin as not to let the ground on which they are laid be seen through them ; for if it be so thin as to let the ground be seen through them , or to run about when it be laid on , it is not good , and will require to be coloured the oftner before your work be perfect and substantial ; whereas if your colour be as stiff as it can well be wrought , your work will be done with more speed ; once doing being more substantial then three times doing with thin colour . here by the way take notice of the fraud and deceit of common painters , who commonly agree to do work by the yard at a certain price , and the work to be coloured three times over , which they commonly paint with such thin colour , ( to avoid the labour of grinding , a little colour serving a great deal of oyl , and besides it works with less pains , and takes up less stuff ) that all three times doing over is not so substantial as one time would be , if the colour had a thick and substantial body : and i 'le maintain , that three times colouring with substantial and well bodied colour , shall last ten times as long as that which is wrought thus sleightly by common painters . in mixing oyl with your colours , take this further note , that if the colour to be mixt be your priming colour , ( that is the first colour you lay on ) it ought to be made very thin , that it may have oyl enough to pierce into the wood , which is much for its preservation ; but after your first colour is laid , let your next be thicker as before is taught . but if your colour to be mixt be for the drawing of the hour lines , or making the figures in a sun-dial , then let it be tempered as stiff as is possible to work it , that it may not presently decay , but may be capable by the quantity laid on , to last as long as any colour on the dial ; to which purpose its being wrought in fat oyl will much conduce to its lasting : how this fat oyl is made , see chap. 7. where you have the manner of it taught at large . chap. vii . how to make gold size to lay gold on when you guild . gold size is made of fat oyl , and yellow oaker ; the oyl is no other than linseed oyl thus ordered ; take what quantity of linseed oyl you judge will serve your turn , put it in a brazen or other vessel that will endure the fire ; when it is in the vessel , put to it a certain quantity of red-lead ; the more you put in , the better will your oyl be ( provided you put not in so much as to hinder its boyling ) for this red-lead adds a drying quality to the oyl , which otherwise being thus ordered , would not dry in any time : when the oyl and lead are thus mingled together , let them gently boyl over a fire of coals without flame a pretty while ; when it 's boyled enough , ( which you may know by taking a little of it , and let it cool , and if it roape like thin treacle , then it is enough ) then with a lighted paper set it on fire , ( which fireing will burn away much of the greasiness of it ) which let burn about a minute or two , or more or less , according as your quantity of oyl is ; and then let it be extinguished ( by clapping a cloath over it ) afterwards let it stand to cool and settle ; and when all the lead be sunk to the bottom , and the oyl be clear , then pour it off , and reserve it in a bladder for use . your yellow oaker must also be thus ordered before it be made into size ; take yellow oaker and grind it on a stone with water till it be very fine , and afterwards lay it on a chalk stone to dry ; this is the common way : but a better , is to wash it as is taught in the fourth chapter . for when it is washed , to be sure nothing but the purest of the colour will be used ; and besides , it 's done with more ease , and less daubing . when your oyl and oaker are thus prepared , you must grind them together , as you do other oyl-colours ; but it 's something more laborious work , and must be ground very fine , even as oyl it self : for the finer it is , the greater lustre will your gold carry that is laid on it . here note , that you must give it such a quantity of your fat oyl , that it may not be so weak as to run when you have laid it on ; nor so stiff , that it may not work well ; but of such a competent body , that after it is laid on , it may settle it self smooth and glasie , which is a chief property of good size . chap. viii . what colours set off best one with another . yellowes set off best with blacks , blews and reds . they set off indifferently well with greens , purples , and whites . blews set off best with yellowes and whites . they-set off indifferently with blacks and reds . but they set not off with greens , purples , and browns . greens-set off best with whites and yellows . they set not off with blacks , blews , or reds . reds set off best with whites , and yellows . they set off indifferently with blews and blacks . blacks and whites set off well with all colours , because they differ so much from all . chap. ix . what colours are sufficient for painting sun-dials . if you are to paint a plain sun-dial , these four colours serve , viz. spanish brown , white lead , vermillion , and lamp-black : the spanish brown is for the priming colour , the vvhite lead is for the last colour of ●●e plain ; the vermillion is for drawing the lines , and the lamp-black is for drawing the figures . but if your dial be more rich , you must have ( besides the four fore-mentioned colours ) gold size to make the figures to lay gold on , and smalt or blew bice for the margin and inner table ; and if you intend to bestow curiosity , then you may use such other colours as your fancy shall direct you may be most suitable to your design ; for which purpose your care must be to observe the ornament and fashion of whatsoever good dial you meet with , and to register your observations : this will be a great help to your fancy on all occasions . chap. x. some instructions for making of plains and boards to draw dials on . dial playns are of two sorts ; first , such as are of the wall of a building it self : or secondly , such as are drawn on tables . the first sort if they are made on brick-work , is done with lime and hair plaistered on the wall , of what bigness the owner pleaseth ; this is the common way . but a better and more durable way , is to temper lime and sand with linseed oyl ; 't is not very chargeable , but exceeding profitable : for this substance will harden to the hardness of a stone , and not decay in many ages . if you cannot have oyl enough to temper a quantity of plaister sufficient for your playn , then temper your lime and sand with scummed milk ; this you will find to last six times as long as your common plaister . now for tables of wood , they being the most common , i shall give such directions for the making of them , as i have alwayes found most profitable and fit for this purpose . the woods that i find best for this use are the clearest oak , and the reddest firr , provided it be not turpentiney ; between these two woods i find little difference as to their alteration by the weather , both being subject to split in case they are bound , and have not free liberty to shrink with dry weather , and swell with wet ; but as to their lasting , i judge oak to be the better : and how long firr will last thus secured and defended with oyl colours , i have not yet experienced ; but we may judge that good red firr that is very roseny , will last the age of any man whatsoever , if it be secured as things of this nature ought to be . in working any of these woods , i advise , that first your boards be cut to such a length as you intend your dial board shall be of , and so many of them as may make up the breadth designed ; then let them be joynted and plained on both sides , and afterwards set to dry ( for 't is observed , that though boards have layn in an house never so long , and are never so dry , yet when they are thus shot and playned , they will shrink afterwards beyond belief , if kept dry ) : when you think they are dry enough and will shrink no more , let them be again shot with good joynts , and every joynt in the glewing doubled together with pins , as coppers do the bottoms of their tubs ; after it is thus glewed , and the joynts be sufficiently dry , then let the face of the board be very well playned and tryed every way , that it may be both smooth and true , and the edges shot true , and all of a thickness , as panels of wainscot are commonly wrought , the edges must be thus true and even , that it may sit into the rabet of a moulding put round it ; just as a panel of wainscot doth in its frame : this will give liberty to the board to shrink and swell without tearing ; whereas mouldings that are nayled round the edge as the common way is , doth so restrain the motion of the wood , that it cannot shrink without tearing : but boards made this way will last a long time without either parting in the joynts , or splitting in the wood . dials are sometimes drawn on playns lined with copper or lead , that they may be free from splitting or tearing ; but i prefer a board ( if it be made as above is directed ) before them in many respects : as first , it is much cheaper : secondly , lead ( and copper too a little ) will swell with the heat of the sun , and grow in time so hollow , and as it were swelled outwards , that the truth of its shadow will be much injured . thirdly , the colours will be apt to peel from the metal , and the dial will be that way more defaced than on woodden playns . chap. xi . how to make the best glew for glewing the joynts of dial-boards . take a quantity of milk that hath stood so long to cream that no more will arise from it ; scum it very clean , and set it over the fire in a leaden pot , and let it boyl a little ; and if any cream arise , take it off , then put in your glew first divided into small pieces , and it will soon melt ; and when you have boyled it to a good body , that it be neither too thick nor too thin ( for in the right observance of this lyes much of the art ) then use it as you do other glew : this binds beyond belief , and will not be subject to resolve with any competent moisture of the weather . 't is certain , that when any sort of glew is burnt to the sides of the pot , the whole is spoiled of its former strength ; to prevent this , let your glew be alwaies melted in balneo maria , which is thus ; take a large skillet , or a little kettle full of water , into which put your glew-pot with a wispe of hay or straw under it , to keep it from the bottom of the vessel ; and as the water in the vessel heats , so will your glew melt : and thus you may do at the first making of your glew , by which means you may boyl it to what body you please , without danger of burning to the sides of the pot . chap. xii . a catalogue of such books as are necessary for him that would be a compleat dialist . dr . record's castle of knowledge . this book ( though something scarce ) is an excellent book for those that would attain the knowledg of the sphere , or motion of the heavens ; which every one that would be a compleat dialist ought perfectly to understand . stirrup's compleat dialist . in this book is contained a brief explanation of the sphere ; as likewise three several wayes to draw dials , two of them geometrically , and the third instrumentally ; all of them as expeditious and true as most . to this book is added an appendix by mr. leybourne , shewing the best wayes for furnishing dials with such lines as shew the suns place in the 12 signs , his declination , right ascension , length of the day and night , the rising and setting of the sun , his azimuth and circles of altitude , with the jewish , babylonish , and italian hours . collins sector on a quadrant . in this book , among other things , are excellent scales and instruments for dialling . to which is added an appendix by john lyon , shewing the way of drawing all manner of dials on the seelings floors , and walls of rooms , to receive the reflection of a small glass . collins dialling . this book among several good geometrical ways for drawing dials , shews also wayes to draw dials from a gnomen stuck into a wall at random , without knowing the declination ; a good book throughout . leybourne's art of dialling . a very ingenious piece , where you will find ( among many other good conceits ) a very easie , exact , and speedy way for drawing fair upright decliners ; and also an instrument the most compendious of all others , especially in drawing small dials . leybourn's introduction to astronomie ; sold by robert mordant at the atlas in cornhill . in this book is shewed how to draw all manner of dials by the globe ; and among the rest , he shews a way to draw an east or west dial geometrically , the best of any extant . the use of sutton 's large quadrant , sold at the atlas in cornhill ; which together with the instrument is very useful for a dialist . phillips's mathematical manual ; wherein are the tables of signs and tangents for calculating the hour distances the arithmetical way . the works of mr. edmund gunter : or , the use of his sector . an excellent piece . foster's azimuth dialling . an ingenious work. oughtred's circles of proportion : in which ( among many other ingenious conceits ) you have the way and manner of drawing the double horozontal dial. blagrave's dialling : a good piece , wherein you have several choyce conceits and explanations of the nature of dials , with the way of drawing the hour lines belonging to , or shewing the hour in any countrey whatsoever . these are the books of greatest note that are yet extant . there is one more yet expected from mr. leybourn , which will be the whole body of dialling , after several most new and easie wayes , which without doubt will be an excellent piece . chap. xiii . the practice of painting sun-dials . when according to the rules given in the books aforementioned , you have drawn on paper the draught of your dial ; and that your board be ready , and your colours prepared according to the directions before given , you must in the painting of your dial proceed thus ; take spanish brown that is well ground and mixed somewhat thin , and with a large bristle brush dipt therein colour your board or playn all over on every side , so that you leave no part uncoloured ; this is called the priming of your dial : when this first colour is dry , do it over again with more of the same colour tempered somewhat thicker ; and when this is also dry , you may if you please do it over again with the same colour , your work will be the substantialler , and last longer . when this last time of colouring with your red lead be dry , then with white lead colour the face of your playn over , and when it is dry work it over again three or four times more successively after each drying , so shall the face of your playn be sufficiently defended against the many years fury and violence of weather . when the last colouring of your white be dry , you must draw on your playn ( with a black-lead pencil ) a horozontal line so far distance from the upmost edge of your dial , as your discretion shall think fit , or your experience finds to be most becoming your playn ; then set out the margin of your dial with boundary lines for the hour , half hour , and quarter divisions of your dial ( as in most dials you see is done ) : when you have thus set out the margin and boundary lines of your dial , then take your paper draught fairly drawn , and place the horozontal line which you before drew on your playn ; in doing of which observe to place the center according as the situation of your playn for convenience sake requires : thus ; if your dial be a full south dial , then let the center be exactly in the middle of your playn : but if your dial decline from the south either east or west , then place not the center of your draught in the center of your playn , but nearer to one side or other of it , according as it declines , having also respect to the quantity of its declination . for example : if your dial decline eastwards , then let the center of your draught be plac't between the center and the eastern side of your playn , the quantity thereof must be according as your dial declines ; if it decline but a little , then place the center of your draught but a little from the center of your playn ; and if it declines much , place the center of your draught the more out of the center of your playn : the reason of my advising this , is , that by so doing you may gain a greater distance for those hour-lines , which in declining playns fall nearer together on one side then they are on the other ; for which reason i alwaies use it in all declining playns , except they decline far , as between 80 and 90 degrees ; for then we commonly draw them without centers , to gain the more distance for the hour lines . when your paper draught is thus artificially placed on the playn , and fastened with pins or small tacks ; then let the draught thereof be transferred to the playn , by laying a ruler over every hour , half hour , and quarter division : and where your ruler shall cut or intersect the boundary lines of your margin , there make marks by drawing lines with a black-lead pencil , of such a length as each division requires ( or is designed by your boundary lines ) observing alwaies to draw the hour , and half hour lines quite through your margin , that they may be guides for the right placing the figures , and for a small spot that is usually placed in the margin , right against the half hour . when your dial draught is thus transferred to the playn it self , you must not forget to draw the substill line according as it lyeth in your draught , to be your guide for the right placing your still or cock ; for you must in every particular be very exact , or else your dial cannot be good . when you have taken every thing that is required from your draught , and have transferred it to the playn , then take your draught off , and with vermillion very well ground and prepared , as before is taught , let the boundary lines of your dial , as also the hour , half hour , and quarter divisions be drawn therewith ; let your colour be as thick and stiff as you can possible work it , so as to draw a clear and smooth line . when your vermillion lines are drawn , then with lamp-black let the figures be drawn , a spot in the middle of the margin right against the half hour line ; and if you please in the margin at the top of your playn you may put the date of the year , your name , or some divine sentence , as is usual in things of this nature : then fit in your cock so as to make right angles with the playn , so shall your dial be drawn and finished in all respects as a plain dial ought to be . chap. xiv . how to guild the figures of sun-dials with gold. if you intend to bestow more cost on a dial then what is expressed in the last chapter , by guilding the figures or other ornaments , you must proceed thus ; whatsoever you would guild must first be drawn with gold size ( of the making of which , see chap. 7. ) according to the true proportion of what you would have guilt , whether figure , letter , or what ever else it be ; when you have thus drawn the true proportion of what you would have guilt , let it remain till it be sufficiently dry to guild upon , which you shall know by touching it with the end of your finger ; for if your finger stick a little to it , and yet the colour come not off , then is it dry enough : but if the colour come off on your finger , then is it not dry enough , and must be let alone longer ; for if you should then lay your gold on , it would so drown it , that it would be worth nothing : but if your size should be so dry as not to hold your finger as it were to it , then is it too dry , and the gold will not take ; for which there is no remedy but new sizing ; therefore you must watch the true time that it be not too wet or too dry ; both extreams being not at all convenient . when your size is ready for guilding , take your book of leaf gold , and opening a leaf of it , take it out with your cane-plyers , and lay it on your guilding cushion , and if it lye not smooth , blow on it with your breath which will lay it flat and plain , then with a knife of cane , or for want of it , an ordinary pocket knife that hath a smooth and sharp edge ; with this ( being wiped very dry on your sleeve that the gold stick not to it ) let your leaf-gold be cut into such pieces or forms as your judgment shall think most suitable to your work . when you have thus cut your gold into convenient forms , then take your tool that was before described in num . 10. of chap. 1. and draw the cloath side of it across your tongue , or breath upon it to make it dampish that the gold may stick to it ; with this tool take your gold up ( by clapping it down on the several pieces you had before cut into forms ) and transfer it to your size , upon which clap it down according to discretion , & your gold will leave your tool , and cleave to your size ; which you must afterwards press down smooth with a bunch of cotton , or a hares foot : and thus you must do piece by piece till you have covered all your size with gold ; and after it is fully dryed , then with your hares foot brush off all the loose gold , so will your guilding remain fair and beautiful . note , that after your guilding is thus perfectly laid on , you may if you please diaper or flourish on it with thin umber whatsoever shall be suitable to your design ; the form and order of which take from examples which are abundant , where painting and gilding are to be seen . chap. xv. how to lay on smalt , the only colour that requires strewing . if you make the margin of your dial blew with strowing smalt , it must be done after the figures are guilt ; thus : take white lead stiffly tempered ( if with fat oyl it will be much the better ) and therewith colour over your whole margin , repairing therein the figures as you come to them ; when you have thus done your margin all over with thick colour , take your smalt , and with a goose-quill-feather cover all your margin with it , and with a piece of cotton dab it down close that it may well take upon the ground laid under it ; and when you imagine the ground to be throughly dry , then wipe off the loose colour with a feather , and blow the remainder of it off with a pair of bellows , so is your work finished . and thus you have a method for colouring any thing else with this colour besides the margins of sun-dials . chap. xvi . the practice of vulgar painting . that which i call vulgar painting , is only the way and manner of colouring wainscot , doors , windows , posts , rayls , pales , gates , and the like ; the method of doing which differs not at all from that of painting sun-dials , that is , in the preparation , mixing and laying on of colours ; and one example i know will be sufficient to direct you in any business of this kind : suppose you have a pair of gates or the like to paint , you must proceed thus ; first prime it with spanish brown ( as you did your dial-board ) twice or three times , when this is dry take white lead well tempered ( as before was taught ) or umber and white , or blew balls , or indico and white , or any other colour you intend your work shall be laid in , and with that colour ( whatsoever it be ) let your gate be coloured four or five times over successively after each drying ( for the oftner you colour any thing without doors , the longer will it last ; wainscotting indeed you need not do over above twice with the last colour , because it 's within doors ) listing , quartering or panelling it as your fancy shall please ; or else letting it go plain , and all of one colour as you shall best like ; so shall your work be finished ; and thus may you do for any other : and if you are minded to gild or the like , repair to the 7. and 14. chapters , where you are sufficiently taught in that also : so that you may see that in this method of painting sun-dials , i have also delivered the whole art of vulgar painting , because they are indeed but one and the same thing . chap. xvii . how to scour , refresh and preserve all manner of oyl paintings . the oyl paintings that i here intend , are only such as are kept from the injuries of weather ; for such paintings as endure the fury of rain and storms ( such as sun-dials , posts , pales , &c. ) are not any waies to be renewed or refreshed , but by being new coloured with the same colour in which it was at first wrought , because that the body and strength of the colour is worn out by the continual assaults of wasting time . but as for such paintings that are sheltered from weather , as all in-door paintings are , they still keep their body and colour , although their beauty may be much impaired by dust , smoak , fly-shits , humid vapors , and the like , which will in time soyl and tawnish them ; to remedy which , take these few rules : if your painting be wainscotting or any other joynary or carpentary work that is painted in oyl , take wood ashes well sifted , which mix with water somewhat thickly , then take a stubbed bristle brush ( like those we call shooe-brushes ) and dip it in the moystened ashes , and therewith rub and scour your painting all over in all places alike , and when you find that all the soyl is taken off , then wash it clean with fair water , and let it dry ; after which take common varnish and therewith varnish your work all over alike , and you will find your painting to be near as fresh as when first laid on . note , that if your varnish be too thick , you must put oyl of turpentine to it , which will make it as thin as you please . but if your painting be more curious , whether figures of men , beasts , landskip , frutage , florage , or the like , then take smalt , ( a sandy colour , to be bought at the colour-shops ) with which and a spunge wet in water , let your picture be gently scowred , and then cleanly washed off with fair water : after it is well dry , let it be run over with varnish , and you will find the beauty and lustre of your picture much recovered . but note , that this scowring ought not to be practised but very seldom ( as when your picture is very much soyled ) because often and too frequent operations in this kind must needs wear off a little of the colours ; therefore strive what you can to preserve their first beauty , by keeping them free from smoak , and by often striking off the dust with a fox tail ; as likewise preserving them from flies , by dressing up your rooms with green boughes , to which the flyes will gather themselves , and so not hurt your pictures . sir hugh platt in the first part of his garden of eden , and 17 page , tells us of an italian fancy for this purpose , by hanging in the roof and sides of the room small pompions or cow combers stuck full of barley , which will sprout into green spiers on which the flyes will lodge . querie , whether vessels of tin made round about full of holes filled with earth , and every hole planted with a sprig of orpen , penyroyal , mints , &c. and watered as need requires , would not be more beautiful and useful for this purpose . another note worth observation is , that all pictures ( especially those that are wrought with mixtures of white lead ) are apt to tawnish and grow rusty , as is seen in all ancient pieces ; to prevent which , in the moneths of may and june let your pictures be exposed to the hot sun three or four dayes ; this will draw off much of the tawnish , and make the colours more fresh and beautiful : and thus doing from year to year will preserve them wonderfully . although in the beginning of this chapter i mentioned dials among those things that are not to be refreshed but by new painting ; yet here take notice , that i think it not convenient at all to lay new colouring upon the old ground of a sun-dial ( that is , to draw the old lines and figures over again in the same posture wherein they were drawn before ) but rather to take the declination anew , and according thereunto make a new draught of your dial , and proceed in the painting of it in all respects as if it were a new dial : for it is observed , that dials which were made many years ago ( which we believe went true when first made ) will not give the true hour now , but go very false and unequal , which is caused by some secret motion of the earth not hitherto taken notice of , which apparently alters the declination of all playns whatsoever . if any one requires more satisfaction herein , let him repair to some old dial that was made many years ago , and according to the distance of the substile from the meridian , let him find out the declination when first made , as any man that is an artist can easily do ; then let him take the declination of the plain by the sun , and he shall find these two declinations to differ considerably according to the number of years contained between your observation and the time of the dials first making ; so that a plain that stood full south 30 , 40 , or 60 years ago , shall now decline some degrees either to the east or west , according to the nature of the earths motion . chap. xviii . some improvements in painting to resist weather , and preserve timber or woodden works from rotting . take the hardest rosin you can get , clarifie it well ; to which rosin add linseed oyl so much as you find by experience to be sufficient ; let them be well melted and incorporated together on the fire , then take either umber , verdigrease , or red lead ( these being extraordinary drying colours ) first ground fine , which put into the oyl and rosin ; and when they are well mixed together , you may use them in colouring timber as you do with other colours : it 's best alwayes to be used hot , least it be too stiff . this is a most excellent thing to preserve timber , it lyeth like the china varnish , and will endure 10 times as long as other painting ( if rightly wrought ) ; this is a most excellent way to preserve the border boards in gardens , and any other thing that we would have last long in wet and moysture ; this colour spread on cloath with a trowel , is a most excellent covering for tents , huts , turrets , houses of pleasure , and the like . and let me add one experiment more that will much commend the use of this mixture ; which is this : let those woodden vessels ( whether hogsheads , barrels , kilderkins , or any other vessel whether upright or decumbent ) that you design to keep any drinkable liquors in , be well and intirely painted on the outside therewith ; which vessel so painted shall keep and preserve all manner of liquors equally to the best bottles whatsoever , by reason that the sponginess and porusness of the wood is intirely closed and shut up by this tough rosinious varnish , thereby keeping the spirits of the liquor from flying away , and so consequently preserving the whole body thereof in its strength and vigour . the best way to make the varnish ( or colour ) for this purpose , is to put no more oyl to the rosin than what shall just serve to toughen it ; nor to mix any colour with it , but burnt umber , because verdigrease and red lead may be objected against by reason of their corroding quality : the best way to lay this colour on , is to heat it hot before you work it , which will make it close the firmer to the wood . postscript . one thing i had forgot when i spake of pencils in the first chapter ; that is , whereas i told you that pencils or brushes were to be cleansed from their colour by washing them with sope and warm water , it is to be understood that this is most convenient in large brushes ; but for smaller pencils , the better way were to dip them in clean sallet oyl , and draw them between your fingers divers times till they are clean ; and when you come to use them again , dip them in a little linseed oyl , and squeeze it out again , and then use them . finis . mechanick dyalling teaching any man, though of an ordinary capacity and unlearned in the mathematicks, to draw a true sun-dyal on any given plane, however scituated : only with the help of a straight ruler and a pair of compasses, and without any arithmetical calculation / by joseph moxon ... moxon, joseph, 1627-1691. 1668 approx. 89 kb of xml-encoded text transcribed from 28 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-12 (eebo-tcp phase 1). a51544 wing m3009 estc r20066 12354007 ocm 12354007 60072 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. a51544) transcribed from: (early english books online ; image set 60072) images scanned from microfilm: (early english books, 1641-1700 ; 643:12) mechanick dyalling teaching any man, though of an ordinary capacity and unlearned in the mathematicks, to draw a true sun-dyal on any given plane, however scituated : only with the help of a straight ruler and a pair of compasses, and without any arithmetical calculation / by joseph moxon ... moxon, joseph, 1627-1691. 49, [6] p. : ill. printed for joseph moxon ..., london : 1668. advertisement: p. [1]-[6] at end. reproduction of original in huntington 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 sundials. mathematical instruments. 2005-02 tcp assigned for keying and markup 2005-03 spi global keyed and coded from proquest page images 2005-04 judith siefring sampled and proofread 2005-04 judith siefring text and markup reviewed and edited 2005-10 pfs batch review (qc) and xml conversion mechanick dyalling : teaching any man , though of an ordinary capacity and unlearned in the mathematicks , to draw a true sun-dyal on any given plane , however scituated : only with the help of a straight rvler and a pair of compasses ; and without any arithmetical calculation . by joseph moxon , hydrographer to the kings most excellent majesty . london . printed for joseph moxon on ludgate-hill , at the sign of atlas . mdclxviii mechanick dyalling . description of dyalling . dyalling originally is a mathematical science , attained by the philosophical contemplation of the motion of the sun , the motion of the shaddow , the constitution of the sphere , the scituation of planes , and the consideration of lines . explanation . the motion of the sun is regular , it moving equal space in equal time ; but the motion of the shaddow irregular in all parts of the earth , unless under the two poles , and that more or less according to the constitution of the sphere and scituation of the plane . and therefore scientifick dyalists by the geometrick considerations of lines , have found out rules to mark out the irregular motion of the shaddow in all latitudes , and on all planes , to comply with the regular motion of the sun. and these rules of adjusting the motion of the shaddow to the motion of the sun may be called scientifick dyalling . but though we may justly account dyalling originally a science , yet such hath been the generosity of many of its studious contemplators , that they have communicated their acquired rules ; whereby it is now become to many of the ingenious no more difficult than an art , and by many late authors so intituled : nay more , by this small treatise it will scarce be accounted more than a manual operation ; for , though ( hitherto ) all the authors i have met with seem to presuppose their reader to understand geometry , and the projecting of the sphere already , or else endeavour in their works to make him understand them , as if they were absolutely necessary to be known by every one that would make a dyal , when as in truth ( the contemplative pains of others aforesaid considered ) they are not ; but indeed are only useful to those that would know the reason of dyalling . thus they do not only discourage young beginners , but also disappoint many gentlemen and others that would willingly either make them themselves , or set their workmen about them , if they knew how to make them . this little piece i have therefore composed for the help of those who understand neither the projection of the sphere , or geometrical operations : only , if they know how to draw a straight line between two points by the side of a ruler , describe a circle with a pair of compasses , erect a perpendicular , and draw one line parallel to another , they may know how to draw a dyal for any given plane , however scituated in any latitude . but perhaps these two last little tricks are not known to all new beginners , therefore i shall shew them . first , how to erect a perpendicular . for example , in fig. 1. upon the line ab you would erect a perpendicular to the point c : place one foot of your compasses upon the point c , and open the other to what distance you please ; for example , to the point a , make there a mark ; then keeping the first foot still in c , turn the other foot toward b , and make there another mark ; then open your compasses wider , suppose to the length ab , and placing one foot in the point a , with the other foot describe a small arch over the point c , and removing the foot of your compasses to the point b , with the other foot describe another small arch , to cut the first arch , as at d. then lay your straight ruler to the point where the two small arches cut each other , and upon the point c , and by the side of the ruler draw the line cd , which shall be a perpendicular to the line ab . another way with once opening the compasses , as by fig. 2. draw the line ab , and place one foot of your compasses upon the point you would have the perpendicular erected , as at the point c , and with the other foot describe the semi-circle a ab b , then placing one foot in b , extend the other foot to b , in the semi-circle ; and keeping that foot in b , extend the other foot to d , and make there a small arch : then remove one foot of your compasses to a , and extend the other foot to a in the semi-circle , and keeping that foot in a , extend the other to d , and make there another small arch , to cut the first small arch ; and laying a straight ruler to the point where these two small arches cut each other , and upon the point c , draw by the side of the ruler the line cd , which shall be perpendicular to the line ab . to erect a perpendicular upon the end of a line , as by fig. 3. on the point b , at one end of the line ab , place one foot of your compasses in the point b , and extend the other on the line towards a , as to b , and with it describe the arch ba c ; then placing one foot in b , extend the other to a in the arch , and make there a mark ; divide with your compasses the arch ba into two equal parts , and keeping the feet of your compasses at that distance , measure in the arch from a to c , then draw a straight line from the point c to the end of the line b , and that straight line shall be perpendicular to the end of the line ab . to draw a line parallel to another line , as by fig. 4. example . if you would draw a line parallel to the line ab , open your compasses to the distance you intend the lines shall stand off each other , and placing one foot successively near each end , describe with the other foot the small arches cd ; lay a straight ruler to the top of these arches , and draw a line by the side of it , and that line shall be parallel to the line ab . definitions . a dyal plane is that flat whereon a dyal is intended to be projected . of dyal planes some be direct , other decliners , others oblique . of direct planes there are five sorts : 1. the horizontal whose plane lies flat , and is parallel to the horizon , beholding the zenith . 2. the south erect , whose plane stands upright , and directly beholds the south . 3. the north erect , whose plane stands upright , and directly beholds the north. 4. the east erect , whose plane stands upright , and directly beholds the east . 5. the west erect , whose plane stands upright , and directly beholds the west . of decliners there are infinite : and yet may be reduced into these two kinds : 1. the south erect plane , declining more or less towards the east or west . 2. the north erect plane , declining more or less towards the east or west . of oblique planes some are direct , others declining ; and are of four sorts : 1. direct inclining planes , which lean towards you , and lie directly in the east , west , north , or south quarters of heaven . 2. direct reclining planes , which lean from you , and lie directly in the east , west , north , or south quarters of heaven . 3. inclining declining planes , which lean towards you , but lie not directly in the east , west , north , or south quarters of heaven : but decline more or less from the north or south , towards the east or west . 4. reclining declining planes , which lean from you , but lie not directly in the east , west , north , or south quarters of heaven : but decline more or less from the north or south , towards the east or west . if the scituation of the plane be not given , you must seek it : for , there are several wayes how to know these several kinds of planes used among artists ; but the readiest and easiest is by an instrument called a declinatory , fitted to the variation of your place : and if it be truly made , you may as safely rely upon it as any other . operation i. the description of the clinatory . the clinatory is made of a square board , as abcd , of a good thickness ; , and the larger the better ; between two of the side is described on the center aa quadrant as ef devided into 90 equal parts or degrees , which are figured with 10 , 20 , 30 , to 90 ; and then back again with the complements of the same numbers to 90 : between the limb and the two semi-diameters is made a round box , into which a magnetical needle is fitted ; and a card of the nautical compass , devided into four nineties , beginning their numbers at the east , west , north , and south points of the compass , from which points the opposite sides of the clinatory receives their names of east , west , north and south . but , note , that the north point of the card must be placed so many degrees towards the east or west sides of the clinatory as the needle varies from the true north point of the world , in the place where you make your dyal ; which your workman that makes your clinatory will know how to fit . upon the center a , whereon the quadrant was described , is fastened a plumb-line , having a plumbet of lead or brass fastned to the end of it , which plumb-line is of such length that the plumbet may fall just into the grove gh , below the quadrant , which is for that purpose made of such a depth that the plumbet may ride freely within it , without stopping at the sides of it . see the figure annexed . with this clinatory you may examine the scituation of planes . as if your plane be horizontal , it is direct : and then for the true scituating your dyal you have only the true north and south line to find : which is done only by setting the clinatory flat down upon the plane , and turning it towards the right or left hand , till you can bring the north point of the needle to hang just over the flower-de-luce , for then if you draw a line by either of the sides parallel to the needle , that line shall be a north and south line . if your plane either recline or incline , apply one of the sides of your clinatory parallel to one of the semi-diameters of the quadrant to the plane , in such sort that the plumb-line hanging at liberty , may fall upon the circumference of the quadrant , for then the number of degrees of the quadrant comprehended between the side of the quadrant parallel to the plane , and the plumb-line shall be the number of degrees for reclination , if the center of the quadrant points upwards ; or inclination , if the center points downwards . if your reclining or inclining plane decline , draw upon it a line parallel to the horizon , which you may do by applying the back-side of the clinatory , and raising or depressing the center of the quadrant , till the plumb-line hang just upon one of the semi-diameters , for then you may by the upper side of the clinatory draw an horizontal line if the plane incline , or by the under side if it recline . if it neither incline or recline , you may draw a horizontal line both by the upper and under sides of the clinatory . having drawn the horizontal line , apply the north side of the clinatory to it , and if the north end of the needle points directly towards the plane , it is then a south plane . if the north point of the needle points directly from the plane , it is a north plane : but if it points towards the east , it is an east plane : if towards the west , a west plane . if it do not point directly either east , west , north , or south , then so many degrees as the needle declines from any of these four points to any of the other of these four points , so many degrees is the declination of the plane . you may find a meridian line another way ; thus , if the sun shine just at noon , hold up a plumb-line so as the shaddow of it may fall upon your plane , and that shaddow shall be a meridian line . operat. ii. to describe a dyal upon a horizontal plane . first draw a north and south line ( which is called a meridian line ) through the middle of the plane : thus , set your declinatory flat upon the plane , and turn it to and fro till the needle hang precisely over the meridian line of the declinatory ; then by the side of the declinatory parallel to its meridian line , draw a straight line on the plane , and if that straight line be in the middle of the plane , it shall be the meridian line , without more ado : but if it be not in the middle of the plane , you must draw a line parallel to it through the middle of the plane for the meridian line , or twelve a clock line : and it shall be the meridian line , and also be the substilar line ; then draw another straight line through the middle of this line , to cut it at right angles for the vi. a clock lines ; and where these two lines cut one another make your centre , whereon describe a circle on your plane as large as you can , which by the meridian line , and the line drawn at right . angles with it will be devided into four quadrants ; one of the quadrants devide into 90 degrees thus , keeping your compasses at the same width they were at when you described the quadrant , place one foot in the twelve a clock line , and extend the other in the quadrant , and make in the quadrant a mark with it ; so shall you have the sixtieth degree marked out : then place one foot of your compasses in the six a clock line , and extend the other in the quadrant , and make in the quadrant another mark with it ; so shall that quadrant be divided into three equal parts ; each of these three equal parts contains 30 degrees : then with your compasses devide one of these three equal parts into three parts , and transfer that distance to the other two third parts of the quadrant , so shall the whole quadrant be devided into nine equal parts . then devide one of these nine equal parts into two equal parts , and transfer that distance to the other eight equal parts , so shall the quadrant be devided into eighteen equal parts . then devide one of these eighteen equal parts into five equal parts , and transfer that distance to the other seventeen equal parts , so shall the whole quadrant be devided into 90 equal parts . each of these 90 equal parts are called degrees . note , that you may in small quadrants devide truer and with less trouble with steel deviders , ( which open or close with a screw for that purpose , ) than you can with compasses . in this quadrant ( thus devided ) count from the substilar or meridian line the elevation of the pole , that is , the number of degrees that the pole of the world is elevated above the horizon of your place , and draw a line from the center through that number of degrees for the stilar line . then on the substilar line choose a point ( where you please ) and through that point draw a line at right angles to the substilar line as long as you can , for the line of contingence , and from that point in the substilar line measure the nearest distance any part of the stilar line hath to that point ; and keeping one foot of your compasses still in that point , set off that distance in the substilar line , and at that distance describe against the line of contingence a semi-circle , which devide from either side the meridian or substilar line into six equal parts thus ; draw a line through the center of this semi-circle parallel to the line of contingence , which shall be the diametral line , and shall devide this semi circle-into two quadrants ; one on one side the substilar line , and the other quadrant on the other side the substilar line : then keeping your compasses at the same distance they were at when you described the semi-circle , place one foot first on one side the diametral line at the intersection of it and the semi-circle , and then on the other side , at the intersection of it and the semi-circle , and extend the other in the semi-circle , and make marks in the semi-circle on either side the substilar line : then place one foot of your compasses at the intersection of the semi-circle and the substilar line , and turn the other foot about on either side the semi-circle and make marks in the semi-circle , so shall the semi-circle be devided into six equal parts : devide one of these equal parts into two equal parts , and transfer that distance to the other five equal parts , so shall the whole semi-circle be devided into twelve equal parts . these twelve devisions are to describe the twelve hours of the day , between six a clock in the morning , and six a clock at night . if you will have half hours you may devide each of these twelve into two equal parts , as before : if you will have quarters you may devide each of these twenty four into two equal parts more , as before . for thus proportioning the devisions in the semi-circle , you may proportion the devisions and sub-devisions of hours upon the dyal plane ; for a straight ruler laid upon each of these devisions , and on the center of this semi-circle , shall shew on the line of contingence the several distances of all the hours and parts of hours on the dyal plane : and straight lines drawn from the center of the dyal plane , through the several devisions on the line of contingence shall be the several hour lines and parts on the dyal plane . but an horizontal dyal in our latitude will admit of four hours more , viz. v , iv , in the morning , and vii , viii , in the evening . therefore in the circle described on the center of the dyal plane transfer the distance between vi and v , and vi and iv , on the other side the six a clock line ; and transfer the distances between vi and vii , and vi and viii on the other side the opposite six a clock hour line , and from the center of the dyal plane draw lines through those transferred distances for the hour lines before and after vi. then mark your hour lines with their respective numbers . the substiler line in this dyal ( as aforesaid ) is xii , from thence towards the right hand mark every successive hour line with i , ii , iii , &c. and from xii towards the left hand with xi , x , ix , &c. the stile must be erected perpendicularly over the substilar line , so as to make an angle with the dyal plane equal to the elevation of the pole of your place . example . you would draw a dyal upon a horizontal plane here at london ; first draw the meridian ( or north and south line ) as xii b , and cross it in the middle with another line at right angles , as vi , vi , which is an east and west line ; where these two lines cut each other as at a , make the center , whereon describe the semi-circle b , vi. vi ; but one of the quadrants , viz. the quadrant from xii to vi , towards the right hand you must devide into 90 equal parts ( as you were taught in fol. 12. ) and at 51 ½ degrees ( which is londons latitude ) make a mark , and laying a straight ruler to the center of the plane , and to this mark draw a line by the side of it for the stiler line . then on the substilar line chuse a point as at c , and through that point draw a line as long as you can perpendicular to the east and west line vi , vi , as ef , ( which is called the contingent line , ) where this contingent line cuts the substilar line place one foot of your compasses , and from thence measure the shortest distance between the point c and the stilar line . and keeping one foot of your compasses still in the point c , set off the shortest distance between the point c and the stilar line on the substilar line , as at d ; which point d shall be a center , whereon with your compasses at the same width you must describe a semi-circle to represent a semi-circle of the equinoctial . this semi-circle devide into six equal parts ( as you were taught fol. 13. ) to each of which equal parts , and to the center of the equinoctial semi-circle lay a straight ruler , and where the straight ruler cuts the line of contingence make marks in the line of contingence . then lay the straight ruler to the semi-circle of the dyal plane , and to each of the marks in the line of contingence , and by the side of it draw twelve straight lines for the twelve fore and afternoon hour lines , viz. from vi in the morning to vi in the evening . then in the quadrant vi b , measure the distance between the vi a clock hour line , and the v a clock hour line , and transfer the same distances from the vi a clock line to vii , and v on both sides the vi a clock hour lines , and through those distances draw from the center of the plane the vii and v a clock hour lines , and measure the distance between the vi a clock hour line and the iv a clock hour line , and transfer the same distance from the vi a clock line to viii and iv , and through those distances draw from the center of the plane the viii a clock and iv a clock hour lines . if you will have the half hours and quarter hours , or any other devision of hours , you must devide each six devisions of the equinoctial into so many parts as you intend , and by a straight ruler laid to the center of the equinoctial , and those devisions in the equinoctial circle make marks in the line of contingence , as you did before for the whole hour lines ; and lines drawn from the center of the plane through those marks shall be the sub-devisions of the hours : but you must remember to make all sub-devisions short lines , and near the verge of the dyal plane , that you may the easier distinguish between the whole hours and the parts of hours ; as you may see in the figure . having drawn the hour lines , set the number of each hour line under it , as you see in the figure . last of all sit a triangular iron , whose angular point being laid to the center of the dyal plane , one side must agree with the substilar line , and its other side with the stilar line ; so is the stile made . and this stile you must erect perpendicularly over the substilar line on the dyal plane , and there fix it . then is your dyal finished . operat. iii. to describe an erect direct south dyal . you may know an erect direct south plane by applying the north side of the declinatory to it ; for then if the north end of the needle hang directly over the north point of the card in the bottom of the box , it is a south plane ; but if it hang not directly over the north point of the card , it is not a direct south plane , but declines either east or west , and that contrary to the pointing of the needle easterly or westerly from the north point of the card : for if the north point of the needle points easterly , the plane declines from the south towards the west : if it point westerly , the plane declines from the south towards the east . you may know if the plane be truly erect or upright , by applying one of the sides ab or ad to it ; for then by holding the center a upwards , so as the plumb-line play free in the grove , if the line falls upon 0 , or 90 , the plane is upright ; but if it hang upon any of the intermediate degrees , it is not upright , but inclines or reclines . if you find it incline , apply the side ab to it , and see what number of degrees the plumb-line falls on , for that number of degrees counted from the side ab , is the number of degrees of inclination . if you find the plane reclines , apply the side ad to it , and see what number of degrees the plumb-line falls on , for that number of degrees counted from the side ad is the number of degrees of reclination . these rules being well understood , may serve you to find the scituation of all other sorts of planes . but for the making a dyal on this plane , you must first draw a meridian line through the middle of the plane , by applying a plumb-line to the middle of it , till the plumbet hang quietly before it : for then if the plumb-line be black't ( for a white ground , or chalked for a dark ground ) and strained as carpenters do their lines , you may with one stroak of the string on the plane describe the meridian line , as a xii : this meridian is also the substilar line . then on the top of this meridian line , as at a , draw another line athwart it to cut it at right angles , as vi , vi , for an east and west line . at the meeting of these two lines on the top , make your center , whereon describe a semi-circle on your plane , as large as you can , which by the meridian line and the east and west line will be devided into two quadrants . one of these quadrants devide into 90 degrees ( as you were taught fol. 12. ) and from the substilar line count the complement of the poles elevation , which ( here at london where the pole is elevated 51 ½ degrees , its complement to 90 ) is 38 ½ degrees , and make there a mark , as at e. then on the substilar line chuse a point ( where you please ) as at f , for the line of contingence to pass through : which line of contingence draw as long as you can , so as it may cut the substilar line at right angles , and from the point f in the substilar line measure the shortest distance between it and the stilar line , and keeping one foot of your compasses still in the point f , transfer that distance into the substilar line , as at g ; then on the point g describe a semi-circle of the equinoctial against the line of contingence , which semi-circle devide into twelve equal parts , ( as you were taught by the example in the horizontal dyal , fol. 13. ) and by a straight ruler laid to each of these devisions , and to the center of the semi-circle make marks in the line of contingence by the side of the ruler : for straight lines drawn from the center of the dyal plane through these marks in the contingent line shall be the 12 hour lines before and after noon . then mark your hour lines with their respective numbers : the substilar or meridian line is xii , from thence towards the right hand with i , ii , iii , &c. and from thence towards the left hand with xi , x , ix , &c. the stile must be erected perpendicularly over the substilar line , so as to make an angle with the dyal plane equal to the complement of the poles elevation , viz. 38 ½ degrees . operat. iv. to make an erect direct north dyal . the erect direct north dyal , stile and all , is made by the same rules , changing upwards for downwards , and the left side for the right , the erect direct south dyal is made : for if the erect direct south dyal be drawn on any transparent plane , as on glass , horn , or an oyled paper , and the horizontal line vi , vi , turned downwards , and the line vii mark't with v , the line viii with iiii , the line v with vii , and the line iiii with viii , then have you of it a north erect direct dyal . all the other hour lines in this dyal are useless , because the sun in our latitude shines on a north face the longest day only before vi in the morning , and after vi at night . operat. v. to describe an erect direct east dyal . hang a plumb-line a little above the place on the wall where you intend to make your dyal , and wait till it hang quietly before the wall : then if the line be rubbed with chalk ( like a carpenters line ) you may by holding the plumbet end close to the wall , and straining it pretty stiff , strike with it a straight line , as carpenters do : this line shall be a perpendicular , as ab . then chuse a convenient point in this perpendicular , as at c , for a center , whereon describe an occult arch , as de ; this arch must contain the number of degrees of the elevation of the equinoctial , counted between d and e , which in our latitude is 38 ½ , or ( which is all one ) the complement of the poles elevation . therefore in a quadrant of the same radius with the occult arch measure 38 ½ degrees , and set them off in the plane from e to d : then from d to the center c in the perpendicular draw the prick't line dc ; this prick't line shall represent the axis of the world. then cross this line at right angles with the line cf , and draw it from c to f , so long as possibly you can : this line shall be the contingent line . then chuse a point in this contingent line , as at vi , draw a line through that point at right angles for the substilar line , as g vi h for the substilar line ; then open your compasses to a convenient width , ( as to vig ) and pitching one foot in the point g , with the other foot describe a semi-circle of the equinoctial against the line of contingence , which semi-circle devide from vi both wayes into six equal parts , as you were taught by the example in the horizontal dyal : and laying a straight ruler on the center of this semi-circle of the equinoctial , and to each of those equal parts mark on the contingent line where the ruler cuts it , for those marks shall be the several points from whence lines drawn parallel to the line cd shall be the respective hour lines . the reason why the contingent line is drawn from vi. to f , so much longer than from vi to c is ; because the hour lines from vi towards xii are more in number towards noon , than they are from vi backward towards iiii : for this dyal will only shew the hours from a little before iv in the morning to almost noon : for just at noon the shaddow goes off the plane ; as you may see if you apply a straight ruler to the center of the equinoctial semi-circle g , and lay it to the point 12 in the semi-circle ; for the straight ruler will then never cut the line of contingence , because the line of contingence is parallel to the line g xii on the equinoctial circle , and lines parallel , though continued to never so great a length never meet . to these hour lines , set figures as may be seen in the scheme . the stile ik of this dyal as well as of all others must stand parallel to the axis of the world ; and also parallel to the face of the plane , and parallel to all the hour lines , and stand directly over the substilar or vi a clock hour-line , and that so high as is the distance of the center of the equinoctial semi-circle from the contingent line . operat. vi. to describe a dyal on an erect direct west plane . an erect direct west dyal , is the same in all respects with an erect direct east dyal : only as the east dyal shews the forenoon hours , so the west shews the afternoon hours . thus if you should draw the east dyal on any transparent plane , as on glass , horn , or oyled paper , on the one side will appear an east dyal , on the other side a west : only the numbers to the hour lines ( as was said before in the north dyal ) must be changed ; for that which in the east dyal is xi , in the west must be i ; that which in the east dyal is x , in the west must be ii ; that which in the east dyal is ix , in the west must be iii , &c. the stile is the same . operat. vii . to describe a dyal on an erect north , or erect south plane declining eastwards or westwards . these four dyals , viz. the erect north declining eastwards , the erect north declining westwards , the erect south declining eastwards , and the erect south declining westwards , are all projected by the same rules ; and therefore are in effect but one dyal differently placed , as you shall see hereafter . first draw on your plane a straight line to represent the horizon of your place , and mark one end of it w for west , and the other end e for east . chuse a point in this horizontal line for a center , as at a , whereon you may describe a circle to comprehend all these four dyals : draw a line as mam perpendicular to the horizontal line we , through the center a for a meridian line , and on that center describe a circle , which by the two lines wae , and mam will be devided into four quadrants , which will comprehend the four dyals aforesaid : for if it be a north declining west you are to draw , the upper quadrant to the left hand serves your purpose : if a south declining west , the same lines continued through the center a into the lower quadrant to the right hand serves your turn ; if a north declining east , the upper quadrant to the right hand serves your turn ; or if a south declining east , the same lines continued through the center a into the lower quadrant to the left hand serves your turn ; and you must draw the declination , complement of the poles altitude , substile , stile and hour lines in it ; but the hour lines must be differently marked as you shall see hereafter . i shall onely give you an example of one of these dyals ; viz. a south declining east . we will suppose you are to draw a dyal that declines from the south 50 degrees towards the east ; here being but one dyal , you need describe but one quadrant of a circle . set off in the lower quadrant wam 50 degrees from the meridian line m towards w , and from the center a draw a straight line through that mark in the quadrant as da , which may be called the line of declination ; then set off from the meridian line the complement of the poles elevation , which in our latitude is 38 ½ degrees , and there draw another line from the center as ap , which we will call the polar line . then take in the horizontal line a convenient portion of the quadrant , as ab , and from the point b draw a line parallel to the meridian line am , and continue that line till it intersect the polar line , as at p , from which point p draw a line parallel to wa , as pc : then measure the distance of ab in the horizontal line , and set off that distance in the line of declination , as from a to d , and from that point of distance draw a line parallel to the meridian am through the horizontal line at r , and through the point d , and continue it through the line pc , as at s ; then laying a straight ruler to the center a , and the intersection of the line pc , at s draw the line as for the substile : then upon the point s erect a line perpendicularly as st ; then measure the distance between r and d , and set that distance off from s to t , and from the center to the point t draw the line at for the stile or gnomon ; and the triangle sat made of iron or brass and erected perpendicularly over the substile sa shall by its upper side ta cast a shaddow upon the hour of the day . but you will say the hour lines must be drawn first : it is true ; therefore to draw them you must chuse a point in the substile line where you think good , and through it draw the line ff as long as you can for the the line of contingence : then with your compasses take the shortest distance between this point and the stile , and transfer that distance below the line of contingence on the substile as at ae , and with your compasses at that distance describe on the center ae a circle to represent the equinoctial ; then ( as you were taught in the example of the horizontal dyal ) devide the semi-circle of the equinoctial into twelve equal parts , beginning at the point in the equinoctial circle , where a straight line drawn from the center of it to the intersection of the line of contingence with the meridian line cuts the equinoctial line , as here at the point g ; then lay a straight ruler to the center of the equinoctial circle , and to every one of the devisions in the semi-circle , and mark where the straight ruler cuts the contingent line ; for straight lines drawn from the center a of the dyal to those several marks on the contingent line shall be the hour lines ; and must be numbred from the noon line or meridian a m backwards , as xii , xi , x , ix . &c. towards the left hand . so is your dyal finished . this dyal drawn on any transparent matter as horn , glass , or an oyled paper , shall on the other side the transparent matter become a south declining west , ( stile and all ) but then the i a clock hour line must be marked ii , the xii xii , the xi a clock hour line i , x , ii , ix , iii , &c. if you project it anew , you must describe the quadrant mw on the other side the meridian line , on the center a from m to e , and then count , ( as before ) the declination , altitude of the pole , substile , and stile in the quadrant , beginning at m towards e , and work in all respects as with the south declining east ; only number this south declining west as in the foregoing paragraph . if you project a north declining east , you must describe the quadrant above the horizontal line from m upwards , towards e on your right hand , and count ( as before ) the declination , altitude , complement of the pole , substile , and stile from the meridian line , and work as with the south declining east : it must be numbred from the meridian line m towards the right hand with xi , x , ix , viii , &c. if this dyal were drawn on transparent matter , the other ▪ side would shew a north declining west : but if you will project it anew , you must describe the quadrant above the horizontal line , from m upwards towards w , and count from the meridian line am the declination , complement , altitude of the pole , substile and stile , and work with them ( in all respects ) as with the south declining east ; but then the xi a clock hour line must be marked i , the x , ii ; the ix , iii , &c. operat. viii . to draw a dyal on an east or west plane reclining , or inclining . draw a straight line parallel to the horizon , to represent a meridian , or xii a clock line , and mark one end n , the other s ; chuse a point in this line , as at a for a center : then if your plane be an east or a west incliner , let fall a perpendicular upon this center , ( that is , the perpendicular must stand above the meridian line ns . ) as ae , and upon the center a describe a semi-circle above the meridian line ns ; ) but if your plane be an east incliner , or a west recliner , let fall a perpendicular from the center a under the meridian line , and upon the center a describe a semi-circle under the meridian line . if your plane be a west incliner , work ( as shall be taught ) in the quadrant on the left hand above the meridian line . if an east recliner , in the quadrant on the right hand above the meridian line . if it be a west recliner , work in the quadrant on the left hand under the meridian . if an east incliner , in the quadrant under the meridian line the right hand . for example , an east dyal reclining 45 degrees . you would draw a dyal on an east plane reclining 45 degrees : therefore in the quadrant on the right hand above the meridian line , set off from the perpendicular ae 45 degrees on the quadrant , for the reclination of the plane ; and set off also in the quadrant 38 ½ degrees from the perpendicular for the complement of the poles elevation , and at these settings off make marks in the quadrant : then lay a straight ruler to the center a , and to the marks in the quadrant , and draw straight lines through them from the center . then chuse in the meridian line ns a convenient point , as at b , and through that point draw a line parallel to the perpendicular ae , which will intersect the line drawn for the complement of the poles elevation ap in p ; from which point p , draw a line parallel to the meridian line ns , to cut the perpendicular ae in c , and also the line of obliquity ao in o. then measure the length ao , and set off that length in the perpendicular ace from a to e , and draw the line eg parallel to the meridian line ns , which will cut the line bp prolonged in g. measure also the length of co , and set that length off from a to q on the line of obliquity ao , and draw the line qr parallel to the perpendicular ace . then measure the distance of ar , and upon the line gpb set it off from g to s ; and laying a straight ruler to the point s and the center a , draw by the side of it the line as ; for the substile line . then measure the length of qr , and from s raise a perpendicular , and in that perpendicular set that length off from s to t ; and laying a straight ruler to the center a and the point t , draw the line at for the stilar line , which stilar line being perpendicularly erected over the substilar line as , will stand parallel to the axis of the world , and cast its shaddow on the hour of the day . to draw the hour lines on this plane , you must ( as you have several times before been directed ) chuse a point in the substilar line , and through that point draw at right angles with the substilar line the line of contingence so long as you can : then measure the shortest distance between that point and the stilar line , and transfer that distance below the line of contingence in the substilar line , as at ae , and with your compasses at that distance describe against the line of contingence the equinoctial circle ; then divide the semi-circle of the equinoctial next the line of contingence into twelve equal parts , ( as you have formerly been taught ) beginning at the point in the equinoctial circle , where a straight line drawn from the center of it to the intersection of the line of contingence with the meridian line ns cuts the equinoctial circle , as here at the point d : then lay a straight ruler to the center of the equinoctial circle , and to every one of the devisions in the equinoctial semi-circle , and mark where the straight ruler cuts the contingent line : for straight lines drawn from the center a of the dyal through these several marks in the contingent line shall be the hour lines , and must be numbred from the meridian or noon-line ns which is the xii a clock line upwards , with xi , x , ix , viii , &c. the center of this dyal must stand downward . if this dyal were turned with its center upwards , it would shew a west inclining 45 degrees , only the numbers to the hour lines must be changed ; for to xi you must set i , to x , ii ; to ix , iii , &c. and the substile over which the stile must stand , must be placed in the semi-circle ( at first described ) as much to the right hand the perpendicular ae , as it doth on the left hand . if this dyal were drawn on glass , horn , or an oyled paper , and you turn the meridian line ns upwards , the backside shall be an east inclining 45 degrees , and the hour lines must be numbred as they are on the east reclining : but the substile over which the stile must stand , must be placed , in the semi-circle ( at first described ) as much to the left hand the perpendicular ae , as it is on the oyled paper to the right hand . if you turn the meridian line ns downwards , the backside shall be a west recliner 45 degrees , and the hour lines must be numbred from the xii a clock line upwards , with i , ii , iii , &c. you must note that all the hour lines of the day will not be described in this single quadrant , nor does the quadrant at all relate to the hour lines ; but is described onely for setting off the complement of the poles elevation and reclination of the plane , that by working ( as hath been shewn ) you may find the place of the substilar line , and the angle the stile makes with it : for having the substilar line , you know how to draw the line of contingence , and to describe the equinoctial circle , by which all the hours are described on the plane . to draw a dyal on a direct south or north plane inclining or reclining . direct reclining or inclining dyals are the same with erect direct dyals that are made for the latitude of some other places ; the latitude of which places are either more than the latitude of your place , if the plane recline ; or less , if the plane incline : and that in such a proportion as the arch of reclination or inclination is . thus a direct south dyal reclining 10 degrees in london's latitude , ( viz. 51 ½ degrees ) is an erect direct south dyal made for the latitude of 61 ½ degrees . and a direct south dyal inclining 10 in the latitude of 51 ½ is an erect direct south dyal in the latitude of 41 ½ degrees : and is to be made according to the direction given in operat . iii. operat. ix . to draw a dyal on a south or north inclining declining , or reclining declining plane . these four sorts of dyals viz. the south inclining declining , and south reclining declining , and north inclining declining , and south reclining declining , are all projected by the same rules ; and therefore are in effect but one dyal differently placed , as you shall see hereafter . first draw on your plane a straight line parallel to the horizon , and mark one end w for west , and the other e for east . on south incliners and recliners , e on the right hand , and w on the left : on north incliners and recliners e on the left hand and w on the right . chuse a point in this horizontal line for a center , as at a ; through this point a draw a line perpendicular to the horizon , and on this point ( as on a center ) describe a semi-circle , one quadrant above , and another below the horizontal line . ( though for this example i describe but one . ) then if the plane respect the south , set off in the lower quadrant from the perpendicular the declination , the inclination , or the reclination , and the complement of the altitude of the pole ; and through these several settings off in the quadrant , draw straight lines from the center a ; then take in the horizontal line towards the semi-circle , a convenient distance from the center a , as b , and through the point b draw a straight line parallel to the perpendicular , and prolong it through the polar line , as bp : through the point p , draw a line parallel to the horizontal line , as pc ; this line will cut the line of obliquity in the point o : then measure the distance of ao , and set off that distance on the perpendicular from a to f , and through the point f draw a straight line parallel to the horizontal line , as fg , for the horizontal intersection . then measure the distance of co , and set off that distance on the perpendicular from a to i ; from the point i , draw the line id parallel to the horizontal line , to cut the line of declination in the point d. then measure the distance of ab , and set off that distance in the line of declination from a to e ; and from the point e draw a straight line parallel to the horizontal line we , to cut the perpendicular in the point k. measure the distance of ek , and set off that distance on the other side the perpendicular in the horizontal intersection , from f to h , and from the point h draw hn parallel to the perpendicular to cut the horizontal line in the point n. then to find the meridian line , substile and stile , do thus . if your plane be a southern incliner , or a northern recliner , measure the distance of ld , and set off that distance in the horizontal intersection from f to m , and through the point m draw the line am for the meridian line . then add the distance of al to ak , thus : measure the distance of al , and place one foot of your compasses in the point k in the perpendicular line , and extend the other to x , and measuring the distance of ax , set it off in the line of obliquity from a to q ; and from the point q draw the line qr parallel to the perpendicular , and cutting the horizontal line in the point r. then measure the distance of ar , and set off that distance from h in the horizontal intersection to s on the line hn , and to the point s draw the line as for the substile . then measure the distance of qr , and set off that distance perpendicularly from the point s to t ; and lastly , from the point a , draw the straight line at for the stilar line , which stilar line being perpendicularly erected over the substilar line as , will stand parallel to the axis of the world , and cast its shadow on the hour of the day . but if the plane be a southern recliner , or northern incliner , measure ( as before ) the distance of ld , and ( as before you were directed ) to set it off from f in the horizontal intersection on the right hand the perpendicular line ; so now , set that distance from f to m in the horizontal intersection on the left hand in the perpendicular line , and draw the line a m for the meridian line . then as before you were directed to add al to ak : so now , substract the distance of al from ak , and the remainder will be lk : set therefore the distance of ik from a to q in the same line of obliquity , and from the point q , draw the line qr parallel to the perpendicular . measure then the distance of a r , and set off that distance in the line hn , from h to s for the substilar line : then erect on the point s a perpendicular , and on that perpendicular set off from s to t the distance of qr : and lastly , from a draw the line a t for the stilar line . if k falls upon l the plane is parallel to the axis of the world , and the dyal drawn upon it will have no center : but s will fall upon h , and ah ( or a s ) will be the substile . i shall give you two examples of these rules : one of a dyal with a center , and the other of a dyal without a center . and first , operat. x. how to draw a dyal with a center , declining 20 degrees , and inclining 30 degrees . having by the foregoing precepts of the last operat . found the substile , stile , and meridian , you must ( as you have often been directed ) chuse a point in the substilar line , through which , at right angles to the substilar line draw the line of contingence as long as you can : then measure the shortest distance between the point of intersection and the stilar line , and transfer that distance on one side the line of contingence upon the substilar line , and so describe the equinoctial semi-circle against the line of contingence : then lay a straight ruler to the center of the equinoctial circle , as at ae , and to the point where the line of contingence cuts the meridian line , as at z , and mark where the straight ruler cuts the equinoctial circle , and from that mark begin to devide the semi-circle into twelve equal parts , and by a straight ruler laid to those devisions and the center of the equinoctial , make marks in the line of contingence . then shall straight lines drawn from the center a of the dyal through every one of those marks in the contingent line be the hour lines of the dyal , and must be numbred from the xii a clock line towards the right hand with i , ii , iii , iv , &c. and the other way with xi , x , ix , &c. operat. xi . how to draw a dyal without a center , on a south plane ; declining east 30 degrees , reclining 34 degrees 32 minutes . having by the precepts of operat . ix found the substile , you must find the meridian line otherwise than you were there taught : for , having drawn the lines of latitude , declination and reclination , and found the substile , measure the distance of bp , and set it off on the line of declination from a to k , and draw from the perpendicular af the line kq parallel to ab : then measure the length of kq , and set it off on the polar line ap , from a to v ; then take the nearest distance between the point v and the line ab , and set it off on the line qk from q to m ; through which point m , draw a line from the center a : then measure with your compasses in the semi-circle wne ( which in this dyal may represent the equinoctial ) the distance of the arch n m , and set off that distance from the intersection of the substile with the semi-circle at s to t in the semi-circle , which point t shall be the point in the equinoctial that you must begin to devide the hours at , for the finding their distances on the line of contingence . then consider ( according to the bigness of your plane ) what heighth your stile shall stand above the substile , and there make a mark in the substile : for the distance between the center a and that mark must be the heighth of the stile perpendicularly erected over the substile , as at i. draw through this point i a line of contingence , as long as you can to cut the substile at right angles , and then laying a ruler to the center a , and successively to each devision of the equinoctial make marks in the line of contingence , and through those marks draw straight lines parallel to the substile , which shall be the hour lines ; and must be numbred from the left hand towards the right , beginning at the xii a clock line with i , ii , iii , &c. and from the right hand towards the left on the xii a clock line with xi , x , ix , &c. the stile to this dyal may be either a straight pin of the length of ai , or else a square of the same heighth , erected perpendicularly upon the point i , in the substile line . operat. xii . to make a dyal on the ceeling of a room , where the direct beams of the sun never come . find some convenient place in the transum of a window to place a small round piece of looking-glass , about the bigness of a groat , or less , so as it may lie exactly horizontal . the point in the middle of this glass we will mark a , and for distinction sake call it nodus . through this nodus you must draw a meridian line on the floor , thus , hang a plumb-line in the window exactly over nodus , and the shadow that the plumb-line casts on the floor just at noon will be a meridian line ; or you may find a meridian line otherwise by the clinatory . having drawn the meridan line on the floor , find a meridian line on the ceeling , thus , hold a plumb-line to the ceeling , over that end of the meridian line next the window ; if the plumbet hang not exactly on the meridian line on the floor , remove your hand on the ceeling one way or other , as you see cause , till it do hang quietly just over it , and at the point where the plumb-line touches the ceeling make a mark , as at b ; that mark b shall be directly over the meridian line on the floor : then remove your plumb-line to the other end of the meridian line on the floor , and find a point on the ceeling directly over it , as you did the former point , as at c , and through these two points b and c on the ceeling , strain and strike a line blackt with smal-coal or any other colour ( as carpenters do ) and that line bc on the ceeling shall be the meridian line , as well as that on the floor : then fasten a string just on the nodus , and remove that string , forwards or backwards , in the meridian line on the ceeling , till it have the same elevation in the quadrant on the clinatory above the horizon that the equinoctial hath in your habitation , and through the point where the string touches the meridian line in the ceeling shall a line be drawn at right angles with the meridian , to represent the equinoctial line . thus in our latitude the elevation of the equator being 38 ½ degrees ; i remove the string fastned to the nodus forwards or backwards in the meridian line of the ceeling , till the plumb-line of the quadrant on the clinatory , when one of the sides are applied to the string , falls upon 38 ½ degrees : and then i find it touch the meridian line at d in the ceeling : therefore at di make a mark , and through this mark strike the line de ( as before i did in the meridian line ) to cut the meridian line at right angles : this line shall be the equinoctial line , and serve to denote the hour distances , as the contingent line does on other dyals , as you have often seen· then i place the center of the quadrant on the clinatory upon nodus , so as the arch of the quadrant may be on the east side the meridian line , and underprop it so , that the flat side of the quadrant may lie parallel to the string , when it is strained between the nodus and the equinoctial , and also so as the string may lie on the semi-diameter of the quadrant , when it is held up to the meridian line on the ceeling . then removing the string the space of 15 degrees in the quadrant , and extending it to the equator on the ceeling , where the string touches the equator , there shall be a point through which the i a clock hour line shall be drawn : and removing the string yet 15 degrees further to the eastwards in the semi-circle of position , and extending it also to the equator , where it touches the equator , there shall be a point through which the ii a clock hour line shall be drawn . removing the string yet 15 degrees f●rther , to the eastwards in the semi-circle of position , and extending it to the equator , there shall be a point through which the iii a clock hour line shall be drawn : the like for all the other after-noon hour lines . so oft as the string is removed through 15 degrees on the quadrant , so oft shall it point out the after-noon distances in the meridian line on the ceeling . having thus found out the points in the equator through which the after-noon hour lines are to be drawn , i may find the fore-noon hour distances also the same way , viz. by removing the arch of the quadrant to the west side the meridian , as before it was placed on the east , and bringing the string to the several 15 degrees on the west side the quadrant ; or else i need only measure the distances of each hours distance found in the equator from the meridian line on the ceeling ; for the same number of hours from xii , have the same distance in the equinoctial line on the other side the meridian , both before and after-noon : the xi a clock hour distance is the same from the meridian line , with the i a clock distance on the other side the meridian ; the x a clock distance , the same with the ii a clock distance ; the ix with the iii , &c. and thus the distances of all the hour lines are found out on the equator . now if the center of this dyal lay within doors , you might draw lines from the center through these pricks in the equator , and those lines should be the hour lines , as in other dyals : but the center of this dyal lies without doors in the air , and therefore not convenient for this purpose : so that for drawing the hour lines , you must consider what angle every hour line in an horizontal dyal makes with the meridian ; that is , at what distance in degrees and minutes the hour lines of an horizontal dyal cut the meridian ; which you may examine , as by operat . ii. for an angle equal to the complement of the same angle , must each respective hour line with the equator on the ceeling have . thus upon the point markt for each hour distance in the equinoctial line on the ceeling , i describe the arches i , ii , iii , iv , as in the figure , and finding the distance from the meridian of the hour lines of an horizontal dyal to be according to the operat . ii. thus , the 1 a clock hour line 11.40 whose complement to 90 is 78.20 the 2 a clock hour line 24.15 whose complement to 90 is 65.45 the 3 a clock hour line 38.14 whose complement to 90 is 51.56 the 4 a clock hour line 53.36 whose complement to 90 is 36.24 i measure in a quadrant of the same radius with those arches already drawn from the equinoctial line for the 1 a clock hour 78.20 for the 2 a clock hour 65.45 for the 3 a clock hour 51.56 for the 4 a clock hour 36.24 and transfer these distances to the arches drawn on the ceeling : for then straight lines drawn through the mark in the arch , and through the mark in the equator , and prolonged both ways to a convenient length , shall be the several hour lines ( aforesaid ; ) and when the sun shines upon the glass at nodus , its beams shall reflect upon the hour of the day . some helps to a young dyalist for his more orderly and quick making of dyals . it may prove somewhat difficult to those that are unpractised in mathematical projections , to devide a circle into 360 degrees ( or which is all one ) a semi-circle into 180 , or a quadrant into 90 degrees ; and though i have taught you in the projecting the horizontal dyal the original way of doing this , yet you may do it a speedier way by a line of chords , which if you will be curious in your practise , you may make your self ; or if you account it not worth your while , you may buy it already made on box or brass of most mathematical instrument-makers . this instrument is by them called a plain scale , which does not only accommodate you with the devisions of a quadrant , but also serves for a ruler to draw straight lines with : the manner of making it is as follows . describe upon a smooth flat even-grain'd board a quarter of an whole circle , as bc , whose radius ab or ac may be four inches , if you intend to make large dyals , or two inches if small ; but if you will , you may have several lines of chords on your scale or rule . devide this quadrant into 90 equal parts as you were taught in the making the horizontal dyal . then draw close by the edge of your straight ruler a line parallel to the edge , and at about 1 / 20 part of an inch a second line parallel to that , and at about ⅛ of an inch a third line parallel to both . then place one foot of your compasses at the beginning of the first degree on the quadrant descibed on the board , as at b , and open the other foot to the end of the first degree , and transfer that distance upon your rule , from b to the first mark or devision , between the two first drawn lines . then place one foot of your compasses again at the begining of the first degree on the quadrant described on the board , as at b , and open the other foot to the end of the second degree , and transfer that distance upon your rule from b to the second mark or devision between the two first drawn lines ; and thus measure the distance of every degree from the first degree described on the quadrant , and transfer it to the rule . but for distinction sake , you may draw every tenth devision from the first line parallel to the edge of the third line , and mark them in succession from the beginning with 10 , 20 , 30 , to 90 : and the fifth devisions you may draw half way between the second and the third parallel lines ; the single devisions only between the two first parallel lines . so is your line of chords made . the vse of the line of chords . as its use is very easie , so its convenience is very great ; for placing one foot of your compasses at the first devision on the scale , and opening the other to the 60 th degree , you may with the points of your compasses ( so extended ) describe a circle , and the several devisions , on the scale shall be the degrees of the four quadrants of that circle , as you may try by working backwards , to what you were just now taught in the making the scale : for as before you measured the distance of the degrees of the quadrant , and transferr'd them to the scale , so now you only measure the d●visions on the scale , and transfer them to the quadrant , semi-circle , or whole circle described on your paper . for example : if you would measure 30 degrees in your described circle , place one foot of your compasses at the begining of devisions on the scale , as at a , and extend the other foot to the divisions marked 30 , and that distance transferred to the circle , shall be the distance of 30 degrees in that circle . do the like for any other number of degrees . you may draw your dyal first on a large sheet of paper , if your dyal plane be so large , if it be not so large , draw it on a smaller piece of paper ; then rub the back-side of your paper-dyal with smal-coal , till it be well black't ; and laying your paper dyal on your dyal plane , so that the east , west , north , or south lines of your paper agree exactly with the east , west , north , or south scituation of your dyal plane . then with wax or pitch fasten the corners of the paper on the plane , and laying a straight ruler on the hour-lines of your dyal , draw with the blunted point of a needle by the side of the ruler , and the smal-coal rub'd on the back-side the paper will leave a mark of the lines on the plane . if you will have the lines drawn red , you may rub the back-side of your paper with vermillion ; if blew , with verditer ; if yellow , with orpment , &c. then draw upon these marked lines with oyl colours , as you please . an explanation of some words of art used in this book . angle . the meeting or joyning of two lines . arch. a part of a circle . axis . the straight line that runs through the center of a sphere , and both ways through the circumference : though in dyalling it is all one with the diameter of a circle . clinatory . see fol. 8 , 9 , 10. chord . see fol. 44 , 45 , 46. complement . the number that is wanting to make up another number 90 degr. or 180 degr. or 360 degrees . contingent . a line crossing the substile at right angles . degree . see fol. 12. diameter . the longest straight line that can be contained within a circle , viz. the line that passes through the center to the circumference both ways . dyal plane . see fol. 7. elevation of the pole. so many degrees as the pole is elevated above the horizon . equinoctial . the equinoctial is a great circle that runs evenly between the two poles of the world. but when we name the equinoctial in this book , we mean a small circle which represents it , and is the circle or arch of a circle which is divided into equal parts to find thereby the unequal parts on the line of contingence . in the horizontal dyal it is that arch of a circle marked gch . horizon . is a great circle encompassing the place we stand upon ; but in dyalling it is represented by a straight line , as in operat . iii. in the south dyal the line vi a vi is the horizontal line . latitude . the latitude of a place is the number of degrees contained between the equinoctial and the place inquired after . line of contingence . see contingent . magnetick needle . the needle touch'd with the loadstone , to make it point to the north. meridian , is a great circle of heaven passing through the north and south points of the horizon ; but in dyalling it is represented by a straight line , as in operat . ii. in the horizontal dyal the line xii a is a meridian line . nadir . the point directly under our feet . nautial compass , is the compass used by navigators , whereon is marked out all the 32 winds or points of the compass . oblique plane . see fol. 7. parallel . see fol. 6 perpendicular . see fol. 5. pole. the north or south points on the globe of the earth , are called north or south pole. quadrant . the fourth part of a circle . radius . half the diameter of a circle . right angle . a straight line that falls perpendicularly upon another straight line , makes at the meeting of those two lines a right angle . semi-circle . half a circle . semi-diameter , the same radius is . sphere . the highest heaven with all its imagined circles is called the sphere . stile . the gnomon or cock of a dyal . substile . the line the stile stands on upon a dyal plane . triangle . a figure consisting of 3 sides and 3 angles . zenith . the point directly over our head. finis . a catalogue of globes coelestial and terrestrial , spheres , mapps , sea-platts , mathematical instruments , and books , made and sold by joseph moxon , on ludgate-hill , at the sign of atlas . globes 26 inches diameter . the price 20 l. the pair . globes , near 15 inches diameter . the price 4 l. globes , 8 inches diameter . the price 2 l. globes , 6 inches diameter . the price 1 l. 10 s. concave hemispheres of the starry orb ; which serves for a case to a terrestrial globe of 3 inches diameter , made portable for the pocket . price 15 s. spheres , according to the copernican hypothesis , both general and particular , 20 inches diameter . price of the general 5 l. of the particular 6 l. of both together 10. spheres , according to the ptolomaick systeme , 14 inches diameter . price 3 l. spheres , according to the ptolomaick systeme , 8 inches diameter . price 1 l. 10 s. gunter's quadrant , 13 inches radius , printed on paper , and pasted on a board , with a nocturnal on the backside . price 5 s. gunter's quadrant , 4 inches radius , printed on paper , and pasted on brass , with a nocturnal on the backside , and a wooden case covered with leather fit for it : a new invention contrived for the pocket . price 6 s. a large mapp of the world , 10 foot long , and 7 foot deep , pasted on cloath and coloured . price 2 l. a mapp of all the world , 4 foot long , and 3 foot deep , pasted on cloath and coloured . price 10 s. in sheets 2 s. 6 d. a mapp of the english empire in america , describing all places inhabited there by the english nation , as well on the islands as on the continent . price 15 s. six scriptural mapps , 1. of all the earth : and how after the flood it was divided among the sons of noah . 2. of paradise , or the garden of eden ; with the countries circumjacent inhabited by the patriarchs . 3. the 40 years travel of the children of israel through the wilderness . 4. of canaan , or the holy land : and how it was divided among the twelve tribes of israel , and travelled through by our saviour and his apostles . 5. the travels of st. paul , and others of the apostles , in their propagating the gospel . 6. jerusalem , as it stood in our saviour's time ; with a book of explanations to these mapps ▪ entituled sacred geography price 6 s. useful to be bound up with bibles . a sea-platt , or mapp of all the world , according to mercator , in two large royal sheets of paper ; set forth by mr. edward wright , and newly corrected by joseph moxon hydrogr . &c. price 2 s. sea platts for sailing to all parts of the world. price 6 d. the sheet . the famous city of batavia in the east-indies , built and inhabited by the duth ; curiously engraved , and printed on four large sheets of royal paper . price 2 s. 6 d. a small mapp of all the world , with descriptions , on one sheet . price 6 d. books . a tutor to astronomy and geography , or the use of both the globes coelestial and terrestrial ; 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wherein the skin , veins , nerves , muscles , bones , sinews and ligaments are accurately delineated . engraven on large copper plates , printed and curiously pasted together , so as at first sight you may behold all the parts of man and woman ; and by turning up the several dissections of the papers , take a view of all their inwards : with alphabetical referrences to the names of every member and part of the body . set forth in latine by remelinus , and michael spaher of tyrol : and englished by john ireton chyrurgeon : and lastly , perused and corrected by several rare anatomists . price 14 s. vignola , or the compleat architect . shewing in a plain and easie way , the rules of the five orders in architecture , viz. tuscan , dorick , ionick , corinthian and composite : whereby any that can but read and understand english , may readily learn the proportions that all members in building have to one another : set forth by mr. james barrozzio of vignola , and translated into english by joseph moxon hydrographer , &c. price 3 s. 6 d. christiologia , or a brief , but true account of the certain year , month , day and minute of the birth of jesus christ. by john butler b. d. and chaplain to his grace james duke of ormond , &c. and rector of lichborough , in the diocess of peterburgh . price 3 s. 6 d. a tutor to astrology , or astrology made easie ; being a plain introduction to the whole art of astrology . whereby the meanest apprehension may learn to erect a figure , and by the same give a determinate judgment upon any question of nativity whatsoever . also new tables of houses , calculated for the latitude of 51 deg . 32 min. also tables of right and oblique ascensions to 6 deg . of latitude . whereunto is added an ephemeris for three years ; with all other necessary tables that belong to the art of astrology . also how to erect a figure the rational way by the tables of triangles , more methodically than hath yet been published ; digested into a small pocket volume , for the conveniency of those that erect figures abroad . by w. eland . price 2 s. the use of a mathematical instrument called a quadrant , shewing very plainly and easily to know the exact height and distance of any steeple , tree , or house , &c. also to know the hour of the day by it ; the heighth of the sun , moon or stars ; and to know the time of the sun-rising and setting , and the length of every day in the year , the place of the sun in the ecliptick , the azimuth , right ascension , and declination of the sun : with many other necessary and delightful conclusions , performed very readily . also the use of a nocturnal , whereby you may learn to know the stars in heaven , and the hour of the night by them . with many other delightful operations . price 6 d. a brief discourse of a passage by the north-pole to japan , china , &c. pleaded by three experiments , and answers to all objections that can be urged against a passage that way . as 1. by a navigation into the north-pole , and two degrees beyond it . 2. by a navigation from japan towards the north-pole . 3. by an experiment made by the czar of muscovy : whereby it appears that to the northward of nova zembla is a free and open sea as far as japan , china , &c. with a mapp of all the discovered land nearest to the pole. by joseph moxon hydrographer &c. price 6 d. regulae trium ordinum literarum typographicarum : or the rules of the three orders of print-letters , viz. the roman , italick , english , capaitals and small . shewing how they are compounded of geometrick figures , and mostly made by rule and compass . useful for writing-masters , painters , carvers , masons , and others that are lovers of curiosity . by joseph moxon hydrographer &c. price 5 s. the use of the astronomical playing cards . teaching an ordinary capacity by them to be acquainted with all the stars in heaven : to know their places , colours , natures and bignesses . also the poetical reasons for every constellation ; very useful , pleasant and delightful for all lovers of ingeniety . by joseph moxon hydrogr . &c. price 6 d. the astronomical cards . by joseph moxon hydrographer , &c. price plain 1 s. coloured 1 s. 6 d. best coloured and the stars gilt 5 s. the genteel house-keepers pastime : or , the mode of carving at the table represented in a pack of playing cards . by w●ich , together with the instructions in this book , any ordinary capacity may easily learn how to cut up , or carve in m●de all the most usual dishes of flesh , fish , fowl , and baked m●●●● ; and how to make the several services of the same at the table ; with the several sawces and garnishes proper to each dish of meat . set forth by several of the best masters in the faculty of carving , and published for publick use. price 6 d. carving cards . by the best carvers at the lord mayors table . price 1 s. compendium euclidis curiosi : or geometrical operations . shewing how with one single opening of the compasses , and a straight ruler all the propositions of euclids first five books are performed . translated out of dutch into english. by joseph moxon . hydrogr . &c. price 1 s. an introduction to the art of species . by sir jonas moore . price 6 d. two tables of ranges , according to degrees of mounture . by henry bond , senior . price 6. d. mechanick exercises : or the doctrine of handy-works , in six monethly exercises ; began january 1. 1677. and monethly continued till june 1678. the first three , viz. the numb . i. numb . ii. numb . iii. teaching the art of smithing . the other three , viz. numb . iiii. numb . v. numb . vi. teaching the art of faynery . accommodated with suitable engraved figures . by joseph moxon hydrographer , &c. price 6 d. each exercise . at the place abovesaid , you may also have all manner of mapps , sea-platts drafts , mathematical books , instruments , &c. at the lowest prizes . advertisement . there is invented by the right honourable the farl of castlemain , a new kind of globe , call'd ( for distinction sake ) the english globe ; being a fix'd and immovable one , performing what the ordinary ones do , and much more , even without their usual appendancies ; as wooden horizons , brazen meridians , vertical circles , horary circles , &c. for it composes it self to the site and position of the world without the marriners compass , or the like forreign help ; and besides other useful and surprising operations ( relating both to the sun and moon , and performed by the shade alone ) we have by it not only the constant proportion of perpendiculars to their shades , with several corollaries thence arising , but also an easie , new , and most compendious way of describing dyals on all planes , as well geometrically , as mechanically : most of which may be taught any one in few hours , though never so unacquainted with mathematicks . to this is added on the pedestal a projection of all the appearing constellations in this horizon , with their figures and shapes . and besides , several new things in it differing from the common astrolabe , ( tending to a clearer and quicker way of operating ) the very principles of all steriographical projections are laid down , and mathematically demonstrated ; as is every thing else of moment throughout the whole treatise . these globes will be made and exposed to sale about august next , ( god willing : ) against which time the book for its use will also be printed , and sold by joseph moxon , on ludgate-hill , at the sign of atlas . dialling made easy, or, tables calculated for the latitude of oxford (but will serve without sensible difference for most parts of england) by the help of which, and a line of chords, the hour-lines may quickly and exactly be described upon most sorts of useful dials : with some brief directions for making two sorts of spot dials / by t.e. edwards, thomas, mathematician. 1692 approx. 212 kb of xml-encoded text transcribed from 37 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-10 (eebo-tcp phase 1). a38104 wing e226 estc r43140 26910403 ocm 26910403 109843 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. a38104) transcribed from: (early english books online ; image set 109843) images scanned from microfilm: (early english books, 1641-1700 ; 1715:6) dialling made easy, or, tables calculated for the latitude of oxford (but will serve without sensible difference for most parts of england) by the help of which, and a line of chords, the hour-lines may quickly and exactly be described upon most sorts of useful dials : with some brief directions for making two sorts of spot dials / by t.e. edwards, thomas, mathematician. 65 p., [3] folded leaves of plates : ill. printed by l. lichfield for sam. clarke, bookseller, oxford : 1692. errata: p. 65. "all instruments for the mathematicks are made by john prujean in oxon"--p. 20. reproduction of original in bodleian 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 sundials -early works to 1800. time -early works to 1800. 2003-05 tcp assigned for keying and markup 2003-06 spi global 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 dialling made easy : or , tables calculated for the latitude of oxford , ( but will serve without sensible difference for most parts of england . ) by the help of which , and a line of chords , the hour-lines may quickly and exactly be described upon most sorts of useful dials . with some brief directions for making two sorts of spot-dials . by t. e. oxford , printed by l. lichfield , for sam. clarke bookseller . 1692. die veneris , 21 april , 1648. ordered by the lords and commons assembled in parliament , that the boursers and treasurers of the colledges in oxforde shall retaine and keepe such monyes as they have received , without making any divident untill they shall receive order from the commitee of lords and commons for the reformation of the universitie of oxon. and that from henceforth , all tenants , and such others as are to pay money or other dutyes to any col●●dge in the universitie of oxford , shall pay the same to the ●eads of houses appointed by authority of parliament , ●●spectively , or to those whom they shall appoint to receive the same : and to no other . and that the acquittance , or acquittances , of such heads of houses , or of such as they shall appoint to receive the same , shall be a sufficient warrant and discharge , to the severall tenants for the payment thereof accordingly : notwithstanding any condition in their leases to the contrary . hen. elsyng , cler. parl. dom. com. to the reader . i think it will not be inconvenient to premise some things concerning these tables , before i give the vses of them. this calculation as it was the product of leisure hours , so it was at first purely design'd for private vse : but understanding that several persons mathematically inclined , were wishing that something of this nature was extant , i was not unwilling that these tables should appear in publick . seeing then they lye open to censure , some will perhaps say , since i had taken the pains of calculating to 60 degr. i should have compleated the whole quadrant of declination . to this i answer , that in dials declining more than 60 deg . the angle which the stile makes with the substile is so small , that the hour-lines will be of no competent distance , and consequently of no , or little use : but to supply this defect , i have laid down a method to draw dials that decline beyond 60 deg . almost as easy as if the tables had been extended to 90 deg . perhaps another objection may be raised , that these tables being calculated only for the latitude of 51 deg . 45 min. they will be of no use any where else . to this i might reply , that they were chiefly design'd for no other place , and printed only for the benefit of young students in this university : but if they serve for any place within 2 deg . on each side their proper latitude ( as i am sure they will , allowing one minutes difference , ) there will be no great reason to complain . dialling made easy , & ' c. fig. i. to draw an horizontal dial. 1. draw the line a , b for 6 and 6 morning and evening . 2. n , s perpendicular to a , b ; this shall be the 12 a clock line . 3. let c the intersection of the foresaid lines be the center of your dial , then take in your compasses 60 degrees of a line of chords , and with that distance upon the center c draw an obscure segment of a circle as a m b. 4. turn to your table of horizontal spaces , and see what degrees and minutes answer to 1 and 11 ; you find they are 11 d 53′ : take therefore 11 d 53′ out of your line of chords , and set it from m to a a , on both sides the meridian . likewise take 24 d 23′ for the 2d and 10th hour , and set it from m to b b. take 38 d 9′ and set it from m to c c ; work after the same manner for the remaining hours . 5. from the center c thro' each of these these points a , b , c , d , e , draw lines , which shall be the hours required . the intermediate spaces , viz. quarter , half , and three-quarters of an hour , are drawn after the same method . 6. lastly , the stile or cock of your dial must always make an angle equal to the lat. of the place . take therefore 51 d 45′ out of your line of chords and set it from m to n , thro' c and n draw the line c n. note . 1. that the stile must stand at right angles with the plane upon the 12 a clock line . 2. that the 12 a clock line must be set exactly north and south . 3. that the whole plane must be laid parallel to the horizon . 4. that the 2 hours above the line of 6 , are drawn by extending the 2 opposite hour-lines thro' the center , as 4 morning is drawn by extending 4 afternoon . fig. ii. directions for a prime-vertical , or , direct south dial. there is little difference between describing a vertical and horizontal dial ; only observe , 1. that the angle which every hour-line makes with the meridian must be taken from the table of prime-vertical spaces , and prickt down on the segment of the circle a , b , m , as before directed in the horizontal dial. 2. that the angle which the stile makes with the meridian must be 38 d 15′ always the complement of your latitude . note . that the stile must stand at right angles with the plane . 2. that the face of the dial must look exactly south , and be placed perpendicular to the horizon . fig. iii. to draw a direct north dial. this is as easy as the former : for a south dial inverted is a north dial. but because this dial looketh toward the north part of the meridian , to which , in these middle latit . without the tropicks , the sun never comes ; therefore must the hours about midnight be omitted ; as 9 , 10 , 11 and 12 at night , and 1 , 2 and 3 in the morn : and the hours 4 , 5 , 7 and 8 must be extended thro the center c , as directed in drawing the horizontal dial : so you will have the hours of 4 , 5 , 6 , 7 and 8 in the morning , and 4 , 5 , 6 , 7 and 8 in the evening . note . that the stile of this dial must point upward to the north pole , as the south dial did downward to the south pole. fig. iv. directions for drawing an east and west dial 1. upon the point c if it be an east dial , or upon the point d if a west , with 60 d of your line of chords , draw an obscure segment of a circle e , f ; then take 38 d 15′ the co-latitude of your place , and set it from e to f , draw c f thro the plane . call this line the equinoctial . 2. assume any two points in this equinoctial at a convenient distance for the hour lines of 11 and 6 , and thro these points g and h draw perpendiculars to the equinoctial . 3. on g with a line of chords draw i , k which shall be 15 d. from g thro k draw the line till it intersects the hour of 6 at l. 4. upon l with 60 d of chords describe an arch of a circle m , n : between the hour line of 6 and g l. divide the arch m , n into five parts with 15 degrees of chords . then turn to your table for east and west dials , and see what numbers stand against each hour , ( and the intermediate spaces , if you please to put them in , ) take the said numbers out of your line of chords , and put them upon the arch n m from n to m. 5. lay a rule from l to each of those divisions o o o , &c. and where the rule intersects the equinoctial line , make marks x x x , &c. lines drawn thro these points x x x , &c. parallel to the hour line of 6 , shall be true hour lines for an east plane from 6 to 11 ; but if you transfer the same distances on the equinoctial before 6 as there be after 6 , and thro those distances draw lines parallel to 6 : you will have also hours before 6 , as 5 , 4 , &c. note . 1. that the stile may be a plate of brass or iron of the same breadth as is the distance between 6 and 9 on the equinoctial , and fixed upon the line 6 6 , perpendicular to the plane . 2. that an east and west dial is the same in all respects , save that the hour lines of 4 , 5 , 6 , 7 , 8 , morn . in an east dial , must be 8 , 7 , 6 , 5 , 4 , even . on the west dial. fig. v. directions for an equinoctial dial. 1. describe a circle a b c d with 60 d of a line of chords . 2. draw the diameters a c and b d at right angles , in the center o. a c shall be the hour of 12 , and b d the hour of 6. 3. turn to your table of equinoctial dials , and see what numbers answer to every hour ; take the said numbers from your line of chords , and prick them down in the circle a b c d on both sides the line of 12 : draw lines from these points thro the center o , and your work is done . note . 1. that the plane of this dial represents the plane of the equinoctial , and must be elevated the same number of degrees toward the south , as the equator in your latitude is . 2. that the line of 12 must stand exactly north and south . 3. you must make a dial on both sides the plane , for the sun shines half the year on one side , and half the year on t'other . 4. the gnomon must be a pin or wire thrust thro the center , and standing perpendicular to the plane on both sides . fig. vi. directions for a polar dial. 1. draw the line a i , then on a the center describe the segment b c d , with 60 deg . of chords . 2. draw e f perpendicular to a i at c , and g h parallel to e f. 3. divide the segment with such numbers as your table for a polar or equinoctial gives you , beginning to divide at c. 4. from a thro each division draw lines till they touch e f. 5. perpendiculars made at these intersections to e f will be the hour-lines . note . 1. that the stile must stand upon 12 , and be equal to the extent between 12 and 3. 2. that g h must be placed horizontally and prime-vertically . 3. the plane must recline so many degrees as the pole doth . fig. xi . directions for finding the declination of a wall. i suppose the benefit and usefulness of these tables may be seen in what has been said already : but their use will much more appear in the inserting of hour-lines upon declining planes , for which they were chiefly intended . before i give directions for decliners , 't will be proper to shew how to find the declination of a wall or window . 1. draw a horizontal line on the wall or plane you would take the declination of . 2. fix a plain board having one streight edge to this line , and parallel to the horizon . 3. draw a line perpendicular to the streight edge , and when the sun shines thereon , hold up a line and plummet , so that the shadow of the line may fall upon the board , crossing that perpendicular line , and make two pricks in the shadow a good way distant from one another , and then instantly take the altitude of the sun with a quadrant . 4. lay a ruler to these two pricks , and draw a line which shall be the sun's azimuth on the board . 5. with the altitude before found , see what the sun's azimuth is on collins's quadrant . 6. take the intersection of the shadow line with the perpendicular for a center , and thereon with 60 deg . of chords describe a segment of a circle i k , then from i toward k set the azimuth found , and thro k and the center draw a right line for the meridian . 7. lastly , the arch intercepted between the meridian on the board , and the perpendicular line apply'd to your line of chords gives the declination . note . that the azimuth must be set on the arch of the circle that way that the true south is . an other way to find the declination . 1. set one side of a quadrant horizontally to the wall ; hold up a string and plummet when the sun shines , and move the string till the shadow of it passes thro the center of the quadrant . 2. see what degree of the limb , the shade falls upon , reckoning from the side of the quadrant which is perpendicular to the wall ; then instantly take the sun's altitude , and find his azimuth by collins's quadrant . 3. if the shade of the string fell between true south and the perpendicular side of the quadrant , add the deg . of the shade to the azimuth , the sum gives the declination ; but if the meridian or south was between the shade and perpendicular , ( which is easily known considering what time of day it is ) then substract the shade from the azimuth , the remainder gives the declination . fig. vii . directions for a declining dial. let the declination given be twenty degrees west-ward . 1. draw the line n s for 12 a clock . 2. in this line choose a center as c , upon which with 60 degrees of chords describe an obscure circle . 3. turn to your table of requisites , and see what is the substile's distance in 20 d of declin . 't is 15 d 5′ . take the said distance out of your line of chords , and set it from m to s : thro c and s draw an obscure line for the substile . 4. now turn to your declination in table of hour distances from substile , and see what numbers stand against each hour ( and ¼ part , if you would insert them in your dial ) transfer these numbers by help of your line of chords into the circle , from s toward a , b , and thro the points draw lines from c the center of your dial ; these are the hours required ▪ 5. and lastly , in your table of requisites , under the stile 's height , and opposite to 20 d of declin . you 'll find 35 d 34′ set this distance from s to t , then draw c , t , and you have the height of your cock or gnomon . note . 1. that the larger the radius of your line of chords is , the better you will insert the distances . 2. if the declination be east , the substile must be placed on the left side the meridian ; and those hours that are next to the numbers in your table must be used : but the contrary , if declination be west . 3. that the gnomon must be fixt upon the substile , and stand at right angles with the plane . fig. viii . directions for a far decliner . because these tables extend no farther than 60 deg . of declination ( and if they had been calculated to 90 deg . they would have been of little use , because the stile 's height being so small , the hour-lines would have been of no competent distance ) and because sometimes there may be occasion to make a dial for a greater declination , i will shew a geometrical way to draw such , by help of a line of chords only . let the declinat . given be 64 d west . 1. make a line b c perpendicular to the horizon of the plane . 2. upon c describe the arch q s , then out of the table of requisites take the substile's distance , and set it from q to n ; take also the stile 's height , and set it from n to h. 3. draw c d for substile , and c e for the stile : then at any convenient distance draw k l parallel to c e the stile . also assume any two points in the substile , and thro them draw the perpend . f g , and h i. 4. take the nearest extent from t to k l , and from u to k l , and set it from t to r , and from u to d. 5. upon d and r , with a line of chords describe two arches of a circle , and set off the inclination of the meridian from m toward p , and from n to o. with 15 d of chord divide the segments into equal parts , beginning at o and p. 6. lay a rule on d , and each division in the arch o n , and mark the intersection of the rule with f g : then lay the rule to r , and work after the same manner in the other arch , and h i. 7. lastly , thro the marks made in h i , and g f , draw the hour-lines . note . 1. when the plane declines westward , the perpendicular c d must be put on the right hand : et vice versâ . 2. that the inclination of meridian must be set on the same side the substile with the perpendicular c b. fig. ix . and xii . directions for making the spot dial. having given some directions for using these tables in most sorts of common plain dials , i should now conclude ; but i suppose it will not be unwelcome to some young practitioners in dialling , if i give a short and easy method for making spot dials . 't is true one clark has treated largly on this subject already , and as he tells the reader that he believes little will be added by him that comes after , to what he shall there say ; so in my judgment he has said a great deal too much , for by endeavouring to make the thing appear more plain , he has made it the more confused and obscure . i shall be as brief as may be . there are two sorts of these dials , one in which the spot moves to the lines , the other in which the lines move to the spot : i will speak of the first chiefly , because more useful , tho the second more curious . 1. then you must get a wood frame , about an inch and ¼ thick , 7 inches long , and about 4 broad : order the joyner to cut this board sloping from a b c d till he meets with the square a b c d , on the other side , and let him make two grooves or rappits as you see in fig. ix to receive two glasses . let the glass a b c d be 5 inches ½ long , and 3 inches broad . the glass a b c d may be of any size , the less the better . f g h i must be taken off , and put on at pleasure , having rappits in it also to keep the glasses fixt . let the distance between the two glasses in the frame be one inch exactly if you can , but there is no great need of being so curious . so much for the frame . 2. having found the declination of your window , if it has any , draw your dial upon paper as above directed : with the substile and stile upon it . 3. now to find where the spot will be ; remember that it will be always in the substile , and as far distant from the stile as is the distance between the two glasses . take therefore the distance of both glasses as they ly in the frame , between the points of your compass , then lay one side of your square on the substile , and the other crossing the stile ; then move the square upon the substile till the extent of the compass does exactly touch both stile and substile , and in that point of the substile which does touch the square , exactly in the angle make the spot ☉ ; then prick the paper thro with a pin , and on the back-side also make a spot with ink. 4. draw the line a b perpendicular to the 1 2 a clock line , and either thro the spot , or a little above it . that part of the dial that is above this line may be cut off , being of no use . 5. when you have finisht your dial on the paper , wet it with gum-water , or any thing that will make the figured side stick to the glass , so as that the hour-lines may appear thro it , and that the line of 1 2 may be perpendicular to the horizon when the frame is laid in your window . 6. place both your glasses in the frame , then laying it flat upon the table with the lesser glass toward you , look thro it till you can see your reflected eye , and the spot of ink on the back-side of the dial in a right line , then while your eye is in that posture , with your pen make a mark upon the glass which will just cover the spot below . next paste a piece of paper upon the other side of this glass , through which paper , when it is dry , cut a little hole opposite to the mark before made ; then dawb the other part of the paper round the hole with ink , that it may admit nolight . having done this , put the glass again into its place , and fastning the top to your frame , set it close to your glass-window , and the sun entring thro the little hole will cast its ray upon the opposite hour-lines , and so give the time of day . note . that if you please you may lay aside the glasses , and use strong dutch paper alone ; first wetting it , and then pasteing the edges of it , strain it upon your frame so stiff as may be without cracking it . paste also another paper on the back-side of your frame , and when both are dry , thrust a streight needle thro the dial spot till it does just touch the paper on the back-side , then by the help of a rule and square , setting the rule upon the substile , and keeping it upright with your square , direct the top of your needle by the side of the square , till it stands exactly upright in the angle which is made by the rule and square , when it is in this posture , push the needle quite thro the lower paper , and that will be the hole thro which the sun must shine . this sort of dial does look very well till the heat of the sun warps the paper , and so makes the hour lines crooked . that sort of dial wherein the hour-lines move to the spot is only the former inverted : but whereas the frame in the former was cut sloping , in this it must be cut streight thro , that the hour-lines may all appear upon the opposite paper ; and note also that the glasses must be of an equal bigness , and the back-side of your paper on which the dial is drawn must be fasten'd to the glass , that you may with a pen-knife cut out the hour-lines and figures , so the sun shining thro these spaces , will form artificial lines upon the oppose paper , and move one after t' other by the spot , as the sun moves in the heavens . thus i have been as short , and ( i hope ) as perspicuous as may be in these directions , and if i can have the happiness to please some ( for i do not expect to satisfy all ) i have my end . all instruments for the mathematicks are made by john prujean in oxon . the three requisites in dialling calculated to every degree of declination in lat. 51.45 . declinat . stile 's heighth . substile's distance a merid. inclination of meridian . declinat . stile 's heighth . substile's distance a merid. inclination of meridian .   ° ′ ° ′ ° ′   ° ′ ° ′ ° ′ 1 38. 14 0. 47 1. 16 46 25. 29 29. 33 52. 50 2 38. 13 1. 34 2. 32 47 24. 58 29. 58 53. 48 3 38. 11 2. 21 3. 48 48 24. 28 30. 22 54. 45 4 38. 8 3. 8 5. 4 49 23. 58 30. 45 55. 41 5 38. 5 3. 55 6. 20 50 23. 27 31. 8 56. 39 6 38. 0 4. 42 7. 37 51 22. 56 31. 30 57. 34 7 37. 55 5. 28 8. 51 52 22. 24 31. 51 58. 29 8 37. 49 6. 14 10. 6 53 21. 52 32. 12 59. 23 9 37. 42 7. 2 11. 25 54 21. 20 32. 32 60. 19 10 37. 34 7. 48 12. 40 55 20. 48 32. 51 61. 12 11 37. 26 8. 33 13. 54 56 20. 15 33. 10 62. 6 12 37. 16 9. 18 15. 8 57 19. 42 33. 28 63. 0 13 37. 06 10. 2 16. 21 58 19. 9 33. 46 63. 52 14 36. 55 10. 48 17. 38 59 18. 35 34. 3 64. 45 15 36. 43 11. 32 18. 51 60 18. 2 34. 19 65. 36 16 36. 31 12. 15 20. 3 61 17. 28 34. 35 66. 30 17 36. 18 12. 59 21. 17 62 16. 54 34. 50 67. 19 18 36. 2 13. 41 22. 28 63 16. 19 35. 5 68. 11 19 35. 50 14. 24 23. 41 64 15. 45 35. 19 69. 2 20 35. 34 15. 5 24. 51 65 15. 10 35. 32 69. 51 21 35. 19 15. 47 26. 4 66 14. 35 35. 46 70. 45 22 35. 2 16. 27 27. 13 67 14. 0 35. 58 71. 34 23 34. 44 17. 7 28. 23 68 13. 44 36. 10 72. 25 24 34. 26 17. 47 29. 33 69 12. 49 36. 21 73. 13 25 34. 8 18. 15 30. 38 70 12. 13 36. 32 74. 4 26 33. 49 19. 4 31. 51 71 11. 38 36. 42 74. 52 27 33. 29 19. 41 32. 58 72 11. 2 36. 51 75. 38 28 33. 8 20. 19 34. 7 73 10. 26 37. 1 76. 32 29 32. 47 20. 55 35. 13 74 9. 34 37. 9 77. 17 30 32. 25 21. 31 36. 20 75 9. 13 37. 17 78. 5 31 32. 3 22. 6 37. 26 76 8. 37 37. 25 78. 57 32 31. 40 22. 40 38. 30 77 8. 9 37. 32 79. 46 33 31. 17 23. 14 39. 35 78 7. 24 37. 37 80. 31 34 30. 53 23. 47 40. 39 79 6. 47 37. 44 81. 19 35 30. 28 24. 19 41. 42 80 6. 10 37. 50 82. 12 36 30. 3 24. 52 42. 47 81 5. 41 37. 54 82. 52 37 29. 36 25. 23 43. 50 82 4. 57 37. 59 83. 47 38 29. 12 25. 53 44. 51 83 4. 19 38. 2 84. 23 39 28. 45 26. 23 45. 53 84 3. 43 38. 6 85. 21 40 28. 18 26. 52 46. 53 45 3. 05 38. 9 86. 12 41 27. 51 27. 21 47. 55 86 2. 29 38. 11 86. 54 42 27. 23 27. 49 48. 54 87 1. 52 38. 13 87. 50 43 26. 55 28. 16 49. 54 88 1. 12 38. 14 88. 26 44 26. 26 28. 42 50. 53 89 0. 37 38. 14 88. 53 45 25. 57 29. 8 51. 51 90 0. 0 38. 15 90. 00   a table shewing what angle every quarter , half and 3 quarters of an hour makes in a horizontal dial.   a table for the same angles in a directsouth dial.   a table for the same angles in an equinoctial and polar dial. in lat. 51. d. 45. m.   ° ′   ° ′   ° ′ 12 00. 00 12 00. 00 12 00. 00 1 02. 57 1 02. 19 1 03. 45 2 05. 54 2 04. 40 2 7. 30 3 08. 52 3 07. 01 3 11. 15 11.1 11. 53 11.1 09. 25 11.1 15. 00 1 14. 55 1 11. 52 1 18. 45 2 18. 01 2 14. 23 2 22. 30 3 21. 10 3 16. 58 3 26. 15 10.2 24. 23 10.2 19. 40 10.2 30. 00 1 27. 41 1 22. 28 1 33. 45 2 31. 04 2 25. 24 2 37. 30 3 34. 33 3 28. 30 3 41. 15 9.3 38. 09 9.3 31. 46 9.3 45. 00 1 41. 51 1 35. 13 1 48. 45 2 45. 40 2 38. 54 2 52. 30 3 49. 36 3 42. 49 3 56. 15 8.4 53. 40 8.4 47. 00 8.4 60. 00 1 57. 53 1 51. 27 1 63. 45 2 62. 11 2 56. 13 2 67. 30 3 66. 37 3 61. 15 3 72. 15 7.5 71. 09 7.5 66. 36 7.5 75. 00 1 75. 38 1 72. 11 1 78. 45 2 80. 29 2 77. 00 2 82. 30 3 85. 14 3 83. 58 3 86. 15 6.6 90. 00 6.6 90. 00 6.6 90. 00 a table of hour-distances , and parts of an hour from the substile . declin . 1   2   3   ° ′   ° ′   ° ′ 6.6 87. 57 3 91. 58 3 89. 55 1 81. 56 6.6 85. 55 6.6 82. 18 2 76. 00 1 79. 54 1 77. 54 3 70. 16 2 74. 02 2 72. 05 5.7 64. 45 3 68. 22 3 66. 30 1 59. 31 5.7 62. 56 5.7 61. 10 2 54. 33 1 57. 46 1 56. 07 3 49. 54 2 52. 57 2 51. 21 4.8 45. 59 3 48. 47 3 46. 54 1 41. 27 4.8 44. 07 4.8 42. 43 2 37. 37 1 40. 07 1 38. 48 3 34. 01 2 36. 22 2 35. 08 3.9 30. 37 3 32. 50 3 31. 41 1 27. 25 3.9 29. 31 3.9 28. 25 2 24. 23 1 26. 22 1 25. 20 3 21. 30 2 23. 24 2 22. 24 2.10 18. 44 3 20. 35 3 19. 36 1 16. 05 2.10 17. 49 2.10 16. 55 2 13. 31 1 15. 12 1 14. 20 3 11. 02 2 12. 40 2 11. 49 1.11 8. 36 3 10. 12 3 9. 22 1 6. 13 1.11 7. 47 1.11 6. 58 2 3. 52 1 5. 25 1 4. 37 3 2. 08 2 3. 40 2 2. 17   substile .         substile . 12 0. 47 3 0. 45 3 0. 02         sub .       1 3. 06 12 1. 34 12 2. 21 2 5. 27 2 3. 34 1 4. 41 3 7. 49 1 6. 14 2 7. 02 11.1 10. 14 3 8. 38 3 9. 26 1 12. 42 11.1 11. 04 11.1 11. 53 2 15. 15 1 13. 33 1 14. 24 3 17. 53 2 15. 46 2 17. 00 10.2 20. 35 3 18. 46 3 19. 41 1 23. 26 10.2 21. 32 10.2 22. 29 2 26. 25 1 24. 25 1 25. 25 3 29. 34 2 27. 26 2 28. 30 9.3 32. 54 3 30. 39 3 31. 46 1 36. 25 9.3 34. 03 9.3 35. 13 2 40. 11 1 37. 39 1 38. 54 3 44. 11 2 41. 30 2 42. 50 8.4 48. 27 3 45. 35 3 47. 00 1 53. 01 8.5 49. 57 8.4 51. 29 2 57. 53 1 54. 37 1 56. 14 3 63. 01 2 59. 34 2 61. 18 7.5 68. 27 3 64. 49 3 66. 38 1 74. 07 7.5 70. 20 7.5 72. 14 2 79. 58 1 76. 05 1 78. 03 3 86. 05 2 82. 00 2 84. 02 declin . 4   5   6   ° ′   ° ′   ° ′ 3 87. 52 3 85. 47 2 88. 07 6.6 81. 49 6.6 79. 48 3 83. 44 1 75. 53 1 73. 55 6.6 77. 45 2 70. 09 2 68. 14 1 71. 55 3 64. 31 3 62. 48 2 66. 18 5.7 59. 23 5.7 57. 39 3 60. 58 1 54. 26 1 52. 48 5.7 55. 55 2 49. 47 2 48 15 1 51. 09 3 45. 26 3 43. 59 2 46. 42 4.8 41. 20 4.8 39. 59 3 42. 33 1 37. 30 1 36. 15 4.8 38. 37 2 33. 55 2 32. 43 1 34. 58 3 30. 32 3 29. 24 2 31. 31 3.8 27. 20 3.9 26. 16 3 28. 16 1 24. 18 1 23. 17 3.9 25. 11 2 21. 25 2 20. 27 1 22. 16 3 18. 40 3 17. 45 2 19. 28 2.10 16. 01 2.10 15. 07 3 16. 49 1 13. 27 1 12. 36 2.10 14. 14 2 10. 58 2 10. 10 1 11. 44 3 8. 33 3 7. 44 2 9. 17 1.11 6. 10 1.11 5. 22 3 6. 55 1 3. 49 1 3. 02 1.11 4. 34 2 1. 30 2 0. 36 1 2. 14   substile .   substile .   substile . 3 0. 49 3 1. 36 2 0. 04 12 3. 08 12 3. 55 3 2. 23 1 5. 28 1 6. 16 12 4. 43 2 7. 50 2 8. 38 1 7. 03 3 10. 15 3 11. 04 2 9. 26 11.1 12. 43 11.1 13. 32 3 11. 52 1 15. 15 1 16. 06 11.1 14. 24 2 17. 52 2 18. 45 1 16. 58 3 20. 35 3 21. 31 2 19. 39 10.2 23. 26 10.2 24. 24 3 22. 27 1 26. 25 1 27. 26 10.2 25. 23 2 29. 33 2 30. 38 1 28. 28 3 32. 53 3 34. 01 2 31. 34 9.3 36. 25 9.3 37. 37 3 35. 11 1 40. 10 1 41. 28 9.3 38. 51 2 44. 11 2 45. 33 1 42. 47 3 48. 28 3 49. 56 2 46. 58 8.4 53. 01 8.4 54. 36 3 51. 27 1 57. 53 1 59. 34 8.4 56. 13 2 63. 03 2 64. 49 1 61. 17 3 68. 29 3 70. 21 2 66. 39 7.5 74. 10 7.5 76. 07 3 72. 16 1 80. 03 1 82. 04 7.5 78. 07 2 86. 04 2 88. 07 1 84. 07 declin . 7   8   9   ° ′   ° ′   ° ′ 2 87. 48 2 84. 46 2 83. 36 3 81. 44 3 79. 43 3 77. 35 6.6 75. 47 6.6 73. 48 6.6 71. 44 1 70. 01 1 68. 05 1 66. 06 2 64. 29 2 62. 38 2 60. 44 3 59. 14 3 57. 29 3 55. 44 5.7 54. 16 5.7 52. 37 5.7 50. 55 1 49. 37 1 48. 03 1 46. 28 2 45. 15 2 43. 47 2 42. 17 3 41. 10 3 39. 48 3 38. 23 4.8 37. 20 4.8 36. 03 4.8 34. 44 1 33. 45 1 32. 32 1 31. 18 2 30. 23 2 29. 16 2 28. 04 3 27. 11 3 26. 07 3 25. 00 3.9 24. 11 3.9 23. 09 3.9 22. 06 1 21. 19 1 20. 20 1 19. 19 2 18. 33 2 17. 38 2 16. 40 3 15. 55 3 15. 01 3 14. 07 2.10 13. 23 2.10 12. 31 2.10 11. 38 1 10. 54 1 10. 04 1 9. 12 2 8. 29 2 7. 40 2 6. 50 3 6. 08 3 5. 20 3 4. 30 1.11 3. 47 1.11 3. 01 1.11 2. 12               sub. 1 1. 29 1 0. 43 1 0. 06   sub.   sub.       2 0. 50 2 1. 36 2 2. 24 3 3. 09 3 3. 54 3 4. 43 12 5. 28 12 6. 14 12 7. 02 1 7. 49 1 8. 36 1 9. 14 2 10. 13 2 11. 00 2 11. 50 3 12 41 3 13. 29 3 14. 20 11.1 15. 12 11.1 16. 02 11.1 16. 54 1 17. 49 1 18. 40 1 19. 34 2 20. 31 2 21. 25 2 22. 21 3 23. 22 3 24. 17 3 25. 16 10.2 26. 20 10.2 27. 18 10.2 28. 21 1 29. 28 1 30. 30 1 31. 35 2 32. 47 2 33. 52 2 35. 02 3 36. 19 3 37. 28 3 38. 44 9.3 40. 04 9.3 41. 20 9.3 42. 39 1 44. 04 1 45. 24 1 46. 50 2 48. 22 2 49. 44 2 51. 19 3 52. 56 3 54. 28 3 56. 06 8.4 57. 48 8.4 59. 26 8.4 61. 12 1 62. 59 1 64. 43 1 66. 35 2 68. 26 2 70. 16 2 72. 14 3 74. 09 3 76. 04 3 78. 07 7.5 80. 03 7.5 82. 02 7.5 83. 36 1 86. 06 1 88. 07 1 89. 44 declin . 10   11   12   ° ′   ° ′   ° ′ 1 87. 41 1 85. 39 1 83. 36 2 81. 34 2 79. 32 2 77. 31 3 75. 34 3 73. 35 3 71. 36 6.6 69. 46 6.6 67. 51 6.6 65. 56 1 64. 13 1 62. 22 1 60. 33 2 58. 56 2 57. 11 2 55. 27 3 53. 58 3 52. 19 3 50. 40 5.7 49. 18 5.7 47. 45 5.7 46. 13 1 44. 57 1 43. 30 1 42. 02 2 40. 52 2 39. 30 2 38. 10 3 37. 04 3 35. 46 3 34. 32 4.8 33. 29 4.8 32. 16 4.8 31. 05 1 30. 08 1 28. 59 1 27. 55 2 26. 57 2 25. 53 2 24. 50 3 23. 57 3 22. 56 3 21. 57 3.9 21. 06 3.9 20. 08 3.9 19. 10 1 18. 22 1 17. 27 1 16. 32 2 15. 45 2 14. 52 2 14. 00 3 13. 14 3 12. 23 3 11. 32 2.10 10. 46 2.10 9. 57 2.10 9. 08 1 8. 23 1 7. 35 1 6. 47 2 6. 01 2 5. 15 2 4. 28 3 3. 43 3 2. 57 3 2. 12               sub . 1.11 1. 25 1.11 0. 40 1.11 0. 05   sub .   sub .       1 0. 52 1 1. 36 1 2. 21 2 3. 10 2 3. 54 2 4. 38 3 5. 35 3 6. 10 3 6. 57 12 7. 48 12 8. 33 12 9. 18 1 10. 18 1 10. 57 1 11. 42 2 12. 43 2 13. 23 2 14. 10 3 15. 15 3 15. 55 3 16. 43 11.1 17. 51 11.1 18. 32 11.1 19. 22 1 20. 33 1 21. 17 1 22. 08 2 23. 22 2 24. 08 2 25. 01 3 26. 21 3 27. 09 3 28. 05 10.2 29. 29 10.2 30. 20 10.2 31. 19 1 32. 38 1 33. 41 1 34. 45 2 36. 20 2 37. 17 2 38. 25 3 40. 05 3 41. 07 3 42. 19 9.3 44. 07 9.3 45. 13 9.3 46. 30 1 48. 32 1 49. 36 1 51. 00 2 53. 00 2 54. 17 2 55. 48 3 57. 55 3 59. 17 3 60. 55 8.4 63. 08 8.4 64. 36 8.4 66. 22 1 68. 38 1 70. 11 1 73. 35 2 74. 23 2 76. 02 2 77. 57 3 80. 21 3 82. 04 3 84. 03 7.5 86. 27 7.5 88. 12 7.5 90. 13 declin . 13   14   15   ° ′   ° ′   ° ′ 1 81. 35 1 79. 27 7.5 83. 34 2 75. 32 2 73. 26 1 77. 25 3 69. 40 3 67. 38 2 71. 26 6 6. 64. 03 6.6 62. 07 3 65. 42 1 58. 45 1 56. 54 6.6 60. 16 2 53. 45 2 52. 00 1 55. 09 3 49. 05 3 47. 26 2 50. 25 5.7 44. 43 5.7 43. 10 3 45. 53 1 40. 38 1 39. 11 5.7 40. 38 2 36. 46 2 35. 28 1 37. 49 3 33. 16 3 31. 59 2 34. 11 4.8 29. 55 4.8 28. 43 3 30. 46 1 26. 46 1 25. 38 4.8 27. 35 2 23. 35 2 22. 42 1 24. 34 3 20. 56 3 19. 56 2 21. 42 3.9 18. 14 3.9 17. 16 3 18. 58 1 15. 39 1 14. 43 3.9 15. 41 2 13. 08 2 12. 15 1 13. 32 3 10. 39 3 9. 51 2 11. 24 2.10 8. 20 2.10 7. 30 3 9. 02 1 6. 08 1 5. 12 2.10 6. 43 2 3. 43 2 2. 56 1 4. 27 3 1. 27 3 0. 40 2 2. 11   sub .   sub .   sub 1.11 0. 49 1.11 1. 35 3 0. 03 1 3. 05 1 3. 49 1.11 2. 18 2 5. 22 2 6. 07 1 4. 33 3 7. 40 3 8. 27 2 6. 51 12 10. 02 12 10. 49 3 9. 10 1 12. 27 1 13. 14 12 11. 32 2 14. 30 2 15. 44 1 13. 58 3 17. 30 3 18. 20 2 16. 29 11.1 20. 11 11.1 21. 03 3 19. 17 1 22. 58 1 23. 52 11.1 21. 51 2 25. 55 2 26. 51 1 24. 43 3 29. 00 3 30. 01 2 27. 45 10.2 32. 18 10.2 33. 22 3 30. 57 1 35. 48 1 36. 56 10.2 34. 22 2 39. 33 2 40. 46 1 37. 23 3 43. 32 3 44. 51 2 41. 55 9.3 47. 49 9.3 49. 14 3 46. 07 1 52. 25 1 53. 57 9.3 50. 36 2 57. 12 2 58. 58 1 55. 25 3 62. 33 3 64. 18 2 60. 33 8.4 68. 04 8.4 69. 57 3 66. 00 1 73. 52 1 75. 51 8.4 71. 45 2 79. 52 2 81. 56 1 77. 44 3 86. 01 3 88. 08 2 83. 54 7.5 92. 14 7.5 94. 23 3 90. 10 declin . 16   17   18   ° ′   ° ′   ° ′ 7.5 81. 33 7.5 79. 28 7.5 77. 27 1 75. 24 1 73. 22 1 71. 22 2 69. 29 2 67. 29 2 65. 31 3 64. 20 3 61. 54 3 60. 03 6.6 58. 28 6.6 56. 39 6.6 54. 53 1 53. 26 1 51. 43 1 50. 04 2 48. 44 2 47. 08 2 45. 33 3 44. 22 3 42 52 3 41. 21 5.7 40. 19 5.7 38. 53 5.7 37. 31 1 36. 30 1 35. 10 1 33. 54 2 32. 56 2 31. 42 2 30. 30 3 29. 36 3 28. 27 3 27. 18 4.8 26. 29 4.8 25. 23 4.8 24. 19 1 23. 31 1 22. 29 1 21. 29 2 20. 43 2 19. 44 2 18. 47 3 18. 02 3 17. 06 3 16. 12 3.9 15. 28 3.9 14. 35 3.9 13. 43 1 12. 59 1 12. 08 1 11. 19 2 10. 30 2 9. 46 2 8. 58 3 8. 15 3 7. 27 3 6. 42 2.10 5. 58 2.10 5. 11 2.10 4. 27 1 3. 42 1 2. 57 1 2. 16 2 1. 28 2 0. 43 2 0. 00   sub .   sub .   sub . 3 0. 47 3 1. 30 3 2. 11 1.11 3. 01 1.11 3. 44 1.11 4. 24 1 5. 16 1 5. 59 1 6. 39 2 7. 32 2 8. 16 2 8. 56 3 9. 57 3 10. 35 3 11. 16 12 12. 15 12 12. 59 12 13. 41 1 14. 42 1 15. 27 1 16. 09 2 17. 14 2 18. 01 2 18. 44 3 19. 53 3 20. 41 3 21. 26 11.1 22. 39 11.1 23. 58 11.1 24. 15 1 25. 33 1 26. 29 1 27. 15 2 28. 38 2 29. 34 2 30. 26 3 31. 54 3 32. 54 3 33. 50 10.2 35. 22 10.2 36. 27 10.2 37. 26 1 39. 06 1 40. 15 1 41. 19 2 43. 05 2 44. 20 2 45. 30 3 47. 22 3 48. 43 3 49. 59 9.3 51. 58 9.3 53. 25 9.3 54. 51 1 54. 26 1 58. 28 1 59. 58 2 62. 09 2 63. 50 2 65. 28 3 67. 43 3 69. 31 3 71. 16 8.4 73. 34 8.4 75. 29 8.4 77. 20 1 79. 39 1 81. 39 1 83. 35 2 85. 53 2 87. 57 2 89. 56 3 92. 11 3 94. 15 3 97. 55 declin . 19   20   21   ° ′   ° ′   ° ′ 7.5 75. 22 3 79. 35 3 77. 12 1 69. 22 7.5 73. 23 7.5 71. 02 2 63. 38 1 66. 56 1 65. 09 3 58. 13 2 61. 45 2 59. 36 6.6 53. 09 3 56. 26 3 54. 24 1 48. 52 6.6 51. 28 6.6 49. 32 2 44. 02 1 46. 51 1 45. 03 3 39. 58 2 42. 33 2 40. 52 5.7 36. 10 3 38. 34 3 37. 00 1 32. 38 5.7 34. 32 5.7 33. 33 2 29. 19 1 31. 25 1 30. 11 3 26. 13 2 28. 10 2 27. 02 4.8 23. 17 3 25. 08 3 24. 03 1 20. 30 4.8 22. 16 4.8 21. 15 2 17. 51 1 19. 33 1 18. 35 3 15. 19 2 16. 57 2 16. 12 3.9 12. 52 3 14. 27 3 13. 35 1 10. 30 3.9 12. 03 3.9 11. 13 2 8. 12 1 9. 43 1 8. 55 3 5. 56 2 7. 26 2 6. 40 2.10 3. 43 3 5. 12 3 4. 27 1 1. 30 2.10 3. 00 2.10 2. 17   sub .             2 0. 42 1 0. 49 1 0. 06         sub .   sub . 3 2. 53 2 1. 22 2 2. 04 1.11 5. 07 3 3. 44 3 4. 13 1 7. 21 1.11 5. 46 1.11 6. 27 2 9. 39 1 08. 00 1 8. 42 3 11. 59 2 10. 18 2 10. 59 12 14. 24 3 12. 39 3 13. 21 1 16. 54 12 15. 04 12 15. 47 2 19. 31 1 17. 36 1 18. 20 3 21. 47 2 20. 13 2 20. 59 11.1 25. 07 3 22. 59 3 23. 49 1 28. 09 11.1 25. 54 11.1 26. 44 2 31. 23 1 28. 58 1 29. 52 3 34. 50 2 32. 16 2 33. 13 10.2 38. 32 3 35. 47 3 36. 48 1 42. 30 10.2 39. 33 10.2 40. 40 2 46. 46 1 43. 36 1 44. 49 3 51. 22 2 47. 59 2 49. 18 9.3 56. 18 3 52. 42 3 54. 11 1 61. 35 3.9 57. 45 9.3 59. 19 2 67. 12 1 63. 09 1 64. 05 3 73. 08 2 68. 54 2 70. 43 8.4 79. 18 3 74. 56 3 76. 51 1 85. 37 4.8 81. 11 8.4 83. 13 2 92. 01 1 87. 35 1 88. 41 3 98. 23 2 94. 02 2 96. 09 declin . 22   23   24   ° ′   ° ′   ° ′ 3 75. 28 3 73. 58 3 71. 21 7.5 69. 20 7.5 67. 53 7.5 65. 21 1 63. 30 1 62. 05 1 59. 40 2 58. 01 2 56. 39 2 54. 23 3 52. 54 3 51. 36 3 49. 28 6.6 4● . 08 6.6 46. 55 6.6 44. 55 1 43. 43 1 42. 34 1 40. 43 2 39. 38 2 38. 34 2 36. 49 3 35. 51 3 34. 50 3 33. 13 5.7 32. 19 5.7 31. 22 5.7 29. 52 1 29. 02 1 28. 08 1 26. 44 2 25. 57 2 25. 07 2 23. 07 3 23. 02 3 22. 16 3 21. 01 4.8 20. 17 4.8 19. 33 4.8 18. 23 1 17. 40 1 16. 59 1 15. 52 2 15. 10 2 14. 31 2 13. 28 3 12. 46 3 12. 08 3 11. 08 3.9 10. 26 3.9 9. 51 3.9 8. 53 1 8. 10 1 7. 36 1 6. 41 2 5. 56 2 5. 25 2 4. 31 3 3. 46 3 3. 15 3 2. 23 2.10 1. 36 2.10 1. 09 2.10 0. 15   sub .   sub.   sub. 1 0. 33 1 1. 02 1 1. 45 2 2. 43 2 3. 16 2 4. 00 3 4. 53 3 5. 25 3 6. 09 1.11 7. 05 1.11 7. 37 1.11 8. 21 1 9. 20 1 9. 58 1 10. 35 2 11. 38 2 12. 16 2 12. 54 3 14. 00 3 14. 38 3 15. 17 12 16 .27 12 17. 06 12 17. 46 1 18. 58 1 19. 41 1 20. 22 2 21. 41 2 22. 24 2 23. 07 3 24. 30 3 25. 15 3 26. 01 11.1 27. 31 11.1 28. 18 11.1 29. 06 1 30. 42 1 31. 32 1 32. 24 2 34. 06 2 35. 01 2 35. 56 3 37. 46 3 38. 45 3 39. 42 10.2 41. 42 10.2 42. 46 10.2 43. 53 1 45. 58 1 47. 08 1 48. 20 2 50. 53 2 51. 50 2 53. 10 3 55. 30 3 57. 09 3 58. 22 9.3 60. 47 9.3 62. 21 9.3 63. 57 1 66. 28 1 68. 09 1 69. 53 2 72. 28 2 74. 26 2 76. 06 3 78. 43 3 80. 38 3 82. 36 8.4 85. 10 8.4 87. 10 8.4 89. 12 1 91. 41 1 93. 45 1 95. 49 2 98. 11 2 100. 15 2 102. 20 declin . 25   26   27   ° ′   ° ′   ° ′ 2 75. 46 2 73. 31 2 71. 29 3 69. 30 3 67. 18 3 65. 20 7.5 63. 30 7.5 61. 26 7 5 59. 32 1 58. 30 1 55. 58 1 54. 10 2 52. 27 2 50. 53 2 49. 11 3 47. 54 3 46. 11 3 44. 36 6.6 43. 27 6.6 41. 50 6.6 40. 22 1 39. 21 1 37. 50 1 36. 29 2 35. 32 2 34. 09 2 32. 17 3 32. 02 3 30. 43 3 29. 33 5.7 28. 46 5.7 27. 33 5.7 26. 20 1 25. 42 1 24. 33 1 23. 32 2 22. 49 2 21. 45 2 20. 47 3 20. 08 3 19. 06 3 18. 10 4.8 17. 31 4.8 16. 35 4.8 15. 43 1 15. 02 1 14. 10 1 13. 21 2 12. 41 2 11. 51 2 11. 04 3 10. 24 3 9. 36 3 8. 51 3.9 8. 11 3.9 7. 26 3.9 6. 43 1 6. 00 1 5. 16 1 4. 37 2 3. 52 2 3. 09 2 2. 30 3 1. 45 3 1. 04 3 0. 26   sub.   sub .   sub. 2.10 0. 21 2.10 1. 02 2.20 1. 38 1 2. 28 1 3. 07 1 3. 43 2 4. 35 2 5. 14 2 5. 49 3 6. 44 3 7. 23 3 7. 57 1.11 8. 55 11.1 9. 34 1.11 10. 08 1 11. 1● 1 11. 49 1 12. 33 2 13. 29 2 14. 08 2 14. 43 3 15. 52 3 16. 33 3 17. 09 12 18. 23 12 19. 04 12 19. 41 1 21. 00 1 21. 43 1 22. 22 2 23. 46 2 24. 31 2 25. 12 3 26. 42 3 27. 30 3 28. 13 11.1 29. 50 11.1 30. 41 11.1 31. 28 1 33. 12 1 34. 07 1 34. 56 2 36. 48 2 37. 48 2 38. 43 3 40. 42 3 41. 48 3 42. 48 10.2 44. 55 10.2 46. 07 10.2 47. 42 1 49. 29 1 50. 49 1 52. 03 2 54. 26 2 55. 53 2 57. 15 3 59. 45 3 61. 21 3 62. 52 9.3 65 28 9.3 67. 13 9.3 68. 52 1 71. 32 1 73. 26 1 75. 13 2 77. 53 2 79. 55 2 81. 49 3 84. 28 3 ●6 . 35 3 88. 35 8.4 91. 08 8.4 93. 19 8.4 95. 26 1 97. 47 1 10. 00 1 102. 03 declin 28   29   30   ° ′   ° ′   ° ′ 2 69. 23 1 73. 49 1 71. 39 3 63. 18 2 67. 28 2 65. 20 7.5 57. 37 3 61. 29 3 59. 24 1 52. 21 7.5 55. 54 7.5 53. 57 2 47. 29 1 50. 44 1 48. 52 3 43. 00 2 46. 00 2 44. 14 6.6 38. 54 3 41. 38 3 39. 59 1 35. 06 6.6 37. 37 6.6 36. 05 2 31. 36 1 33. 55 1 32. 30 3 28. 21 2 30. 30 2 29. 10 5.7 25. 19 3 27. 20 3 26. 05 1 22. 29 5.7 24. 21 5.7 23. 13 2 19. 48 1 21. 35 1 20. 31 3 17. 16 2 18. 58 2 17. 58 4.8 14. 51 3 16. 28 3 15. 33 1 12. 32 4.8 14. 06 4.8 13 13 2 10. 18 1 11. 49 1 11. 01 3 8. 07 2 9. 36 2 8. 50 3.9 6. 00 3 7. 27 3 6. 44 1 3. 55 3.9 5. 22 3.9 4. 41 2 1. 51 1 3. 17 1 2. 38   sub.             3 0. 12 2 1. 15 2 0. 37         sub.   sub. 2.10 2. 15 3 0. 48 3 1. 23 1 4. 19 2.10 2. 50 2.10 3. 24 2 6. 25 1 4. 54 1 5. 27 3 8. 32 2 7. 00 2 7. 31 1 ▪ 11 10. 44 3 9. 08 3 9. 39 1 12. 59 1.11 11. 20 1.11 11. 49 2 15. 19 1 13. 36 1 14. 05 3 17. 45 2 15. 57 2 16. 26 12 20. 19 3 18. 25 3 18. 55 1 23. 01 12 20. 55 12 21. 31 2 25. 54 1 23. 45 1 24. 17 3 28. 58 2 26. 40 2 27. 14 11.1 32. 16 3 29. 47 3 30. 24 1 35. 49 11.1 33. 09 11.1 33. 49 2 39. 40 1 36. 47 1 37. 31 3 43. 51 2 40. 43 2 41. 33 10.2 48. 24 3 45. 01 3 45. 56 1 53. 20 10.2 49. 40 10.2 50. 44 2 58. 42 1 54. 44 1 55. 57 3 64. 28 2 60. 14 2 61. 35 9.3 70. 37 3 66. 08 3 67. 40 1 77. 06 9.3 73. 22 9.3 74. 05 2 83. 49 1 79. 00 1 80. 53 3 90. 40 2 85. 48 2 87. 50 8.4 97. 30 3 92. 42 3 94. 49 1 104. 11 4.8 ●9 . 31 8.4 101. 42 declin 31   32   33   ° ′   ° ′   ° ′ 1 69. 34 1 67. 32 8.4 71. 59 2 63. 19 2 61. 21 1 65. 28 3 57. 29 3 55. 38 2 59. 23 7.5 52. 07 7.5 50. 22 3 53. 46 1 47. 11 1 45. 33 7 5 48. 37 2 42. 40 2 41. 09 1 43. 55 3 38. 32 3 37. 07 2 39. 38 6 34. 44 6.6 33. 25 3 35. 43 1 31. 14 2 30. 01 6.6 32. 08 2 28. 01 2 26. 53 1 29. 01 3 25. 00 3 23. ●8 2 25. 46 5.7 22. 12 5.7 21. 14 3 22. 56 1 19. 34 1 18. 39 5.7 20. 16 2 17. 05 2 16. 13 1 17. 45 3 14. 43 3 13. 54 2 15. 23 4.8 12. 26 4.8 11. 41 3 13. 07 1 10. 07 1 9. 32 4.8 10. 56 2 8. 08 2 7. 27 1 8. 50 3 6. 04 3 5. 26 2 6. 47 3 9 4. 02 3.6 3. 25 3 4. 47 1 2. 02 1 1. 27 3.9 2. 49         sub .       2 0. 02 2 0. 32 1 0. 52   sud .         sub . 3 1. 57 3 2. 29 2 1. 05 2.10 3. 58 2.10 4. 29 3 3. 02 1 5. 59 1 6. 30 1 5. 00 2 8. 03 2 8. 34 2 7. 01 3 10. 10 3 10. 40 3 9. 04 1.11 12. 21 1.11 12. 52 1.11 11. 10 1 14. 38 1 15. 08 1 13. 22 2 16. 38 2 17. 30 2 15. 39 3 19. 28 3 20. 03 3 18. 01 12 22. 06 12 22. 40 12 20. 33 1 24. 54 1 25. 30 1 23. 14 2 27. 54 2 28. 32 2 26. 06 3 31. 07 3 31. 48 3 29. 10 11.1 34. 36 11.1 35. 21 11.1 32. 31 1 37. 45 1 39. 13 1 36. 08 2 42. 30 2 43. 27 2 40. 05 3 47. 00 3 48. 04 3 44. 25 10.2 51. 56 10.2 53. 07 10 2 49. 10 1 57. 18 1 58. 38 1 54. 21 2 63. 06 2 64. 36 2 60. 02 3 69. 20 3 70. 59 3 66. 10 9.3 75. 57 9.3 77. 45 9.3 72. 44 1 82. 50 1 84. 46 1 79. 39 2 89. 52 2 91. 48 2 86. 48 3 96. 56 3 98. 59 3 94. 01 8.4 103. 49 8.4 105. 53 8.4 101. 08 declin . 34   35   36   ° ′   ° ′   ° ′ 8.4 69. 53 8.4 67. 46 8.4 65. 37 1 63. 25 1 61. 24 1 59. 20 2 57. 26 2 55. 31 2 53. 41 3 51. 57 3 50. 08 3 48. 19 7.5 46. 54 7.5 45. 14 7.5 43. 32 1 42. 2● 1 40. 47 1 39. 13 2 38. 09 2 36. 43 2 35. 17 3 34. 21 3 33. 00 3 31. 42 6.6 30. 52 6.6 29. 39 6.6 28. 25 1 27. 39 1 26. 31 1 25. 23 2 24. 47 2 23. 38 2 22. 35 3 21. 55 3 20. 57 3 19. 58 5.7 19. 20 5.7 18. 25 5.7 17. 31 1 16. 53 1 16. 02 1 15 ▪ 11 2 14. 34 2 13. 46 2 12. 59 3 12. 21 3 11. 36 3 10. 52 4.8 10. 13 4.8 9. 3● 4.8 8. 49 1 8. 09 1 7. 30 1 6. 50 2 6. 09 2 5. 31 2 4. 54 3 4. 11 3 3. 35 3 3. 00 3.9 2. 14 3.9 1. 40 3.9 1. 07         sub.   sub . 1 0. 08 1 0. 14 1 0. 46   sub .             2 1. 37 2 2. 08 2 2. 39 3 3. 33 3 4. 03 3 4. 33 2.18 5. 31 2.10 6. 00 2.10 6. 29 1 7. 30 1 7. 59 1 8. 27 2 9. 33 2 10. 00 2 10. 29 3 11. 40 3 12. 07 3 12. 35 1.11 13. 51 1.11 14. 18 1.11 14. 47 1 16. 08 1 16. 36 1 17. 05 2 18. 32 2 19. 01 2 19. 31 3 21. 04 3 21. 35 3 22. 06 12 23. 47 11.1 24. 19 12 24. 52 1 26. 41 1 27. 15 1 27. 51 2 29. 49 2 30. 26 2 31. 05 3 33. 12 2 33. 53 3 34. 36 11.1 36. 54 12 37. 40 11.1 38. 31 1 40. 57 1 41. 48 1 42. 43 2 45. 23 2 46. 22 2 47. 24 3 50. 16 3 51. 23 3 52. 34 10.2 55. 37 11.1 56. 53 10.2 58. 15 1 61. 40 1 62. 53 1 64. 26 2 67. 42 2 69. 23 2 71. 07 3 74. 30 3 76. 17 3 78. 13 9.3 81. 34 9.3 83. 31 9.3 85. 34 1 88. 50 1 90. 54 1 93. 04 2 96 07 2 98. 14 2 100. 28 3 103. 16 3 105. 24 3 107. 37 declin . 37   38   39   ° ′   ° ′   ° ′ 3 70. 12 3 68. 05 3 65. 55 8.4 63. 30 8.4 61. 28 8.4 59. 23 1 57. 19 1 55. 24 1 53. 26 2 51. 4 2 49. 52 2 48. 02 3 46. 32 3 44. 52 3 43. 11 7.5 41. 54 7.5 40. 22 7.5 38. 48 1 37. 42 1 36. 17 1 34. 55 2 33. 53 2 32. 35 2 31. 16 3 30. 24 3 29. 13 3 28. 00 6.6 27. 13 6.6 26. 07 6.6 25. 00 1 24. 17 1 23. 16 2 22. 14 2 21. 33 2 20. 37 3 19. 40 3 19. 01 3 18. 09 1 17. 16 5.7 16. 38 5.7 15. 49 5.7 14. 59 1 14. 22 1 13. 37 1 12. 51 2 12. 13 2 11. 30 2 10. 48 3 10. 09 3 9. 29 3 8. 49 4.8 8. 09 4.8 7. 31 4.8 6. 54 1 6. 12 1 5. 37 1 5. 02 2 4. 18 2 3. 45 2 3. 11 3 2. 26 3 1. 54 3 1. 23               sub. 3.9 0. 35 3.9 0. 05 3.9 0. 26   sub .   sub.       1 1. 16 1 1. 45 1 2. 14 2 3. 08 2 3. 36 2 4. 03 3 5. 01 3 5. 28 3 5. 54 2.10 6. 56 2.10 7. 22 2.10 7. 48 1 8. 54 1 9. 19 1 9. 44 2 10. 55 2 11. 21 2 11. 45 3 13. 01 3 13. 26 3 13. 51 1.11 15. 13 1.11 15. 40 1.11 16. 03 1 17. 31 1 17. 58 1 18. 22 2 19. 58 2 20. 25 2 20. 51 3 22. 34 3 22. 03 3 23. 31 12 25. 23 12 25. 53 12 26. 23 1 28. 24 1 28. 57 1 29. 30 2 31. 41 2 32. 18 2 32. 55 3 35. 17 3 35. 58 3 36. 40 11.1 39. 14 11.1 40. 01 11.1 40. 49 2 43. 36 1 44. 30 1 45. 25 1 48. 25 2 49. 27 2 50. 31 3 53. 45 3 54. 56 3 56. 10 10.2 59. 35 10.2 60. 58 10.2 62. 24 1 65. 58 1 67. 32 1 69. 11 2 71. 52 2 74. 36 2 76. 26 3 80. 07 3 82. 02 3 84. 03 9.3 87. 02 9.3 89. 42 9.3 91. 50 1 95. 13 1 97. 21 1 99. 33 2 102. 40 2 104. 49 2 107. 02 declin . 40   41   42   ° ′   ° ′   ° ′ 3 63. 48 2 68. 31 2 66. 20 8.4 57. 22 3 61. 37 3 59. 32 1 51. 32 8.4 55. 18 8.4 53. 21 2 46. 17 1 49. 37 1 47. 47 3 41. 33 2 44. 31 2 42. 49 7.5 37. 18 3 39. 55 3 38. 23 1 33. 28 7.5 35. 49 7.5 34. 22 2 30. 00 1 32. 06 1 30. 48 3 26. 50 2 28. 45 2 27. 33 6.6 23. 56 3 24. 59 3 24. 35 1 21. 19 6.6 22. 52 6.6 21. 52 2 18. 46 1 20. 16 1 19 21 3 16. 15 2 17. 51 2 17. 01 5.7 14. 13 3 15. 35 3 14. 47 1 12. 08 5.7 13. 26 5.7 12. 43 2 10. 07 1 11. 23 1 10. 42 3 8. 10 2 9. 26 2 8. 47 4.8 6. 18 3 7. 33 3 6. 56 1 4. 28 4.8 5. 42 4.8 5. 08 2 2. 40 1 3. 55 1 3. 24 3 0. 53 2 2. 09 2 1. 40   sub .         sub . 3.9 0. 54 3 0. 24 3 0. 04         sub.       1 2. 40 3.9 1. 22 3.9 1. 48 2 4. 29 1 3. 08 1 3. 32 3 6. 19 2 4. 54 2 5. 18 2.10 8. 11 3 6. 43 3 7. 06 1 10. 07 2.10 8. 34 2.10 8. 57 2 12. 8 1 10. 31 1 10. 52 3 14. 13 2 12. 3● 2 12. 53 1.11 16. 15 3 14. 37 3 14. 57 1 18. 46 1.11 16. 50 11.1 17. 11 2 21. 19 1 19. 11 1 19. 33 3 23. 57 2 21. 41 2 2● .04 12 26. 52 3 24. 24 3 24 .48 1 30. 01 12 27. 31 12 27. 49 2 23. 29 1 30. 34 1 31. 04 3 37. 20 2 34. 07 2 34. 42 11.1 41. 31 3 38. 02 3 38. 43 1 46. 18 11.1 42. 24 11.1 43. 11 2 51. 34 1 47. 17 1 48. 13 3 57. 24 2 52. 42 2 53. 48 10.2 63. 49 3 58. 44 3 60. 02 1 70. 49 10.2 65. 23 8.4 67. 14 2 78. 17 1 72. 35 1 74. 20 3 86. 04 2 89. 16 2 82. 13 9.3 93. 58 3 88. 13 3 90. 21 1 101. 45 9.3 96. 13 9.3 98. 26 2 109. 13 1 104. 03 1 106. 17 declin . 43   44   45   ° ′   ° ′   ° ′ 2 64. 06 2 61. 53 2 66. 51 3 57. 24 3 55. 18 3 59. 38 8.4 51. 21 8.4 49. 24 8.4 53. 14 1 45. 57 1 44. 09 1 47. 30 2 41. 08 2 39. 29 2 42. 24 3 36. 50 3 35. 20 3 37. 54 7.5 32. 59 7 5 31. 36 7.5 33. 52 1 29. 31 1 28. 15 1 30. 17 2 26. 22 2 25. 13 2 27. 02 3 23. 30 3 22. 27 3 24. 07 6.6 20. 52 6.6 19. 54 6.6 21. 26 1 18. 26 1 17. 32 1 18. 58 2 16. 09 2 15. 19 2 16. 41 3 14. 00 3 13. 19 3 14. 32 5.7 11. 58 5.7 11. 16 5.7 12. 31 1 10. 02 1 9. 23 1 10. 36 2 8. 11 2 7. 34 2 8. 46 3 6. 22 3 5. 48 3 6. 59 4.8 4. 37 4.8 4. 05 4.8 5. 16 1 2. 53 1 2. 24 1 3. 37 2 1. 11 2 0. 43 2 1. 56   sub .   sub.       3 0. 34 3 0. 57 3 0. 17               sub . 3 9 2. 13 3.9 2. 38 3.9 1. 21 1 3. 56 1 4. 19 1 3. 01 2 5. 41 2 6. 03 1 4. 41 3 7. 28 3 7. 49 2 6. 23 2.10 9. 17 2.10 9. 39 3 8. 08 1 11. 13 1 11. 32 2.10 9. 57 2 13. 12 2 13. 32 1 11. 50 3 15. 18 3 15. 37 2 13. 49 1.11 17. 31 1.11 17. 51 3 15. 55 1 19. 54 1 20. 14 1.11 18. 09 2 22. 28 2 22. 49 1 20. 34 3 25. 14 3 25. 37 2 23. 09 12 28. 16 11.1 28. 42 3 25. 58 1 31. 36 1 32. 06 12 29. 08 2 35. 18 2 35. 52 1 32. 34 3 39. 25 3 40. 06 2 36. 26 11.1 44. 01 12 44. 50 3 40. 47 1 49. 11 1 50. 10 11.1 45. 39 2 54. 59 2 56. 10 1 51. 10 3 61. 25 3 62. 50 2 57. 22 10.2 68. 31 11.1 70. 11 3 64. 17 1 76. 11 1 78. 05 10.2 71. 53 2 84. 16 2 86. 22 1 80. 02 3 92. 32 3 94. 47 2 88. 31 9.3 100. 43 9.3 103. 02 3 97. 03 1 108. 35 1 110. 52 9.3 105. 21 declin . 46   47   48   ° ′   ° ′   ° ′ 1 64. 32 1 62. 10 1 59. 54 2 57. 29 2 55. 15 2 53. 08 3 51. 12 3 49. 08 3 47. 10 8.4 45. 37 8.4 43. 43 8.4 41. 56 1 40. 41 1 38. 57 1 37. 20 2 36. 20 2 34. 45 2 33. 17 3 32. 27 3 31. 01 3 29. 41 7.5 28. 59 7.5 27. 41 7.5 26. 28 1 25. 52 1 24. 41 1 23. 35 2 23. 02 2 21. 57 2 20. 57 3 20. 27 3 19. 27 3 18. 32 6.6 18. 04 6.6 17. 09 6.6 16. 19 1 15. 51 1 15. 01 1 14. 14 2 13. 46 2 13. 00 2 12. 18 3 11. 49 3 11. 06 3 10. 28 5.7 9. 57 5.7 9. 18 5.7 9. 9 1 8. 09 1 7. 33 1 6. 59 2 6. 25 2 5. 52 2 4. 40 3 4. 44 3 4. 14 3 3. 45 4.8 3. 06 4.8 2. 38 4.8 2. 11 1 1. 28 1 1. 02 1 0. 37   sub.   sub .   sub . 2 0. 08 2 0. 33 2 0. 56 3 1. 45 3 2. 08 3 2. 29 3.9 3. 23 3.9 3. 44 3.9 3. 58 1 5. 02 1 4. 16 1 4. 07 2 6. 44 2 7. 02 2 7. 20 3 8. 28 3 8. 45 3 9. 02 2.10 10. 16 2.10 10. 33 2.10 10. 48 1 12. 09 1 12. 25 1 12. 40 2 14. 08 2 14. 24 2 14. 39 3 16. 14 3 16. 30 3 16. 45 1.11 18. 29 1.11 18. 45 1.11 19. 01 1 20. 53 1 21. 11 1 21. 27 2 23. 31 2 23. 50 2 24. 08 3 26. 24 3 26. 45 3 27. 05 12 29. 33 12 29. 58 12 30. 22 1 33. 06 1 33. 38 1 34. 03 2 37. 04 2 37. 38 2 38. 12 3 41. 31 3 42. 13 3 42. 56 11.1 46. 34 11.1 47. 25 11.1 48. 18 1 52. 15 1 53. 20 1 54. 25 2 58. 41 2 60. 00 2 61. 21 3 65. 51 3 67. 26 3 69. 04 10.2 73. 43 10.2 75. 34 10.2 77. 29 1 82. 06 1 84. 13 1 86. 23 2 90. 47 2 93. 05 2 95. 33 3 99. 25 3 101. 41 3 104. 15 9.3 107. 44 9.3 109. 06 9.3 112. 33 declin . 49   50   51   ° ′   ° ′   ° ′ 9.3 95. 5 9.3 62. 35 9.3 60. 13 1 57. 38 1 55. 18 1 53. 05 2 51. 02 2 48. 53 2 46. 50 3 45. 02 3 43. 17 3 41. 26 8.4 40 11 8.4 38. 25 8.4 36. 44 1 35. 45 1 34. 09 1 32. 38 2 31. 51 2 30. 24 2 29. 02 3 28. 23 3 27. 04 3 25. 51 7.5 25. 17 7.5 24. 06 7.5 22. 59 1 22. 30 1 21. 25 1 20. 24 2 19. 58 2 18. 59 2 18. 04 3 17. 39 3 16. 45 3 15. 54 6.6 15. 30 6.6 14. 41 6.6 13. 54 1 13. 30 1 12. 44 1 12. 02 2 11. 36 2 10. 55 2 10. 16 3 9. 49 3 9. 11 3 8. 35 5.7 8. 06 5.7 7. 31 5.7 6. 58 1 6. 27 1 5. 55 1 5. 26 2 4. 52 2 4 22 2 3. 54 3 3. 18 3 2 50 3 2. 25 4.8 1. 45 4.8 1. 57 4.8 0. 57         sub .   sub . 1 0. 14 1 0. 10 1 0. 35   sub .             2 1. 17 2 1. 40 2 1. 59 3 2. 50 3 3. 10 3 3. 28 3.9 4. 23 3.9 4. 42 3.9 4. 57 1 5. 58 1 6. 15 1 6. 31 2 7. 34 2 7. 52 2 8. 06 3 9. 17 3 9. 31 3 9. 45 2.10 11. 03 2.10 11. 18 2.10 11. 30 1 12. 55 1 13. 09 1 13. 20 2 14. 52 2 15. 06 2 15. 18 3 16. 59 3 17. 13 3 17. 24 1.11 19. 15 1.11 19. 29 1.11 19. 41 1 21. 43 1 21. 58 1 22. 11 2 24. 25 2 24. 43 2 24. 57 3 27. 25 3 27. 45 3 28. 03 12 30. 45 12 31. 08 12 31. 30 1 34. 31 1 35. 01 1 35. 27 2 38. 47 2 39. 24 2 39. 58 3 43. 39 3 44. 25 3 45. 09 11.1 49. 12 11.1 50. 11 11.1 51. 08 1 55. 33 1 56. 48 1 58. 00 2 62. 45 2 64. 17 2 65. 48 3 70. 46 3 72. 37 3 74. 27 10.2 79. 28 10.2 81. 38 10.2 83. 47 1 88. 36 1 91. 00 1 93. 19 2 97. 48 2 100. 20 2 102. 50 3 106. 40 3 109. 14 3 111. 43 declin . 52   53   54   ° ′   ° ′   ° ′ 3 65. 31 3 63. 15 3 60. 38 9.3 57. 49 9.3 55. 27 9.3 53. 01 1 50. 51 1 48. 40 1 46. 28 2 44. 49 2 42. 50 2 40. 51 3 39. 36 3 37. 49 3 36. 02 8.4 35. 05 8.4 33. 29 8.4 31. 53 1 31. 09 1 29. 42 1 28. 17 2 27. 42 2 26. 24 2 25. 07 3 24. 38 3 23. 27 3 22. 18 7.5 21. 54 7.5 20. 50 7.5 19. 47 1 19. 25 1 18. 28 1 17. 30 2 17. 10 2 16. 17 2 15. 25 3 15. 05 3 14. 17 3 13. 30 6.6 13. 09 6.6 12. 26 6.6 11. 43 1 11. 21 1 10. 41 1 10. 02 2 9. 38 2 9. 02 2 8. 26 3 8. 01 3 7. 27 3 6. 55 5.7 6. 27 5.7 5. 57 5.7 5. 27 1 4. 54 1 4. 30 1 4. 01 2 3. 27 2 3. 02 2 2. 38 3 2. 00 3 1. 38 3 1. 15               sub . 4.8 0. 35 4.8 0. 14 4.8 0. 07   sub.   sub.       1 0. 51 1 1. 10 1 1. 29 2 2. 17 2 2. 34 2 2. 51 3 3. 45 3 4. 00 3 4. 15 3.9 5. 13 3.9 5. 27 3.9 5. 41 1 6. 44 1 6. 57 1 7. 10 2 8. 19 2 8. 30 2 8. 42 3 9. 57 3 10. 08 3 10. 18 2.10 11. 41 2.10 11. 51 2.10 12. 00 1 13. 31 1 13. 40 1 13. 49 2 15. 28 2 15. 37 2 15. 46 3 17. 35 3 17. 42 3 17. 53 1.11 19. 52 1.11 20. 02 1.11 20. 12 1 22. 23 1 22. 34 1 22. 45 2 25. 11 2 25. 23 2 25. 37 3 28. 19 3 28. 33 3 28. 51 12 31. 51 12 32. 12 12 32. 32 1 35. 54 1 36. 19 1 36. 48 2 40. 32 2 41. 06 2 41. 44 3 45. 54 3 46. 39 3 47. 30 11.1 52. 07 11.1 53. 06 11.1 54. 14 1 59. 16 1 60. 34 1 62. 02 2 67. 25 2 69. 03 2 70. 54 3 76. 24 3 78. 25 3 80. 38 10.2 86. 02 10.2 88. 21 10.2 90. 42 1 95. 51 1 98. 22 1 101. 04 2 105. 23 2 107. 58 2 110. 41 declin 55   56   57   ° ′   ° ′   ° ′ 3 58. 08 3 63. 57 3 61. 12 9.3 50. 43 9.3 55. 34 9.3 53. 00 1 44. 22 1 48. 22 1 46. 03 2 38. 58 2 42. 16 2 40. 11 3 34. 21 3 27. 05 3 35. 15 8.4 30. 23 8.4 32. 41 8.4 31. 02 1 26. 56 1 28. 53 1 27. 26 2 23. 54 2 25. 36 2 24. 18 3 21. 13 3 22. 42 3 21. 32 7.5 18. 48 7.5 20. 08 5.7 19. 05 1 16. 37 1 17. 50 2 16. 53 2 14. 37 2 15. 44 1 14. 53 3 12. 46 3 13. 49 2 13. 03 6.6 11. 03 6.6 12. 03 3 11. 20 1 9. 25 1 10. 23 6.6 9. 44 2 7. 53 2 8. 49 1 8. 14 3 6. 25 3 7. 20 3 6. 48 3.7 4. 59 3.7 5. 54 3.7 5. 26 1 3. 36 1 4. 32 2 4. 06 2 2. 15 2 3. 11 1 2. 48 3 0. 55 3 1. 53 3 1. 31   sub .             4.8 0. 26 4.8 0. 34 4.8 0. 15         sub .   sub . 1 1. 45 1 0. 41 1 1. 01 2 3. 06 2 2. 02 2 2. 17 3 4. 29 3 3. 21 3 3. 34 3.9 5. 53 3.9 4. 42 3.9 4. 53 1 7. 21 1 6. 05 1 6. 15 2 8. 51 2 7. 31 2 7. 39 3 10. 27 3 9. 00 3 9. 08 2.10 12. 08 2.10 10. 35 2.10 10. 41 1 13. 56 1 12. 15 1 12. 21 2 15. 53 2 14. 03 2 14. 07 3 18. 00 3 15. 58 3 16. 04 1.11 20. 19 1.11 18. 06 1.11 18. 11 1 22. 54 1 20. 26 1 20. 31 2 25. 48 2 23. 02 2 23. 09 3 29. 05 3 25. 58 3 26. 07 12 32. 51 12 29. 19 12 29. 32 1 37. 13 1 33. 10 1 33. 28 2 42. 19 2 37. 40 2 38. 07 3 48. 18 3 42. 56 3 43. 35 11.1 55. 20 11.1 49. 10 11.1 50. 05 2 63. 28 1 56. 30 1 57. 46 1 72. 44 2 65. 03 2 66. 43 3 82. 51 3 74. 43 3 76. 51 10.2 93. 23 10.2 85. 15 10.2 87. 47 1 103. 43 1 96. 03 1 98. 50 2 113. 19 2 106. 29 2 109. 21 declin . 58   59   60   ° ′   ° ′   ° ′ 2 58. 30 1 64. 26 1 61. 59 3 50. 32 2 55. 44 2 53. 04 9.3 43. 50 3 48. 01 3 45. 37 1 38. 13 9.3 41. 35 9.3 39. 28 2 33. 30 1 36. 14 1 34. 22 3 29. 29 2 31. 45 2 30. 06 8.4 26. 03 3 27. 56 3 26. 29 1 23. 04 8.4 24. 40 8.4 23. 23 2 20. 26 1 21. 50 1 20. 41 3 18. 05 2 19. 20 2 18. 18 7.5 15. 59 3 17. 06 3 16. 11 1 14. 04 7.5 15. 05 7.5 14. 16 2 12. 18 1 13. 16 1 12. 31 3 10. 41 2 11. 35 2 10. 54 6.6 9. 08 3 10. 01 3 9. 24 1 7. 42 6.6 8. 32 6.6 8. 00 2 6. 19 1 7. 09 1 6. 39 3 4. 58 2 5. 39 2 5. 23 5.7 3. 42 3 4. 33 3 4. 08 1 2. 26 5.7 3. 18 5.7 2. 56 2 1. 11 1 2. 05 1 1. 24   sub .             3 0. 02 2 0. 53 2 0. 35         sub .   sub . 4.8 1. 16 3 0. 19 3 0. 34 1 2. 31 4.8 1. 31 4.8 1. 44 2 3. 46 1 2. 44 1 2. 55 3 5. 04 2 3. 57 2 4. 07 3.9 6. 24 3 5. 13 3 5. 21 1 7. 47 3.9 6. 32 3.9 6. 38 2 9. 14 1 7. 53 1 7. 58 3 10. 46 2 9. 18 2 9. 23 2.10 12. 25 3 10. 50 3 10. 53 1 14. 11 2.10 12. 28 2.10 12. 30 2 16. 07 1 14. 13 2 14. 14 3 18. 14 2 16. 09 2 16. 09 1.11 20. 35 3 18. 16 3 18. 16 1 23. 14 1.11 20. 37 1.11 20. 39 2 26. 15 1 23. 18 1 23. 21 3 29. 43 2 26. 22 2 26. 23 12 33. 46 3 29. 54 3 30. 03 1 38. 32 12 34. 03 12 34. 19 2 44. 13 1 38. 58 1 39. 24 3 50. 59 2 44. 53 2 45. 31 11.1 59. 02 3 51. 57 3 52. 57 1 68. 27 11.1 60. 26 11.1 61. 52 2 79. 03 1 70. 19 1 72. 17 3 90. 21 2 81. 25 2 83. 53 10.2 101. 38 3 93. 57 3 95. 57 1 112. 11 10.2 104. 37 10.2 107. 35 a table for east and west dials .   ° ′ 11 00. 00 1 03. 45 2 07. 30 3 11. 15 10 15. 00 1 18. 45 2 22. 30 3 26. 15 9 30. 00 1 33. 45 2 37. 30 3 41. 15 8 45. 00 1 48. 45 2 52. 30 3 56. 15 7 60. 00 1 63. 45 2 67. 30 3 72. 15 errata . page 39. line 7. for 47° . 08′ . read 48° . 08′ . p. 40. l. 19. for 3° . 57′ . r. 63° . 57′ . p. 42. l. 23. for 46° . 35′ . r. 86° . 35′ . p. 42. l. 25. for 10° . 00′ . r. 100° . 00′ . p. 44. l. last , for 69° . 31′ . r. 99° . 31′ . p. 45. l. 11. for 23° . 28′ . r. 23° . 58′ . p. 47. l. 18. for 9° . 3′ . r. 9° . 31′ ▪ 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 purposes, all without asking permission. early english books online. (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 . 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 ☞ ☞ elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by samuel foster ... foster, samuel, d. 1652. 1654 approx. 341 kb of xml-encoded text transcribed from 111 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2003-11 (eebo-tcp phase 1). a40031 wing f1632 estc r7034 12251532 ocm 12251532 57111 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. a40031) transcribed from: (early english books online ; image set 57111) images scanned from microfilm: (early english books, 1641-1700 ; 143:14) elliptical or azimuthal horologiography comprehending severall wayes of describing dials upon all kindes of superficies, either plain or curved, and unto upright stiles in whatsoever position they shall be placed / invented and demonstrated by samuel foster ... foster, samuel, d. 1652. twysden, john, 1607-1688. wingate, edmund, 1596-1656. [6], 204, [3] p. : ill. printed by r. & w. leybourn for nicholas bourn ..., london : 1654. "to the reader" signed: john twysden, edmund wingate. in four parts, paged continuously; pts. 2-4 each have special t.p. title page vignette. advertisement on p. [3] at end. reproduction of original in harvard university libraries. includes index. imperfect: imprint mutilated. 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 horology -early works to 1800. sundials. 2003-06 tcp assigned for keying and markup 2003-07 spi global keyed and coded from proquest page images 2003-08 mona logarbo sampled and proofread 2003-08 mona logarbo text and markup reviewed and edited 2003-10 pfs batch review (qc) and xml conversion elliptical , or azimuthal horologiography . comprehending severall wayes of describing dials upon all kindes of superficies either plain or curved : and unto upright stiles in whatsoever position they shall be placed . invented and demonstrated by samvel foster , late professor of astr●●●mie in gresham-colledge . london , 〈…〉 for nicholas t 〈…〉 to the reader . courteous reader , amongst other treatises of this deceased author , of which , in their due time , we intend to make thee partaker , we have , in the first place , made choice of this , as well in some measure to keep our word with thee , as also to stay thy expectation till other things can be made ready . we intend not to spend time either in the commendation of it or the author , being w●ll assured that our words will not , where the matter it self is 〈…〉 capable to , satisfie therein the judicious reader . onely let 〈◊〉 say thus much , that though the generall subject of this book 〈◊〉 dialling , yet 't is handled in a way which no man whosoever , that we know of , hath hitherto fully traced . for by it thou wilt see , that the representing the true houre by the shadow made by the axis of the world is but one of those infinite wayes which may be invented , and that it is as possible to do the same thing by the shadow of the axis of one of the vertical circles , and by the projection of one ellipsis upon a plain , supply the place of all the parallels comprehended within the tropicks . 't is true , that mr. va●lezard , a learned mathematician , we think yet living in france , hath some yeers since published a short treatise in that language , in which he shew●th by the projection of an ellipsis upon the plain of the horizon , and by the help of an upright moveable stile to finde the he●re and azimuth , with some other uses of the same . but this treatise of our author is very different from that , and most of the things here handled , such as are not appliable to his , and in themselves wholly new . in the next place , if it shall seem strange to any that amongst other things , as well of this , as of different natures , which we intend shortly to publish ; we begin first with this of dialling ( a subject upon which something too hath already been published by our author , and from whence some might perhaps take occasion to carp at him , as either unable for other things , or too much busied in this ) we desire them first to con●●der how difficult this subject hath been thought by the 〈◊〉 , and withall what large volumes have been writ by som● of the best rank of later mathematicians , ( such were clavius , maigran , and others ) and then compare their wayes with what they shall here and hereafter ( god willing ) finde in our author , and we doubt not , but they will be forced to yield him it this honour , to have made that art , in all the cases of it , and all circumstances thereunto belonging , more easie and expedite both in the understanding and practice , and with much more brevity than any that have gone before him either of our own or other nations . lastly , we advertize thee ( reader ) that our authour , in regard of his great and long infirmities , could not fit either this , or any other of his treatises for the presse , as he desired and intended ; and therefore they must needs want much of that accomplishment which otherwise they would have had . but we hope , notwithstanding , thou wilt finde so much of worth in them as they now are , whereby thou wilt judge them ( as we do ) fitter to be made publick , then wholly suppressed . john twysden , edmund wingate . errata . page 85. line 11. for pag. 81. read pag. 70. there are divers other literall faults , but this is most materiall . elliptical or azimvthal horologiography . of the elliptical dial , whose index stands perpendicular to the plain : how to draw and divide the ellipsis upon an horizontal plain , or any other plain that declineth not . 1. let the two right lines b c and a d , cut each other at right angles in a. then making a c or a b as a radius , let a d be the right sine of the poles elevation above the plain ; and make up the rectangled parallelogram b e f c , and continue it further to g and h. 2. divide the radius a c into the sines of 15 , 30 , 45 , 60 , 75 gr . at a b c d e , which stand for the six equall houres : ●o that a a may be the sine of one houre , a b of two houres , a c of three houres , a d , a e of four and five houres . and in the same sort divide the other lines which are equall to a c , namely d f , a b , d e , into the same parts , at a b c d e , and at l m n o p : and draw the parallel lines a a , b b , &c. l l , m m , &c. then taking a d , or c f , or b e , for a radius , set on again the sines of the six equall houres from b and c , at f g h i k , and as many as shall be needfull on the other side , at q r s : and draw the parallel lines f f , g g , &c. and q q , r r , &c. 3. first , if a curved line be begun at d or 12 , and carried both wayes diagonally through the succeeding parallelograms , the same line shall be the ellipsis required . thus you see it carried from d to 11 , from 11 to 10 , from 10 to 9 , and so through the successive opposite angles of the following parallelograms , at 8 , 7 , 6 , 5 , 4 , 3. and on the other side in the same manner , from d , diagonally through 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9. this line thus regularly described without angles , and extravagant inordinate flexures , is the ellipsis here intended . secondly , if from the center a , through each of the angular points in the small parallelograms , certain short lines be drawn , ( as in the figure you see done at 11 , 10 , 9 , &c. and at 1 , 2 , 3 , &c. ) the same lines shall be the requisite horary divisions of the elliptical line . 4. if these divisions into single houres shall not be enough , but that more exactnesse is required , into halves and quarters ; then must you ( besides the sines of whole houres at a , b , &c. l , m , &c. ) put in the right sines of the intermediate halves and quarters of the equall houres ; which will be easie to do by the former directions . declinations of the sun from the aequinoctiall .   ianu. febru . march aprill may iune iuly augu. septē . octob. novē . decēb. 1 21 49 13 57 3 34 8 30 18 01 23 11 22 09 15 15 4 29 7 10 17 36 23 07 2 21 39 13 37 3 11 8 52 18 16 23 14 22 00 14 57 4 06 7 33 17 52 23 12 3 21 29 13 17 2 47 9 13 18 31 23 18 21 51 14 39 3 43 7 56 18 08 23 16 4 21 18 12 57 2 23 9 34 18 46 23 21 21 42 14 21 3 20 8 18 18 24 23 19 5 21 07 12 36 2 00 9 56 19 00 23 24 21 33 14 02 2 57 8 40 18 40 23 22 6 20 56 12 15 1 36 10 17 19 14 23 26 21 23 13 43 2 34 9 02 18 55 23 25 7 20 44 11 54 1 12 10 38 19 27 23 28 21 13 13 24 2 10 9 24 19 10 23 27 8 20 32 11 33 0 48 10 59 19 40 23 29 21 02 13 05 1 47 9 46 19 24 23 29 9 20 19 11 12 0 24 11 20 19 53 23 30 20 51 12 45 1 24 10 08 19 38 23 30 10 20 06 10 50 aequin .   11 40 20 06 23 31 20 40 12 25 1 00 10 30 19 52 23 31 11 19 53 10 28 0 24 12 01 20 18 23 31 20 29 12 05 0 37 10 51 20 05 23 31 12 19 39 10 06 0 48 12 22 20 30 23 31 20 17 11 45 aequinoct . 14 11 12 20 18 23 31 13 19 25 9 44 1 12 12 42 20 42 23 31 20 05 11 25   10 11 33 20 31 23 31 14 19 10 9 22 1 36 13 02 20 53 23 30 19 53 11 04 0 34 11 54 20 43 23 30 15 18 55 9 00 2 00 13 21 21 04 23 28 19 40 10 43 0 58 12 15 20 55 23 28 16 18 40 8 38 2 23 13 40 21 14 23 26 19 27 10 22 1 22 12 36 21 06 23 25 17 18 25 8 15 2 47 14 00 21 24 23 24 19 13 10 01 1 46 12 57 21 17 23 22 18 18 10 7 52 3 11 14 19 21 34 23 20 18 59 9 40 2 10 13 17 21 27 23 19 19 17 54 7 29 3 34 14 38 21 43 23 17 18 45 9 19 2 33 13 37 21 37 23 16 20 17 37 7 06 3 57 14 57 21 52 23 14 18 30 8 58 2 56 13 57 21 47 23 12 21 17 20 6 43 4 20 15 15 22 01 23 11 18 15 8 36 3 19 14 16 21 57 23 08 22 17 03 6 20 4 43 15 32 22 10 23 06 18 00 8 14 3 42 14 35 22 06 23 03 23 16 46 5 57 5 06 15 49 22 18 23 01 17 45 7 52 4 06 14 54 22 15 22 58 24 16 28 5 34 5 29 16 06 22 25 22 56 17 30 7 30 4 29 15 13 22 23 22 52 25 16 10 5 11 5 52 16 23 22 32 22 51 17 14 7 08 4 52 15 32 22 32 22 45 26 15 52 4 47 6 15 16 40 22 39 22 45 16 58 6 46 5 15 15 51 22 38 22 38 27 15 34 4 24 6 38 16 57 22 45 22 39 16 42 6 24 5 38 16 09 22 45 22 31 28 15 15 4 01 7 00 17 13 22 51 22 32 16 25 6 01 6 01 16 27 22 51 22 24 29 14 56     7 22 17 29 22 57 22 25 16 08 5 38 6 24 16 45 22 57 22 16 30 14 37     7 45 17 45 23 02 22 17 15 51 5 15 6 47 17 02 23 02 22 08 31 14 17     8 08     23 07     15 33 4 52     17 19     21 59 tangents of the suns declination from the aequinoctiall .   ianu. febru . march aprill may iune iuly augu. septē . octob. novē . decēb. 1 4005 2484 0623 1498 3251 4281 4071 2727 0784 1257 3170 4272 2 3971 2422 0554 1562 3300 4295 4042 2671 0717 1325 3222 4287 3 3936 2360 0485 1626 3348 4307 4012 2615 0650 1392 3275 4300 4 3900 2297 0416 1690 3396 4317 3981 2558 0584 1458 3326 4311 5 3863 2234 0347 1754 3443 4326 3949 3500 0517 1524 3377 4321 6 3825 2170 0278 1818 3489 4334 3916 2442 0450 1590 3427 4330 7 3785 2106 0209 1881 3534 4341 3882 2383 0383 1656 3476 4338 8 3744 2042 0140 1944 3576 4347 3847 2323 0316 1722 35●3 4345 9 3702 1978 0070 2007 3618 4350 3811 2263 0249 1788 3568 4350 10 3659 1913 aequin . 2069 3659 4351 3776 2202 0181 1853 3613 4352 11 3615 1847 0070 2131 3700 4352 3739 2141 0112 1917 3657 4352 12 3570 1781 0140 2193 3740 4352 3700 2079 aequin . 42 1981 3700 4351 13 3524 1715 0209 2254 3779 4351 3660 2017 29 2045 3742 4350 14 2477 1649 0278 2315 3817 4348 3618 1955 0099 2109 378● 4347 15 3429 1582 0347 2375 3854 4343 3575 1893 0169 2172 3821 4341 16 3380 1515 0416 2434 3889 4336 3532 1831 0239 2235 3859 4333 17 3330 1448 0485 2493 3922 4327 3487 1768 0309 2298 3896 4324 18 3280 1381 0554 2552 3952 4317 3442 1705 0378 2360 39●2 4314 19 3230 1314 0623 2610 3982 4306 3396 1642 0446 2422 3907 4302 20 3177 1246 0692 2668 4012 4294 3349 1578 0514 2483 4000 4288 21 3123 1178 0761 2725 4042 4281 3301 1513 0582 2543 4031 4273 22 3068 1110 0829 2780 4071 4266 3253 1448 0650 2603 4061 4256 23 3013 1042 0897 2834 4099 4250 3204 1382 0718 2662 4090 4238 24 2957 0974 0965 2888 4125 4232 3154 1316 0786 2721 4118 4217 25 2900 0906 1033 ●941 4149 4213 3103 1250 0854 2779 4146 4195 26 2842 0837 1101 2994 4171 4193 3052 1184 0921 2837 4172 4172 27 2784 0768 1168 3047 4192 4172 3000 1118 0988 2895 4196 4148 28 2726 0699 1●35 3099 4212 4149 2947 1052 1055 2952 4218 4123 29 2667   1301 3150 4231 4124 2893 0985 1122 3008 4238 4096 30 2607   1367 3201 4249 4098 2838 0918 1189 3063 4256 4068 31 2546   1433   4266   2783 0851   3117   4038 sect . i. of the elliptical dial , where the index stands perpendicular to the plain : how to draw it for an horizontal plain , or any other plain that declineth not . because here is no declination supposed , therefore the draught will be the more easie . 1. make b c for the longer diameter of your ellipsis , and count one halfe of it , that is a b or a c as the radius , and through the point a , draw an infinite line ( as d e ) at right angles to b c. then for the shorter diameter , you must consider the elevation of the pole above your plain . looke therefore to make a d or a e equall to the sine of the poles elevation , which sine must be estimated to the radius a b. thus you have the two extream diameters . 2. through the four points d b e c you may describe the ellipsis , either by elliptical compasses , or by finding many points through which it is to passe : for effecting which there are multitudes of wayes prescribed . one may be this . describe two circles upon the two extream diameters b c and d e. divide them both into any like parts , as b f h n p q c , and v i k e o t x ; and from each couple of those like divisions ( as f and i ) draw f g i g , ( the first parallel to a n , the second parallel to a b ) cutting each other at right angles in g , so the point g shall be one of the points through which the ellipsis is to be described . in the same manner , from h and k must be drawn two lines concurring at right angles in m , which will therefore be another point through which the ellipsis must passe . so again from p and o , t and q , lines are to be drawn , concurring at right angles in s and r , which are two other points of passage for the ellipsis . after this manner may the other halfe b d c be described . how to divide the ellipsis into its requisite parts . 1. if the parts b f h n p q c were all equall , and 12 of them in number , then would the points g m , &c. be the houre points . and accordingly , if those houres were subdivided into equall halves and quarters , &c. there would be found points for the halves and quarters , &c. of the houres . and if the ellipsis be first drawn , then the exteriour circle it selfe by lines issuing out of f h p q , drawn parallel to a n ) will divide the ellipsis : or , where these lines cut it more obliquely ( as at g and r , &c. ) there the lesser circles equall parts ( at i and t , &c. ) will cut the same ellipsis at more right angles , so that one of them may help the other in this division . 2. or , if the quadrants b n c , or v e x ( or of any other circle described upon the center a ) were divided into horizontall spaces , such as are proper for this diall ( as you see them done upon the quadrants b y c ) and a ruler laid from the center a did transfer those divisions into the ellipsis ( as is done upon that halfe of it which is noted with b d c ) this division would be the same with the former . the proportion whereby to make the table of horizontall spaces for this ellipsis , is inverse to that which is for common horizontall dials : thus , as the sine of the poles elevation above the plain , is to the radius ; so is the tangent of 15 gr . 30 , 45 , &c. to the tangent of the spaces of every houre from 12 , upon the plain whereon the ellipsis is described . and according to these spaces must every houre be set on from n or y towards b or c. how to make the zodiac , or dayes of the yeare , whereby the ellipsis and index are to be set in a right position , that they may daily stand true to shew the houre . whether the 12 signes or the 12 moneths be fittest for use , is left to every mans choise . bu● in both these it is required that the declinations due to the signes or dayes be known , because these are to be inscribed by them . so that now it only remains to be shewed what kinde of scale that must be , out of which the foresaid declinations are to be taken . in generall , the scale must be a scale of naturall tangents . in particular , there are these two rules , one of which will with due convenience serve for all kindes of latitudes , or elevations . you may use which is most sutable to your purpose . the first for great latitudes . make the lesser semidiameter a d or a e , to be a tangent of the poles elevation above the plain ; and that scale is the scale of tangents out of which the declinations ( before mentioned ) must be taken . the second for lesser latitudes . to the radius b a , finde the co-sine of the plains latitude or elevation ; and make that co-sine a radius , or tangent of 45 gr . and this will be the same scale with the former ▪ by these wayes the declinations of these two tables ( one for the signes , the other for the dayes of the moneths ) may be inserted from the equinoctiall line , or middle of the scale , by the graduations of such a tangent line as is before mentioned . signes . declin . signes . ♑ ♋ 0 23 31 30     5 23 26 25     10 23 9 20     15 22 41 15     20 22 2 10     25 21 12 5 ♊ ● ♒ ♌ 0 20 13 30     5 19 5 25     10 17 48 20     15 16 24 15     20 14 52 10     25 13 14 5 ♉ ♏ ♓ ♍ 0 11 31 30     5 9 43 25     10 7 51 20     15 5 56 15     20 3 58 10     25 2 00 5 ♈ ♎ moneth declin . moneth declin . moneth declin . moneth declin . ianu. 5 21 7 aprill 5 9 56 iuly 5 21 33 octob. 5 8 40 10 20 6 10 11 41 10 20 41 10 10 29 15 18 55 15 13 20 15 19 39 15 12 15 20 17 37 20 14 55 20 18 30 20 13 56 25 16 10 25 16 23 25 17 14 25 15 32 31 14 17 30 17 45 31 15 33 31 17 18 febr. 5 12 36 may 5 19 0 aug. 5 14 1 nove. 5 18 40 10 10 50 10 20 6 10 12 24 10 19 51 15 9 00 15 21 4 15 10 42 15 20 54 20 7 6 20 21 53 20 8 56 20 21 47 25 5 11 25 22 32 25 7 7 25 22 30 28 4 1 31 23 6 31 4 52 30 23 2 mar. 5 2 0 iune 5 23 24 septe . 5 2 56 dece . 5 23 22 10 ae 1 10 23 31 10 1 0 10 23 31 15 2 0 15 23 28 15 ae 57 15 23 28 20 3 57 20 23 14 20 2 55 20 23 11 25 5 52 25 22 50 25 4 51 25 22 45 31 8 7 30 22 17 30 6 46 31 21 58 or if it be required rather to set them on by a decimal scale equally divided , then must the tangents of these declinations ( here specified ) be used ; and the decimal scale must be equal to the co-sine of the plains latitude , which co-sine must be limited to b a , the greater radius of the ellipsis . and for this purpose here are two tables more which do expresse the said tangents . signes . tang signes . moneth tang moneth tang moneth tang moneth tang ♑ ♋ 0 4348 30   ianu. 5 2862 aprill 5 1751 iuly 5 3949 octob. 5 1524   5 4334 25   10 3659 10 2068 10 3775 10 1850   10 4276 20   15 3427 15 2370 15 3571 15 2171   15 4180 15   20 3175 20 2664 20 3346 20 2481   20 4047 10   25 2899 25 2940 25 3102 25 2780   25 3879 5 ♊ ● 31 2546 30 3201 31 2783 31 3115 ♒ ♌ 0 3683 30   febr. 5 2235 may 5 3443 aug. 5 2496 nove. 5 3378   5 3460 25   10 1914 10 3659 10 2199 10 3610   10 3211 20   15 1584 15 3852 15 1890 15 3819   15 2943 15   20 1246 20 4017 20 1572 20 3996   20 2655 10   25 0907 25 4149 25 1249 25 4142   25 2352 5 ♉ ♏ 28 0702 31 4265 31 0851 30 4252 ♓ ♍ 0 2038 30   mar. 5 0349 iune 5 4327 septe . 5 0512 dece . 5 4320   5 1712 25   10 aeq 3 10 4348 10 0175 10 4348   10 1379 20   15 0349 15 4341 15 ae166 15 4341   15 1039 15   20 0690 20 4293 20 0509 20 4283   20 0693 10   25 1028 25 4210 25 0849 25 4193   25 0349 5 ♈ ♎ 31 1426 30 4098 30 1187 31 4033 another way to describe and divide the same zodiac , or scale of moneths . the former descriptions do suppose that the yearly course of the sun is to be set in a streight scale , in which the parts neerest to the two tropicks will be exceeding close together ; and those at the aequinoctiall or middle part of the scale will be wide . but if it shall better be liked that the parts should stand distinctly one from the other ; it will then be most expedient , first , to limit out the whole length of the scale : then upon that length ( as a diameter ) to describe a circle , and to divide it as is here under shewed . how to divide the annuall circle into its requisite parts . you are first to divide the circle ( it selfe , or one equall to it ) into 360 equall degrees , and by them to divide the said circle into such unequall parts as the numbers of degrees and minutes expressed in the tables will require . the two tables are the right ascensions of such parts of the ecliptic as are due to the suns place or declination , in any of the signes , or of any dayes of the 12 moneths . the tables are here placed . the manner of dividing the circle is the same with other like things of this kinde .   mon r. ascen . mon r. ascē mon r. ascen mon r. ascen . ia 10 302 43 a 10 28 36 i. 10 119 47 o. 10 205 06 20 313 16 20 37 56 20 129 40 20 214 53 31 324 21 30 47 33 31 140 15 31 225 50 fe 10 334 03 m 10 57 23 a 10 149 37 n 10 236 11 20 343 30 20 67 27 20 158 48 20 246 51 28 351 07 31 78 45 31 168 45 30 257 57 m 10 0 14 i. 10 89 05 s. 10 177 44 d 10 269 02 20 9 18 20 99 27 20 186 47 20 280 11 31 19 19 30 109 43 30 195 53 31 292 08   signs r. ascē signs r. ascen . signs r. ascen signs r. ascen . ♈ 10 9 11 ♋ 10 100 53 ♎ 10 189 11 ♑ 10 280 53   20 18 27   20 111 39   20 198 27   20 291 39   30 27 54   30 122 12   30 207 54   30 302 12 ♉ 10 37 34 ♌ 10 132 28 ♏ 10 217 34 ♒ 10 312 28   20 47 32   20 142 26   20 227 32   20 322 26   30 57 48   30 152 06   30 237 48   30 332 06 ♊ 10 68 21 ♍ 10 16● 33 ♐ 10 248 21 ♓ 10 341 33   20 79 07   20 170 49 20 259 07   20 350 49   30 90 00   30 180 00   30 270 00   30 360 00 the beginning of the equall parts or degrees , must be upon that middle diameter of the circle , which lies parallel to the two six a clock lines , or perpendicular to the twelve a clock line . the best manner of dividing the circle is as in the preceding figure , especially if the fiduciall circle be cut through with small divisions , that so the intersections may be the more discernable , and the circle more distinguishable from the rest ; and more cleere from mistakes . concerning the index . the index must stand perpendicular to the plain , and must , according to the time of the year , be neerer or further from the ellipsis . now , whether the index should move upon the plate lying still , or the elliptical plate move towards the index fixed upon another plate , must be left to every mans judgement and best liking . but the mover with its peculiar index ( called here the zodiacall index ) must move according to the length of the plains meridian line , either in it , or parallel to it , alwayes so that the fiduciall edge of the dials index may ever stand in the meridian line . but if to this ellipticall diall it be thought fit to joyn an ordinary horizontal dial ( fitted to the elevation of the pole above the plain ) that so it may set it selfe true north and south , which by this double kinde of diall it will do ; for the houres in both dials will never agree to be the same till it stand right , and the best time for setting it , is when the sun is in the east or west azimuth , the worst time is at noon : then the common horizontal dial may be the standing plate , the index of the ellipsis being fixed to it , the elliptical plate may move too and fro upon the horizontal fixed plate , according to the graduations of the zodiac . of the place for the suns annual course or zodiac . whichsoever of the two ( the ellipsis or index ) is moveable or fixed , it matters not . in both wayes it is necessary to set the zodiac right , which must thus be done . let the fiduciall edge of the index be placed exactly in the center of the ellipsis . and being there set , let the place of the line or threed ( or whatever it be that serves for the zodiacall index ) be marked upon the standing plate . then through that mark or point draw a streight line parallel to the line of the two sixes , or perpendicular to the line of twelve . the same line is to be esteemed for the aequinoctiall : and from thence are all the parts of the scale ( or circle ) to be insctibed by help of the former tables . the vses of the elliptical dial. the index and ellipsis being used alone , and set to their true distance , by the help of the zodiac , the ellipsis it selfe with its upright index , will shew the true houre of the day . but it is supposed hitherto , that the plain be either direct , looking full north or south , or else horizontal : and in every such plain , there must be drawn the meridian line of the place or plain , which in direct flats are one and the same . and according to the coast of this meridian line must the moving or sliding be , as also the line of 12 upon the ellipsis must lie in or parallel to it . this way therefore requires , first , a meridian line to be drawn , whereby to place the ellipsis . and in all direct plains the meridian line is the same with the vertical line of the plain . but if to this elliptical dial , be adjoyned a common horizontal dial with an axis , then there needs no finding of a meridian line before hand , because they two betwixt them will finde one , and consequently will place themselves in a true position . only with this proviso , that if the plain be not horizontal , it must yet be such as looketh directly north or south . this was intimated before . there may other uses be made of it , if it have other scales adjoyned to the zodiac . 1. if a tangent line of 23½ gr . be inscribed according to the length of the zodiac , then when the situation of the ellipsis to the index is fitted , the zodiacal index will shew ( upon this tangent scale ) the suns declination . 2. in the 11 page there are two rules given whereby to finde that scale of tangents out of which the zodiacal scale is to be divided . if then out of that scale of tangents so found and limited , you take the co-tangent of your latitude , and divide that length as a radius or whole line of sines into houres and parts of houres , and put in so many of them as will reach the suns greatest declination , or the length of the zodiac both wayes from the equinoctiall ; then the former rectification of the zodiacal index to the day of the moneth or suns place , will perfectly shew ( in this scale , and for that latitude to which all the work is made ) the ascensional difference , with what depends thereon : namely , the length of the day and night , and the time of sun rising and setting . 3. the dial being fixed in a true position , the place or coast of the sun rising and setting upon the plain , may be discerned in the heavens . for when the ellipsis is rectified , and the time of the sun rising or setting is known ( as before ) then with your eye project the fiduciall edge of the upright index upon that time of sun rising or setting counted in the ellipsis , and the same edge will shew in the heavens whereabout the sun will ascend or descend upon the plain . that is , it shews the amplitude of sun rising upon the plain . 1. note , that if the plain be horizontall , then the two last uses serve for the place where a man lives . but if the plain be not horizontal , then it serves not for the place , but is proper to the plain it selfe , or to that horizon or latitude which the plain represents , according as they are set down in the former rules : because things are done to the plains latitude , and not to the places . 2. note again , that if upon plain● that are not horizontal , you would yet have them ( to shew the second of these last preceding conclusions , namely , ) to shew the time of sun rising or setting for the place in which you are , and not for the plains , then must you take ( not the co-tangent of the plains latitude , but ) the co-tangent of the latitude of the place , just as you did in the horizontall plain , and then the zodiacal index being rectified will effect the thing required for the place , and not for the plain . 3. note , that both in the horizontal and the other plains , there may be a peculiar scale put , either for the amplitude proper to those plains , or else for the amplitude proper to the place , if a table of amplitudes be computed for either of them , by this proportion . as the radius , to the sine of 1 , 2 , 3 , &c. or 5 , 10 , 15 , &c. degr . of amplitude ; so the co-sine of the poles elevation , to the sine of the competent declination : and then by the said competent declination ( taken in the scale of tangents of 23½ equal to the halfe zodiac ) the forenamed amplitudes be inscribed . or having the declinations , you may look out their tangents in the canon , by which tangents the said amplitudes may be inferred with help of the decimall scale made to the co-sine of the plains latitude , as was before mentioned . the amplitudes proper to the plain are of no great use , unlesse the plain do justly represent the horizon of some known place , whose amplitude you desire to be acquainted withall . i have therefore here added a particular table belonging to the latitude of london 51 gr . 30 min. that by it the said amplitudes may be inserted into such direct recliners and incliners as shall any way stand in the said latitude . a table of amplitudes , with their answerable declinations , for the latitude of 51 gr . 30 min. a. declin a. declin . a. declin . a. decl●n ▪ 1 0 37 11 6 43 21 12 53 31 18 42 2 1 14 12 7 26 22 13 29 32 19 16 3 1 52 13 8 03 23 14 04 33 19 49 4 2 29 14 8 40 24 14 40 34 20 22 5 3 07 15 9 16 25 15 15 35 20 55 6 3 44 16 9 53 26 15 50 36 21 28 7 4 21 17 10 29 27 16 25 37 22 0● 8 4 58 18 11 05 28 17 00 38 22 32 9 5 36 19 11 41 29 17 34 39 23 04 10 6 12 20 12 18 30 18 08 40 ●3 35 another table of amplitudes , with the tangents of their answerable declinations , for the latitude of 51 gr . 30 mi. a. tan. a. tan. a. tan. a. tan. 1 0109 11 1196 21 2289 31 3385 2 0217 12 1305 22 2398 32 3495 3 0326 13 1414 23 2507 33 3●04 4 0434 14 1524 24 2617 34 3712 5 0543 15 1632 25 2726 35 3822 6 0652 16 1741 26 2836 36 3932 7 0761 17 1851 27 2946 37 4040 8 0869 18 1960 28 3056 38 4149 9 0981 19 2069 29 3166 39 4258 10 1087 20 2180 30 3275 40 4365 so far of this elliptical dial , as it is to be described upon any direct plain which lies under the same meridian with the meridian of the place , and which declineth not from it . sect . ii. how to frame the elliptical dial to other plains which are not direct but declining : to an index that standeth perpendicular to the plain . it is not here to be enquired whether the plain declining , be erect or leaning ; for one rule serves both these kindes . but then it must first be supposed , that the plains situation is in all respects known , how much and which way it declines and reclines or inclines . secondly , these three things must further be found ( either by calculation or otherwise ) namely , 1. the poles elevation above the plain . 2. the plains difference of longitude . 3. the departure of the substylar ( which is the plains proper meridian ) from the vertical line of the plain . these are pre-requisites to that which comes after , in which you must proceed by the following directions . how to limit and draw the ellipsis . having the substylar or proper meridian drawn upon the plain , in its true position , and also knowing the elevation of the pole above the plain , you may , upon the substylar line , set out your shorter diameter , and on a line drawn perpendicular thereto , you may set off the longer diameter , in this proportion : let the longer diameter be as the radius , the shorter as the sine of the plains latitude taken to the same radius . or , the longer semidiameter may be the radius , and the shorter semidiameter must then be the sine of the elevation of the pole above the plain . upon these two extream diameters thus limited , may the ellipsis be described with elliptical compasses . or otherwise it may be done by two circles , the one circumscribed , the other inscribed , both divided into like parts , and so points found through which the ellipsis is to passe , as in the former direct plains was prescribed , and through those points the ellipsis may , with a stedfast hand be described . in right horizons or polar plains there will be no ellipsis at all , but it vanisheth into a streight line only . how to divide the ellipsis into such requisite parts as the plain shall require . if you should put in the houres proper to the plain there will be no difficulty , for then the substilar being taken for the line of 12 , the houres must be drawn as before was shewed in direct plains . for in this case the dial is horizontal , shewing the houre of the day to all places that lie under that meridian under which the plain it selfe lyeth ▪ but if you would put in the houres of the place ( as most frequently is desired ) then you are to work ( for the division of the ellipsis ) either by calculation or by protraction . if you calculate for the ellipsis , you must first forme all the angles at the pole as the manner is in these kindes of dials : then you are to invert the terms of proportion from the common way of calculating ordinary dials , and say thus : as the sine of the poles elevation above the plain , is to the radius ; so is the tangent of each angle at the pole , to the tangent of the angular distances of all the houres from the substylar line of the plain . supposing therefore that you had your ellipsis before described , and the substilar line set off from the plains vertical line in its true position , you may from that substylar line ( and by the horary spaces found by calculation ) set in every houre point into the ellipsis , by help of any circle described upon the center a. as you see here are two circles , the circumscribed and the inscribed , either of which , or any other circle else , will serve to prick on the houres by . the houre points are here set upon the circumscribed circle , and transferred into the ellipsis by a ruler laid to a , the common center of the ellipsis and of the circle . this is one way , and is best done by calculation : here you may also put in the halves , and quarters , and halfe quarters of houres , or what other parts you shall best like of . how to limit the ellipsis , and divide it , without calculation . but if you are desirous to do it rather by protraction , then must you work somwhat as formerly was done , the manner i will in briefe shew , wherein you may see the way both how to describe the ellipsis , and also how to divide it , all under one . 1. having prepared all necessary requisites before hand , you must first set off the substilar line in its true position from the vertical line , which to do , i here suppose already known . so in the following figure you see a c b drawn for the substilar line , or the proper meridian of the plain . 2. upon this plain , and upon some point of the substylar line , as at a , as upon a center , describe the circle b m e , and quarter it , and let the semidiameter of that circle be counted as the radius 3. upon one of the quarters , set off the poles elevation as from e to f , and draw a f , and from e to a f take the least distance , as e g , and with that distance , upon the former center a , describe another lesser circle as c h. 4. from b ( where the greater circle cuts the substylar ) set b m equall to the plains difference of longitude : and from m divide that exterior circle into 24 equal parts , which signifie the 24 houres ; and by a ruler laid to the common center a , transfer them into the lesser circle , as m h , 10 i , 6 k , &c. 5. from every point in the great circle , draw lines parallel to the substylar , and from every point of the lesser circle draw lines perpendicular to the substylar , or parallel to the longest diameter p a r , and so each couple of these lines shall cut each other at right angles . 6. note where every line thus drawn from any point of the greatest circle , meets with the other line which is drawn from the like point in the lesser circle . for the concourses of these answerable lines are the points here required . that is , those are the points which shew both where the ellipsis is to be drawn , and also where it is to be divided into its requisite parts . thus in the precedent figure : m o and h o meet in the point o , 10 l and i l meet in the point l , 6 n and k n meet in the point n , and so the rest will meet in their due places . these points shew the way where through the ellipsis is to be drawn , and the same points shew where the houres are to be marked out . the like may be done for quarters and halfe quarters of houres , or any other division that shall be best liked of . there are many other wayes to do the same things , but i suppose this to be most expedient . note : that 1. the index in these elliptical dials must stand perpendicularly upright upon the plain , making right angles with it every way , as it was ordered to do upon the former sort of direct plains . 2. the annual course of the sun must be limited as formerly in the other plains , that is , making the greater semidiameter as a badius , you must finde the co-sine of the poles elevation above the plain . this co-sine is to be made either as a tangent of 45 gr . whereof you are only to use 23½ : or else it is to be made a decimal scale . by both these you have tables and rules how to compleat the suns annual course , either in degrees of the signes in the zodiac , or by the dayes of the 12 moneths . 3. either the index must move and the ellipsis lie still , or contrarily . every man in this must do as his invention shall best suggest . and that motion must be made either upon the substylar line , or else parallel to it , which way soever it be , it must be precisely and punctually ordered . 4. all other things concerning the time of sun rising and setting , the suns amplitudes and declinations , may here ( in the same manner as before ) be inserted either proper to the plain , or proper to the place , as shall be desired . 5. in right horizons or polar plains , the ellipsis closeth quite up into a streight line : and so the division of it is only by the exterior circles parts projected upon it by those lines that are drawn parallel to the substylar . thus far of the elliptical dial , as it is to be described upon declining plains whose indexes stand perpendicular to the plains , and which do not lie under the meridian of the place but swerve from it . sect . iii. another way to prick down the ellipsis upon an horizontal plain . latitude 51 gr . 30 min. hor ang. altit . hor xii 00 ●0 38 30 xii   04 47 38 24     09 33 38 7     14 16 37 38   xi . 18 54 36 58 i.   23 27 36 7     27 53 35 7     32 13 33 56   x. 36 25 32 37 ii.   40 29 31 10     44 26 29 36     48 15 27 54   ix . 51 57 26 7 iii.   55 32 24 14     59 1 22 16     62 24 20 14   viii 65 41 18 8 iv.   68 54 15 59     72 2 13 47     75 7 11 33   vii 78 9 9 17 v.   81 9 6 59     84 7 4 39     87 4 2 20   vi. 90 0 0 0 vi. for this purpose here are two tables joyned together , both of them made for the latitude of london , 51 gr . 30 min. the like to which every man may calculate for his own place . the first of them is a table of such angles as are made by the hour lines ( coming through the center of the ellipsis ) with the meridian line or line of 12. and it was made by that rule which was given ( in this case ) at the beginning of this treatise . namely , as the sine of your poles elevation ( which is here 51 gr . 30 m. ) is to the radius ; so is the tangent of each houre and their quarters ( counted from 12 a clock , ) to the tangent of the angle required . the second table is of the altitudes of each houre and quarter , in the equinoctial circle , above the horizon ; and it is calculated by this rule . as the radius , to the co-sine of your latitude ( which is here the sine of 38 gr . 30 min. ) so is the sine of each houre and their quarters , ( counted from 6 a clock , ) to the sine of the altitude which is here sought . then having computed these two tables , you may ( by help of them ) both draw and divide the ellipsis into its true houres and quarters by these following directions . first , draw the two lines 6 6 and d e , crossing each other at right angles in a , and let a d be the meridian , and 6 a 6 the two six a clock lines . then upon a as a center describe the circle 6 d 6 of any convenient bignesse ; and upon the same circle from d ( on both sides of it ) set on the houres and their quarters , as 11 and 1 , 10 and 2 , &c. by help of the degrees and minutes of the angles set to every houre in the former table . and through every of these points inscribed into the circle , draw lines from the center a , as a 11 , a 10 , a 1 , a 2 , with the rest of the houres and their quarters if you will. fourthly , looke in the second table for the altitudes of the equinoctiall at every houre . then count those altitudes in the line of sines , and take with your compasses the several distances of them from a , & transfer the said distances from the center a to every houre respectively , so shall those intersections give you the points through which the ellipsis must be drawn . thus the altitude of 7 and 5 a clock ( which is 9 gr . 17 m. ) being taken upon the line of sines from a towards s , is inserted into 4 houres at the note m. and the altitude at 8 and 4 ( which in the table is 18 gr . 8 min. ) being taken and transferred to the four houres at n , do give four points more . the rest of the altitudes give ( each of them ) two houres only , as at o , i , and r , is done . and the last of all at 12 , gives one only point at t. the like may be done for the quarters . and so through these points thus found the ellipsis may easily be drawn , and the lines formerly drawn give the divisions that are due unto it . note . 1. this is propounded only for horizontal plains , but it may without difficulty be applyed to any other direct plain . the trouble that is , comes by reason that the foresaid plains have a different elevation of the pole , from that elevation that belongs to the horizon or place where they are to stand : and consequently there will be required two new tables for that elevation which is proper to the plain ; the calculation of which will easily be done by the two former proportions set down for that end . note further that by these two tables you may prick down an ellipsis upon any leaning ( not upright ) plain whatsoever . but the index must then lie in the zenith line of the place , ( not of the plain ) and the ellipsis ( or index ) must move in the meridian of the place ( not of the plain ) and the zodiac must be like or proportional to the zodiac of the horizon , but augmented , for the most part , in this proportion . as the radius , to the secant of that angle that the meridian line ( upon the plain ) makes with the horizon ; ( which angle must be gott●n by a clinatorie ; ) so the radius of the horizontal zodiac , ( which is the co-sine of the latitude , ) to the radius of the zodiac proper to the plain : which zodiac must be set according to the line of 12 upon the plain . 2. this way of delineation may likewise be applyed to all other plains which decline , and are not direct . but in these will be found more difficulty then in the former plains which declined not , unlesse it be required to put in the houres that are proper to the plain ; for in that case the work is the same which was in them . but if the houres of the place are to be inscribed , as most usually they are , then there will be some trouble , by reason that the difference of the plains longitude seldome falls upon any just houre . i purpose not here to shew the way , it being such as will prove un-pleasant to the unskilfull , and such as have knowledge will finde it out quickly . perhaps the pleasure of the thing done , will recompense their labour . sect . iv. here follow some vses and varieties of this ellipticall diall . uses . first , if the index stand still , and the ellipsis be made moveable ; you may upon that plate which is immoveable , and on which the index standeth , describe a circle , whose center must be the foot of the index . and drawing a meridian line upon the said immoveable plate ( just under the meridian of the ellipsis , and ) from the foot of the index you may divide that circle ( beginning from the meridian line ) into 360 gr . or so many of them as shall be needfull . by these degrees you shall finde the suns azimuth by the shadow of the upright index , and by the divisions of the ellipsis you may know the houre of the day . so both azimuth and houre are shewed upon the two plates by one upright index . this kinde of diall must be set in its true position ( for it will not set it selfe ) and must there be fixed . if it be upon an horizontall , or any direct plain , you may draw a meridian line thereon , and either draw the azimuths true upon it , and so make the ellipsis to move to and fro according to that meridian : or else , if both your plates be to be fixed upon any base prepared for them , you may , first , upon that base draw a true meridian line , and to that line make the meridian of the diall to agree . the like must be done in all declining flats , onely you must be carefull to make the shortest di●meter of the ellipsis ( which is the proper meridian of the plain ) to agree with the same proper meridian , as it shall first be drawn upon the base in its true position . 2. a second excellent use may be for the setting of moveable dials , such as are made to stand out of the weather , and are only to be set in the sun at pleasure , when the houre shall be required . of this kinde are polyhedrall bodies made of some light matter , and concaves , &c. but it is supposed that some of the superficies of such bodies , ( &c. ) have the houres of the p●ace described upon them the ordinary way , by the axis of the equinoctiall , and that this ellipsis is fitted to some one plain ( which principally is the horizontal plain ( because that alone will shew all the houres of the day ) or in some horizontal position to the concave , or any such diall ) in its just position and alwayes with an upright index , ( that is , an index perpendicular to the plain whereon it stands ) which is not alwayes upright in respect of us , but only in respect of the plain ) and with the houres also of the place . these things being so fitted , you may place your body or concave in the sun , and turn it about till you discern that both kindes of dials do agree to shew one houre , which when they do , then all stand right in their true situation , which will never else happen but only when the houres do in both agree . — to this place is referred the double horizontal diall , the one of them giving the houre by the axis , the other by the upright index . — and here it must be noted , that the worst time of the day for this setting the dials in a true position , is at the neerest to noon , and the best time is about the suns b●ing in the east and west azimuth . 3. this kinde of diall may be placed ( in a moveable posture ) before a casement in any window that hath a flat board upon which it may move . and the index to it may be an upright threed ( precisely upright it must be ) that is fixed at the top and bottome of the window , to and from which the ellipsis must move in a meridian line that is drawn upon the board of the window , & from or justly even upon the threed . — and where i say the window must be flat , it is to be understood that there is no extraordinary exactnesse required in that . for if it should rise or incline 5 gr . to the horizon , it would not erre one minute of time about our latitude ; provided that the meridian be truly drawn , and the index perfectly upright , in which all possible exactnesse is required . 4. if any choise plain reclining , or however situated plain that shall represent the horizon of some notable place , be fixed as a base whereon to place such a diall thus fitted , and drawn to the proper houres of that same place : this double diall being placed upon that base , and turned about thereon , will shew the true houre at that place when both of them do agree , and the meridian of it will shew how the proper meridian of that place is to be placed , and which way drawn . 5. if such a recliner ( or incliner , or erect ) shall be furnished with such a double diall to some intended declination , though they stand and be framed properly to the plain in respect ( chiefly ) of the upright index , and shall ( notwithstanding ) have the proper houres of the place upon both dials : and if further such a recliner as was mentioned , have an horizontal base , upon which it may be turned round : i say , though this body be horizontally moveable , and may be turned any way , yet the two dials will never agree to shew the same houre , untill the base stand to the true declination to which the double diall was drawn . sect . v. some varieties of the structure of it do here follow . first , instead of the ellipsis drawn to the proper meridian and latitude of the plain , there may be used a true circle equally divided : but then let these things be observed . first , that the index must be perpendicular every way to the plain on which the circle is elevated , standing upright upon it . secondly , that the motion ( either of index or circle ) must be according to the proper meridian of the plain . thirdly , that the zodiac must also be proper to the plain , and must lie upon the plain or base it selfe , and must be the same with that which should serve for the ellipsis . but then , fourthly , this equally divided circle must be elevated over the proper meridian of the plain ( that is , the base of it must rest upon the proper east and west line of the plain ) so much it must be elevated as the elevation of the equinoctiall above that plain comes unto . and the elevation of it may be either way ; that is , either upon the south coast , and then it will be answerable to the equinoctiall in the heavens ; or else it may be towards the north , but still so much as the equinoctials ( not the poles ) elevation comes unto , and in some cases it must be done both wayes that the sun may come at it , and it must then have two indexes . both wayes it is alike true . it must be divided into equall houres and parts , and so much of it must stand above the base or plain on which it moves , as the longest day comes unto . the beginning of the division must be from the line of 12 , and if the plain be not direct , then must the difference of longitude be counted in the circular degrees , from the highest point , or from the verticall line of this equinoctiall circle . 2. this diall is not so tyed to a plain , but that it may also be brought to an un-even superficies , yet so , that the motion of it , or the zodiacal scale must be upon some plain . but to shew the manner how this should be performed , is more proper for those projecting wayes which i have in another place had some assayes of . only remember that the horizontal diall must be projected , not by the axis , but by the zenith . 3. a double diall depending upon this kinde , may be made to set it selfe , and to shew the houre in a craticular forme , which is also more properly to be handled in my other wayes . there remains yet one structure more , by which it is fitted for some uses that follow after , and is made a portable instrument , not at all to be used in observation . 1. make a right angled parallelogram , as a b c d , whose length may be about three times the breadth : and let the limbe ( upon which the graduations stand ) be higher then the plain or area of it . 2. describe a semiellipsis to your own latitude , whose longest radius let be m o , a little longer than a b , that when the center m is left out and cut off , so much as m n comes to ( which may be about halfe a quarter of an houre from 12 in the ellipsis ) the remaining breadth n o , or e f , or g h , may be just fit to a b the breadth of the hollow area , in which this lesser rectangled parallelogram e f g h may be moved to and fro as there shall be occasion . and let the thicknesse or depth of this moving plate be the same with the thicknesse of the limbe of the lower plate . the division of the ellipsis may be made according to the rules before given . 3. in the midst of this lesser plate , let a hole be pierced quite through it ; and in that hole let a threed be extended , as i k , which is to serve for an index , by which the said semiellipsis is to be rectified . and therefore it is best that the threed i k be laid just in the longer diameter , which is the 6 a clock line . 4. take the lesser parallelogram and put it into its place , as close to the end a b as it will go , and when it is there laid , observe where the threed lies upon the lower area , and draw the line upon which it lies , which must be the line 90 , 90 ; upon which the scales ( that are inscribed ) must take their beginning . and likewise you are to take the longer radius o m , and as the plate there lies , set one foot of that radius upon o , and let the other reach as far as it will , which must be upon the rising limbe beyond the inner edge of it , so much as m n comes to : in that point you are to make one center : so again , turning your lesser plate about , you must lay it in the contrary way , close to a b , as you did before , and from the point o you may finde the second center : both these centers ( in each of which must a threed be fastned ) will happen to fall in the former line 90 , and 90 , neer the points t and v. being thus far prepared , you may take away the lesser plate , and divide the scales upon the greater plate , in this manner . 5. first , for the two limbs of it , they are nothing else but the degrees of a circle projected thereon from the two centers : which may therefore without difficulty be done either from the degrees of a circle , or else by tangent lines , as any man will easily perceive . 6. secondly , for the other scales you must make them parallel to the edge of the limbs . there are three of them in number : the longest is a scale of tangents : the middlemost is a scale of houres , made in the form of a line of sines : the third is a zodiac , or scale of the year . the tangents are made thus . taking m o as radius , finde the co-sine of your latitude agreeable thereto : and making that co-sine as a radius , or tangent of 45 degrees , all the other tangents are to be fitted accordingly . then to know how far of this line will be usefull , put the lesser plate again into its place , as close to the end c d as it will go , so shall the threed i k shew the furthest point of it that will be usefull . the middle scale of sines is made thus . count the complement of your latitude upon the new made tangent line , and from the beginning of the scale to that point , make a line of sines for six houres and their quarters , so shall this scale be fitted . the zodiac or annual scale is equall in length to the tangent of 23 gr . 30 min. and the manner of inscription with tables whereby to do it , are set down before , and need not be here repeated . the reason why this instrument is thus contrived . the intent of the contrivance is , that it should shew both azimuth and houre , and that therefore there should be a ci●●● and an ellipsis sliding over it , both of them made to one center , so that the center should be in the midst , and the circle should be a whole entire circle , and the ellipsis also whole . but because the ellipsis sliding upon the nether plain , doth somtimes of necessity cover the center , so as that there can be no threed fixed for use , i was therefore compelled to order it in this manner that is before exprest , that the motion of the upper plate and the centers migh● stand cleere without hin●●ing each other . then further it must be noted , that the instrument is ( not generall to all , but ) particular to one la●●tude . the uses of it do here follow , which by reason of the semiellipsis will not be so easie , because many changes are required . in generall note thus much . 1. in using this instrument you must alwayes suppose that you looke upon the north. 2. if the declination of the sun or star be north , hold a b towards you . if south , then hold c d towards you . 3. if the sun or star be eastward of the meridian , let the morning houres be towards you . if they be westward , let the evening houres be towards you . 4. alwayes use that threed which is on the same side with the center of the ellipsis . and that side lin● of the larger plate is ever to be taken for the meridian , and consequently the other side must be esteemed for one of the two coasts of east or west . 5. becau●e north is furthest from you , therefo●● the furthest quarter of the limbe must be either north-east or north-west : the neerest south-east or south-west . these cautions being observed , the conclusions that are to be wrought will be easie . 1. by having the azimuth to finde the houre . first , order your instrument by the former cautions , according to the present case of your observation . and rectifie the upper ●late ( by help of the threed index ) either to the declination of the star , counted in the tangent line , or to the suns place in the zodiac . then count the azimuth given upon the right quarter of the limbe , and apply the threed thereto : so shall the threed shew ( upon the ellipsis ) the houre in which the sun or star is . if the work be for the sun , then the houre thus shewed is the suns houre , or houre of the day . if the work be for a star , it then shews the stars houre only ; which must afterwards be reduced into the suns houre , that it may be the true houre of the night . this change may most easilie be done by a common nocturnall . 2. by having the houre in which the sun or any star is , the azimuth thereof may be found . the instrument being rectified ; lay the threed upon the houre ( counted in the ellipsis ) the same threed will shew the azimuth in the graduations upon the limbe . 3. to finde the ascensionall difference , the semidiurnall and seminocturnall arkes , with the time of their risings and settings . in generall note this : if the sun or star decline northward , then the semidiurnall arke is greater than six houres ; if southward , then lesse . the upper plate therefore being rectified , either to the suns place , or to the stars declination , the threed will shew upon the middle scale , the ascensionall difference , if you count the houres how many they are from 6. it shews the seminocturnall arks , by the houres of setting and rising being rightly taken according to the condition of north or south declination . it shews the time of sun rising and setting , according to the houres upon which it lies . but for the stars , it shews only upon what houre-circles they rise and set ; but the true houre or time thereof must be found by turning the stars houre into the suns houre , which ( as i said before ) will be done with most ease by an ordinary nocturnall , in which the stars ( that you deal with ) are placed . 4. to finde the amplitude of their risings and settings . when your upper plate lies so rectified , lay the thre●● upon the ellipsis , to the ascensionall difference , or s●midiurnall arke before found . the threed so laid , will shew the amplitude in the limb of the instrument . note here . in the first of these propositions i suppose the azimuth , in the second i suppose the houre to be given by observation . for the observing of these two , you may make in any convenient window , both an horizontall diall , with a threed axis lying in the axis of the world , and an azimuth circle with a threed axis , standing upright in the zenith line . the divisions sutable to both these , may be inscribed upon the board of the window ; and for the sun , the shadow of the threed will give the houre of it upon the horizontall diall ; or the azimuth upon the azimuthall circle . but for the stars that cast no shadows , you may make sliding sights , that may move upon the edge of the window board , or may be upon the bo●rd it selfe , ( which must in these respects be level , or at least neer to it . ) and those moving sights must be furnished with short wires or pins which may best stand upon the edge of the sight , and go down to the window board , and to the divisions that are drawn upon it , which may be both for sights by which to looke at the standing index and star both together , and must serve as indexes to shew the houre or azimuth on which a star is at any time . only this generall caution must be given , that th●se pins be set parallel to the threed axis that is set in the window ( that is ) for the azimuths they must stand upright , but for the houre they must lie aslope , parallel to the axis of the world . this i have sometimes done , and this any man may at his pleasure do , whereby either the houre or a●●muth may be observed , and when any one of them is known ; this instrument ( now described ) will finde the other . in such like cases this instrument will be delightfull and of good use . 5. to finde the declination of the sun. set the threed of the upper pla●e upon the suns place in the zodiac , or upon the day of the moneth in the annual scale , the same th●eed will give the declination thereto belonging , upon the tangent scale . 6. how to make an horizontall diall by this instrument . the meaning is to know how many degrees are intercepted between the meridian line , and any other houre line . there are two wayes to do it . the first is the converse of the making of the instrument . for if the threed index be laid to the line where the scales of the lower plate begin , so that the line of six do lie evenly between the two ●●nters where the threeds are fastned , t●en the center threed applyed to the houres severally will shew how much they are gradually distant from the line of 12. but you must here remember to take 6 for 12 , and 7 or 5 for 11 or 1 , and so the rest of the houres in that order . and the graduall space● or distances from the meridian or line of 12 , must be counted from 90 contrary to the numeration of the limb . bu● this is not so artificiall as this that now follows . set the moving plate to any houre upon the middle scale , and apply the center threed to the like houre in the ellipsis taken on that side of it that is furthest from ●he beginning of the three scales , and it will s●ew you ( upon the limb ) the angle or horizontall space that is du● to that houre . the like is to be done in every other of the houres . sect . vi. an advertisement concerning s●me ot●er uses of the instrument that was last described . since the writing of that which is gone before , other things there are which came into my thoughts concerning the further use of this instrument . it were more expedient therefore that the ellipsis were divided into houres and degrees , rather then into quarters and half quarters of houres . which division ( as also the description of it ) may by protraction be performed by those rules that are given before . but if it be thought better to do it by tables ; then by the former rules of calculation given for this work , you may frame two tables , the one of angles , the other of altitudes to your own latitude , such as in the following table are computed to ●he latitude of london . b● these tables you may put in every houre and third degree , the intermediate degrees may be equally divided . the numeration will be either by 15 , 30 , 45 , &c. according as the signing of it by hours wil requir● ; or ●lse besides the houres you may set on 10 , 20 , 30 , &c. in small figures , which will stand without hindring the numbers ( or numerall letters ) that are set for the houres . ad latitudinem 51 gr . 30 min. hor gr . angles altit . gr . hor xii 00 00 00 38 30 00 xii   12 3 51 38 26 3     9 7 39 38 16 6     6 11 26 37 57 9     3 15 12 37 31 12   xi . 00 18 54 36 58 00 i.   12 22 33 36 18 3     9 26 08 35 32 6     6 29 38 34 40 9     3 33 04 33 41 12   x. 00 36 25 32 37 00 ii.   12 39 41 31 27 3     9 42 52 30 14 6     6 45 59 28 56 9     3 49 00 27 33 12   ix . 00 51 57 26 07 00 iii.   12 54 50 24 37 3     9 57 38 23 04 6     6 60 23 21 28 9     3 63 04 19 49 12   viii 00 65 41 18 08 00 iv.   12 68 16 16 25 3     9 70 47 14 40 6     6 73 17 12 53 9     3 75 44 11 05 12   vii 00 78 09 09 17 00 v.   12 80 33 07 26 3     9 82 56 05 35 6     6 85 18 03 44 9     3 87 39 01 52 12   this ellipsis thus divided , may represent either the horizon , or else any of the ( almicantars or ) parallel circles to the horizon ; & the divisions of it must then signifie the azimuths . and if it be too big ( especially when it comes to represent any of the higher almicantars ) you may adde another sliding plate of the same breadth that the former was , na●●ly m o here equall to m o in the former , and m n must in both be justly equall , that it may both slip in the former cavity , so as just to fill it , and that the same limbs , centers , and threeds , may in both agree . a lesser ellipsis ( i say ) must be described , which though it be lesse , yet it must be like and proportionall to the former . therefore assuming any length as m p , for the longer radius , you must ( to that radius ) make m r and m s equall● to the sine of your latitude , and so describe and divide it as formerly was ordered . then again , to this new ellipsis there must a new scale of tangents be made , whose length must be limited as the former was , in this manner . to m p as radius , finde the co-sine of your latitude , and make that co-sine a radius or tangent of 45 gr . and according thereunto , continue the scale of tangents as far as it will go . it must begin upon the same line that the other scale of tangents began , and must go the same way with it . and in this new way of ordering the scales and the ellipses , it must be considered what the other parts of the instrument do signifie the end of the lower plate , upon which the scales begin , must be accounted for the north , the other end c d is to be taken for south . the side lines are the meridian . the degrees of the limbs are to be used as the degrees of the equinoctiall . the threed applyed to those degrees are the meridians comming from the centers , which are the north pole. the tangents are the degrees of altitude . the ellipsis notes out that almicantar which the threed index stands unto upon the tangent scale . and further . it is to be here noted , that if the meridians ( or threeds comming from the center ) were rightly divided , those divisions or parts of it should signifie the degrees of declination from the equinoctiall . but the inconvenience of it is , that because one ellipsis is to represent every almicantar , therefore the radius cannot possibly be at all times the same in length , but must varie according to the removing or severall positions of the said ellipsis . that is , when the ellipsis signifies the horizon ( or stands at the beginning of the scales , without any altitude at all ) then the radius m o in the first moving plate , or m p in the second moving plate , is the scale of declinations , but at other times , when the ellipsis signifies any almicantar , then must the secant of that almicantar ( of altitude ) be taken as the radius for that particular work . but alwayes the radius ( of what length soever it be ) must be divided as a whole line of sines , the greatest parts beginning towards the center , the least parts ending at the remotest end from the center , and yet then again , the numeration of the parts or declinations of it must begin at the remotest end , and must end in the center at the pole. it might be applyed to use in the motions of the stars , but that would be troublesome ; therefore it shall suffice to make it usefull only for the sun. the best way that ( for the present ) i know whereby to divide this radius ( thus of severall lengths ) is this . divide the length from the center t to c into 90 sines ( the greatest parts beginning at the center , but numbered the contrary way . ) this line may be drawn from the center t into some part of the limbe , where it may have room to receive divisions . then with your compasses take the radius t v , equall to m o , and from the end of the former scale , open the threed to the least distance , and where the threed stayes , there make a mark upon the limbe v d , for there is a new scale to begin , for the secants that were mentioned before . then lay the threed from t to 80 in the opposite limb , and move the ellipsis till the point o fall just under the threed , and take t o in your compasses , and put in that distance as before , so shall you put in 10 gr . into this new scale . in like manner lay the threed to 70 , 60 , &c. and bring the ellipsis till the point o lies under it , so shall t o ( in these severall positions ) give the lengths to be inscribed in this new scale , for 20 gr . and 30 gr . &c. all the rest must be done in like manner , till you have put in as many as will come within the reach of the limb , which will be upon 60 gr . or thereabout . the like should also be done for the lesser ellipsis , working in the same manner by the threed and the point p. these are called the new inscribed scales . the former line of sines serves to them both indifferently . now when things are thus fitted , the uses of them will be such as in particular astrolabes is vulgarly known . some of the uses are here mentioned . 1. having the suns declination and altitude to finde the houre and azimuth . first , for north declinations . set the ellipsis to the altitude counted in the tangent scale . then count the same altitude in the new inscribed scale of secants , and thereto lay the threed . afterwards , take the least distance from the suns declination , ( counted in the fore-mentioned inscribed scale of sines ) to the threed . set one foot of that distance in the center , and extend the other till it crosse the ellipsis , and where it crosseth , thither apply the threed , which ( in the limb ) will shew the houre and scruple ( in degrees ) required . the same point of the compasses doth also immediately shew ( upon the ellipsis ) the azimuth sought for . if the compasses do crosse the ellipsis twice ( as somtimes it will ) take that crossing that is furthest from the meridian . secondly , for south declinations . the work is in a manner the same : only you must note that the threed will crosse the ellipsis twice , and you are in this case to take that crossing which is neerest to the meridian . by supposing the altitude to be 00 , that is , by laying the ellipsis upon the line t v , and using the declination as before , you may finde both the amplitude and ascensionall difference as before , and what else doth thereon depend . but because these things being done neer the horizon , will not prove good , especially when the declination is little from the equinoctiall : and because the work at the best is but troublesome , i shall here break off , supposing that i have already written too much . a demonstration of the ellipticall diall upon an horizontall plain : shewing the reason why the same diall , by an upright index should shew the true houre . the reason is principally deduced from the sphere it self : and secondly , from the orthographicall projection of the sphere upon the plain of the horizon ; which as it doth represent the sphere it selfe , so it doth perform the same conclusions with the like certainty that the sphere doth ; and that upon this generall ground , that looke what circles , and in what parts they cut each other upon the former , the same circles and the same parts of those circles do upon this latter cut each other in the same manner . i shall here take for granted , that the projecture is ( for truth ) in all respects answerable to the sphere it selfe , which is abundantly made good by those that have treated upon this subject . then from th●s projection i shall shew how the ellipticall diall upon the horizon must be of the same truth with it . and therefore these things following must be considered , as so many lemmaes to prove what is here required . note , that in no form of projection besides the orthographicall , this kinde of diall can be true ; that is , that no figure but an ellipsis will do it , nor yet all ellipses , or ellipses divided into all 〈◊〉 , as in some projections the● are , when one halfe or 180 gr . on one side , are greater than the other halfe or 180 gr . on the other side : these will not do here , but only such as whose extream diameters are the sixes and twelves , that is , such as are produced by infinite orthographicall projection . lemma i. first , the equinoctiall circle , and all the parallels of declination , are projected upon the plain of the horizon into like ellipses . and that they are all of them divided ( by the meridians ) alike or proportionally one to the other . the reason of this may be conceived upon this ground . since the equinoctiall and all the parallels are circles of a just parallel situation one to the other ( as their name imports : ) therefore on what point soever of the horizontall axis the eye is placed , the said circles must be ellipses ( excepting only if the eye be placed upon the superficies of the sphere , for then , and only then , they will be all perfect circles , ) as all that write upon this subject do sufficiently make good : and consequently , when the eye is placed upon the same axis , in an infinite distance , as this projection supposeth . but the second thing is more to our purpose , and that is , that all their degrees and quarters , and other like parts , art like and proportionall one to the other ( not all the parts of one and the same ellipsis are like , that is equall , one to the other , as in the circles themselves upon the sphere they are , but each one of these ellipses is justly like any other of them , both in the whole , and likewise when any of the like parts of one be compared with the like parts of the other , as one quarter , &c. with another . ) and the reason briefly is , because the eye is supposed ( in this orthographical projection ) to stand in an infinite distance from the plain on which the projection is made ; and because it doeh stand in an infinite distance , therefore all these circles are alike situated to the eye , and consequently must make a like or proportionall projection of every of those parallels with their like situated parts . but if the eye should stand at a finite distance , then these parallels would not have a like situation to the eye , neither in respect of distance , nor in respect of position , and so the like parts of the parallels could not be like one to the other in the projection , because of their different position . but in the orthographicall projection at an infinite distance all things will be alike , being alike projected from a like position . i say in this it will be so , and in none besides this . corollary . whence will follow : that the equinoctiall circle , or ellipsis rather , may represent both it selfe , and likewise any other of the parallels , if it be so contrived to stand from the zenith point , as that it may have a proportionall situation from it , suiting with the position of any other parallel which it is to represent . and this is the main ground upon which this diall will be proved to be true . lemma ii. secondly , any two like plain figures may be alike situated . to any point assigned ; so as that from that point any two right lines being drawn infinitely , shall cut proportionall parts from those two like figures . this is generall to all like figures , whether they be two or more ; plain or solid : but for our purpose it will be enough to shew it to be true in like ellipses . one way for the making of one ellipsis like to another , is by assuming of any point , as a in the greater ellipsis , and from thence drawing as many lines as you wil , as a b , a c , you must divide them into proportionall parts at d & e , that as a b to a d ; so a c to a e : & then through those points the lesser ellipsis may be drawn , which must be like to the greater ▪ and alike situated to the point a , because all those lines are proportionall , and the subtenses b c , d e , will likewise be parallel and proportionall . and if the lines a b , a c , had been augmented proportionally ( as here they were diminished ) there would have been a greater ellipsis described , like to the other . corollary . hence will follow ▪ that if the centers of two like ellipses be laid upon one line , as upon one of their common diameters , and from any point assigned ( unto which they are to be alike situated ) there be an infinite line drawn to any point of one of the ellipses , and the like point of the other ellipsis be brought to that ●●finite line ( the center of it still keeping upon the common diameter , ) those ellipses in that situation shall stand alike posited to the assigned point . because there are two lines ( the common diameter and the infinite line ) that from the point assigned do cut like parts of both ellipses . the common diameter makes them alike situate in respect of one dimension ( suppose the length ) of the locus planus whereon they lie , and the infinite line limits them in respect of the other dimension ( suppose the breadth ) of the same locus planus . so that they are quite limited in respect of their like situation to the point , and can ( in this respect ) lie in no other place for this individuall position . lemma iii. thirdly , all the azimuths of the horizon are ( in this orthographicall projection ) cast into streight lines meeting all in the zenith point . the reason is , because all these circles do crosse each other upon the axis of the horizon ( or upon the zenith line , ) upon which line the eye is imagined to be placed at an infinite distance . therefore they all ( that is , the plains of them infinitely continued ) do crosse through the eye , and consequently must all be projected into streight lines . and their concourse being upon the zenith line , and the same zenith line comming into the eye , the said whole line , and their whole concourse will be cast or resolved into one single point , answering to the zenith point of the sphere . the application of these things to the purpose intended . first , in the projection it selfe we are to consider these things . if it be furnished with parallels and houres , as the manner is , and laid horizontally with the meridian line of it just north and south : and if further , there be an upright index set in the zenith point at z , then shall the shadow of that erect index represent the azimuth in which the sun is , and if you note where it cuts the parallel wherein the sun ( for that day ) is , the same shadow will shew ( among the houres of that parallel ) the time of the day . this conclusion is true upon any horizontal projection ( the stereographicall as well as the orthographicall ) for the streight lines or azimuths whether shadow or index comming from the zenith , and cutting through any houre in any parallel , doth shew the azimuth in which the sun must be at that houre for the day of that parallel . and further , because all the projection is made upon the plain of the horizon ( which therefore must be the fundamentall circle of the projection , and ) which for that cause is , and must be ( as all fundamental circles of any projection are ) equally divided ; therefore an upright index standing in z the center of it , and so answering to the zenith line of the sphere , and the degrees of the horizontall circle answering to the degrees of the horizon in the sphere . i say the shadow of that index doth really shew the true azimuths of the sun , or the true angles of position that the sun at any time maketh with the meridian line . and further , the same shadow or azimuth where it crosseth the parallel of the day ( which parallel is divided into its proper parts , like to the parts of the equinoctiall , by the meridian circles issuing out of the poles ) it notes out also the houre of the day . so that the projection made in this manner with an upright index and divided parallels , will ( by the shadow of that index ) shew the true houre of the day upon that parallel that is proper for the day , if the meridian of the projection lie in the true meridian of the horizon . the next thing to be shewed , is how the ellipticall equinoctiall may supply the use of all the parallels . that is to say , how the equinoctiall , being made movaable , and the index standing alwayes still , may be fitted to represent any of the suns parallels . and because ( as was said before ) it is every way like to each of the parallels , it will therefore be only required to give some rule how the said equinoctiall may be at any time placed in a like position to the zenith point at z , that any parallel hath to the same point , so as that any right line ( or azimuth ) being drawn from thence may cut a like part or point in the equinoctiall that it doth in the parallel . let the parallel be h e , i would remove the equinoctiall ae a so , as that it might have a like position to the point z , that the parallel h e hath . by the corollary of the first lemma , it may represent it . by the corollary of the second lemma , the way of it will be easie . for first , the two ellipses h e and ae a , have their centers upon the meridian line ae h z p , as upon one of their common diameters . secondly , the parallel h e cuts the six a clock circle , or meridian p e a , in the point e , making e a equall in radiall degrees to h ae ( for the parallel is every where equidistant to the equator , and the meridional arks , therefore , intercepted between them , such as are e a and h ae , must be equall . ) so that e a is the declination of that parallel from the equinoctiall . if therefore the line ( or azimuth ) z e be drawn infinitely through the point of 6 in the parallel , or through the declination ( of that parallel ) counted in the circle of six ; and then the equinoctiall ( still keeping its center upon the common diameter ae p ) be slipped up till the point a thereof do concur with the infinite line ( drawn before ) at g ; in thus doing , i say that the equinoctiall is in a like situation to the point z , that the parallel h e is in , to the same point . and consequently , that any right line ( or azimuth ) from z , will cut the same ( rather like ) parts in one that it doth in the other . as by the corollary of the second lemma will appear . the next thing to be enquired is , what the line a g is , or how it must be found and estimated . for this purpose , consider , that z p e and z c o are two triangles rectangled at p and c , and having a common angle at z ; therefore , as z p co-sine of the latitude , to z c the radius ; so p e , the co-tangent of the declination , to the tangent of c o. or , as the radius z c , to the co-sine of the latitude z p ; so is the tangent of e a , the parallels declination , to the tangent of a o , that is , to the line a g , which is the tangent of a o , because the arke a o is expressed in its full quantity ▪ without any enlarging or fore-shortning , and the right line a g stands to it as a tangent thereof , standing at the end of the radius z a. now consider , that if z c be considered as radius , then must the tangent of e a be the tangent of the parellels declination , estimated to the same radius . and because z p is in the selfe same proportion to a c , therefore : if z p ( the co-sine of the latitude ) be estimated for a radius , then must a g be esteemed as the tangent ( of this parallels declination ) to the said radius z p. and hence it will follow that if z p the co-sine of the latitude [ being taken ( as that co-sine ) to the radius z c or z a , which is the longer radius of the ellipticall equinoctiall ] be counted as a radius or tangent of 45 gr . and so be divided and continued as occasion shall be , it is to be noted i say , that this scale thus limited will be a right scale for the requisite motion of the ellipticall equinoctiall to such a position in respect of the zenith point , as that it may represent any parallel , whose declination is known , if it be removed to the same declination counted upon that scale . and being set in that like posture , any azimuth or shadow of the upright index that would passe through any houre point of the parallel , will also passe through the same point of the equinoctiall , and so this one may serve for them all . this gives the reason of making the zodiac or annuall scale , mentioned in the former treatise , i say it gives the reason of the second way mentioned for lesser latitudes . for if either the signes of the zodiac , or the moneths of the year be put in according to their declinations taken in this scale , it will be all one to set your ellipsis to the signe or moneth , that it is to set it to the declination , since they are made one to answer to the other . then for the first way for greater latitudes : because , as the co-sine of the latitude , is to the sine of the latitude ; so the radius , ( or tangent of 45 gr . ) to the tangent of the latitude . you see that if you divide the co-sine of your latitude into 45 tangents ; or the sine of your latitude into the tangent of your latitude , the scale will be all one according to the former proportion . what is before spoken of north parallels of declination , is the same in south also , and one scale ( for the kinde of it ) serves both ; only in north declinations the equinoctiall goes neerer to the zenith point , in south it goes further of . all other scales annexed to the ellipsis have their dependance wholly upon this scale of declinations , and will not need any further explication in this place . then again , it matters not where the little scale or zodiac stands , so that it give the just quantity of removall for the equinoctiall ellipsis . and so also , it is no matter whether the index move , and the equinoctiall stand , or contrarily : it is only required that their situation be such as may make the foot of the index stand in a proportionall position to the station of the parallel , as was formerly shewed . further. for the circular year or zodiac , the ground of it is thus . the circle is supposed to be divided into equall parts or degrees : and so every sine ( as is d f ) is a proper sine to any arke ( as d c ) to which it is annexed . again , a b in another consideration is to be esteemed ( as formerly for the semidiameter of the circle , so now again ) for the tangent of 23½ degrees , whence this proportion will stand : as the radius a b , is to the sine f d ; so the same a b , tangent of 23½ ▪ to the tangent a e. which is the tangēt belonging to the right ascension c d. [ because in the sphere , as the radius , to the sine of any right ascension , so the tangent of 23½ to the tangent of the declination answering to that right ascension . ] if therefore you have the right ascension belonging to any day , and put it into the circle of equall degrees , as c d here is , the sine of that right ascension , viz. f d , will be equall to a e the tangent belonging to that right ascension , if b a be taken for the tangent of 23½ gr . so that this way of putting in moneths and signes is the same in effect with the former zodiac or annuall course of the sun. thus far for demonstration of what was hard in the first section , pag. 8 , &c. concerning horizontall plains . what is added more in the first section of framing it to other plains that are direct , is the same with this . for there is no difference but only in the latitude of the plain , which is no re●ll discrepance from the former , they both going upon one ground , and therefore no more to be said here of such plains . sect. 2. pag. 22 , &c. of framing it to declining pl●ins . declining plains , if they had their proper houres upon them , would al●o ( in this ellipticall respect ) be the same with the former direct , or horizontall plains . the distinction is , because they are made to another meridian than is their own , that is , to the meridian of the place , and all the account of houres is deduced from it . the difference therefore of these from the former , is only this , that the houres in the ellipsis are not evenly fitted to the quarters or diameters of the ellipsis , but fall to stand as casually they may . now if in the former case of horizonta●l plains , you do but suppose the equinoctiall and parallels to be divided from any casuall meridian as p m ( not p ae ) into houres and parts , then will follow still the same things in substance that was before . namely , that if the equinoctiall circle be removed according to a g ( which must be supposed to be suited to the latitude of the place as well as to the suns parallel of declination ) with those divisions now spoken of , then still , when the equinoctiall by that motion hath got a like situation to the parallel , the parts of one will be answerable to the parts of the other , in respect of the shadows or azimuths that are cast from the zenith point and upright index standing in it at z. so that even in these plains also , the same houre will be shewed by the equinoctiall that would be by the parallel . and so the ground is ( in these ) like ( rather the same ) in substance with the former direct plains . so much of these . sect. 3. pag. 29 , &c. that pricking down of the horizontall diall mentioned in this section , is the pricking down of the equinoctiall with its houres or parts , just as the orthographicall projection it selfe hath them . the first table shews what angles or upon what azimuths from the meridian every houre point lies . the second gives the altitudes , or rather the distances of the same houre points from the zenith , and therefore no more will be required of this . the fourth section requires no explication nor demonstration . sect. 5. pag. 37 , &c. for the varieties it must be known , that the substance is the same with that which went before , which being well understood will give light enough to this . but in this section there is mention made of a circle ( insteed of the horizontall ellipsis ) elevated to the height of the equinoctiall . the reason of that will best be seen out of the sphere it selfe , for there we know , that the equinoctiall and all the parallels to it are both equally divided , and equally elevated , and so being all alike , the equinoctiall may supply the room of each and all of them . only it must be required ( whether the index move toward the circle ; or the circle towards it ) that this motion ( if the zodiac be upon the horizontall plain ) be limited by an horizontall zodiac , such as was used for the ellipsis upon any plain , ( which plain , what ever it be , i now count as an horizontall plain . ) but if the index be made to move upon the equinoctiall ( which though it do , yet it must still lie in the zenith line , not perpendicular to the circular plain , but making an angle with it equall to the latitude of the place , and not the latitude of the plain . ) if , i say , the index be made to move upon the equinoctiall plain , then must a new scale be made like to that upon the horizontal plain , but somewhat larger ; that is , it must be augmented so as that the parts and whole of the horizontall scale , to the parts and whole of this equinoctiall scale ; must be as the radius , to the co-secant of the latitude . or thus . because , as the radius , to the co-secant of the latitude ; so the sine of the latitude to the radius : and so again , the co-sine of the latitude , to the co-tangent of the latitude . therefore , whereas on the horizontall plain ( for the ellipsis thereon described ) you looke the co-sine of your latitude , and make it a radius or tangent of 45 gr . you must in this take the co-tangent of your latitude , and make it a radius or decimall scale ; and by it you may put on the zodiac or moneths as was prescribed before . this co ▪ sine mentioned before was taken to the greater radius , of the ellipsis , and so this co-tangent here mentioned must be taken to the ( same greater radius of the ellipsis , if the ellipsis were here used , or to the ) radius of the equinoctiall circle , whose radius must be conceived to be the same with the greater radius of the ellipsis . for from this circle projected orthographically ( that is , by perpendiculars let fall from the periphery to the subjacent plain ) is the ellipsis deduced , and from that ellipsis ( again ) is this circle raised or restored . and so the reason of this circular diall with an upright index will be understood well enough . then whereas it is said that it makes no matter which way this circle be raised , that is , whether toward north or south , there is no difficulty in this , for which way soever it is turned , if the plain and center of it lie in a just distance from the index or zenith line , the same index must shew the same point or houre , because in both wayes they are situated from it upon the same coast. sect. 6. pag. 47 , &c. that which is here done will not be difficult , if it be considered that what is aimed at , hath relation to stofflerius astrolabe : and that here one ellipsis must serve to represent every almicantar . i purpose not to repeat any thing that is before said : and the rather because this structure will not be so expedient and ready in some uses as could be wished . they that desire the reason of it , may fetch it out of this that hath been already said : which may be done without any great labour . sect . vii . 1. how to draw and divide the ellipsis upon any plain , to an index that stands upright ( not to the plain , but ) to the zenith line of the place , or perpendicular to the plain of the horizon . this may be done artificially by calculation , or by a lineary way , but not without too much trouble and incumberance , which would deter any man quite from putting it in practise . i shall therefore let that way alone , and fall upon another more feasible and delightfull . the way that i intend , is partly by projection of such lines as are usefull , and partly by inscription of such points as the ellipticall line is to passe through , as shall be seen hereafter . and therefere we shall not need to looke after any position of the plain in respect of declination or leaning . 2. the manner of the work . 1. let the plain be a b c , and the upright index a d. first then , assume any point in the index as at e , and from thence ( by the projecting quadrant , or some levell ) cast an horizontall levell line as f g upon the plain ; or if it cannot be cast upon the plain it selfe , set any board ( for a time ) that may receive it ; lying in the same levell with the assigned point at e. and further , the same horizontall line must be continued and returned about the point e , by help of threeds stretched out from some points or other of the horizontall line before drawn , as from f or g , unto two supporters set up ( for a while ) for this purpose ( either upon the plain or otherwise neer to it ) as is to be seen at h and i : so that now the points h f g i are all in one and the same horizontall levell to one another , and likewise to the first point of the index assumed at e. 2. at some convenient time of the day , when the azimuth may be found distinctly , observe where the shadow of the index cutteth the horizontall line , suppose at k , and there make a mark , and immediately take the suns altitude . 3. by the suns altitude observed , compute what azimuth the sun was then in , which will tell you what azimuth the observed point k ( in the horizontall line ) doth represent . 4. take a pastboard as e l , and fit it to the horizontall line of the plain f g , and to the assumed centerat ● : and applying it to its proper place ( as in the figure is represented ) draw a line from the point k to the center e , quite over the pastboard . and then knowing what azim●●h it is , you may ( from it ) set off the meridian in the true co●st of it , such as will be answerable to the heavens , which meridian line suppose to be p e upon the pastboard . 5. you are next of all , from this meridian line p e , to set of all the other houres of an horizontall diall ( not the common horizontall diall , but ) according to su●h numbers or arks as are expressed in the table of angles , pag. 29. ) and when this is done upon the pastboard , as you see done at o p q κ s t u : then , 6. apply the pastboard to its former position upon the plain , and from the center e project the houres from the pastboard to the horizontall line and threeds . and then taking away your pastboard , you may draw lines from a to such points as are in the horizontall line i● selfe , as a o , a p , a q , a κ , a s , a t , a u. and for the others that stand not upon the horizontall line of the plain , but upon the threeds which are separate from the plain ( such as are ε ρ π ) you may project them by the index a d , reposing the same upon the said points ( ε ρ π ) of the threed ; and marking where the shadow or appearance of the threed traceth the plain , there draw the lines a m , a n , a u. and so all the lines ( that the plain is capable of ) may be drawn . and it is to be noted , that these lines so drawn are not such houre lines as usually are upon other dials , but they are those azimuth lines in which the sun is at every houre of the equinoctiall circle . so the houres and quarters may be put in by the table , pag. 29 , and the column of angles in the same table . 7. looke again into the table pag. 29 , for the column of altitudes ; for by that must the ellipsis it selfe be desc●ibed , for the former azimuthall-ho●re l●●es will give it the true divisions into its requisite parts . the manner of the division is this . make a scale of 90 right sines , but number them versedly , as is here expressed . and let the scale be of a fit ●ignes●e for your plain , which your own judgement must direct you to do . out of this scale ( setting alwayes one foot of your compasses in the point of 90 ) take such altitudes as are ( in the table ) appropriated to every houre , and put them into those severall houres in this wise . suppose the altitude for 12 a clock were to be put in , which altitude is 38 gr . 30 min. with my compasses i take ( in this scale of sines ) from 90 to 38 gr . 30 min. and one foot of th●t length i put into the line of 12 , namely p a upon the plain , and ( alwayes keeping it upon some part of that line ) i remove it thereon , neerer to , or further from the center a , till the other foot of the compasses being turned about will justly touch the edge ( or fiduciall line ) of the index . then i diligently observe where , or in what point of the line p a the first foot of my compasses stayeth , for that is the point through which the ellipsis must passe upon the houre-line of 12. so for any other houre ( and the intermediate quarters ) taking their altitudes , ( set down in the table ) out of the same scale of sines , and inserting them in the same manner as was shewed before , you shall finde all the points ( for every houre and quarter ) through which the ellipsis is to be drawn . through these points therefore you must draw it carefully , as one continued line , without any breaches or angles in it . 3. concerning the motion that is to be made either by the index , or by the ellipsis it selfe . the motion must be made according to the 12 a clock line , which is alwayes the proper meridian to the index or zenith line . that is , the removall or sliding must be either in the meridian it selfe , or else parallel to it . now if the diall it selfe be made to slide upon the plain , that is , if the ellipsis be to be described upon another plate that s●all slip over the former plain , then must this plate be first of all laid fast upon the plain , and the houres and ellipticall line must be described thereon in the same manner that was shewed before . and this plate must move in the meridian line , that is , the 12 a clock line of the plate ( being imagined to be continued forth right ) must move through the fiduciall edge of the index . and so again , if the diall be drawn upon the plain it self , and the index be made to move ( the diall it selfe standing still ) then must care be had that the fiduciall edge of the index do precisely move in the meridian or 12 a clock line , and that the same index move alwayes precisely upright in the zenith line . 4. concerning the place of the suns annuall course or zodiac . the place of it may be best upon the plain . it must stand ( if it be inscribed into a streight line ) parallel to the line of 12 , and needs not to be in or upon the said line . the mover ( whether plate or index ) must have another peculiar index in it , called he●e the zodiacall index , by which it is to be rectified according to the time of the year , if the zodiac be put into a circle ( as is mentioned pag. 15. ) then that diame●er ( of the annuall circle ) which passeth from tropick to tropick , must be either in or parallel to the line of 12. 5. of what limitation or length the zodiac must be , and how to be described , and where to be set . it must be regulated by the former scale of sines mentioned pag. 73. for you may take from 90 ( in that scale ) to the number of the latitude of your place ( not complement as that scale is numbered ) and count that length as a new radius . unto th●s new radius finde the secant of the meridians inclination ( which what it is shall be presently shewed in the 6 proposition following ) and then make this secant to be a tangent of 45 gr . and out of that scale of tangents take 23½ gr . for that length being set both wayes from the equinoctiall , will give the length of the zodiac ( in a streight line , or the diameter of the zodiac put into a circle : ) this work is to be understood for the describing of the zodiacall scale upon the plain it selfe . but first , you are to pitch the equinoctials place , which is that place only upon which the zodiacall index lyeth , when the mover ( whether plate or index ) is placed in the same position that it had when the ellipticall line was described . now when this tangent of 45 gr . is thus found , you may take the declinations ( in the tables of the 4 and 5 pages : ) out of it , and so prick on the suns yearly course . or else make the forementioned secant a decimall scale , and out of it take the tangents noted in the tables pag. 6 , 7. so shall you make the same annuall course of the sun that was produced the former way . note , that if the index be made to move , it is not necessary that it should move upon or parallel to the superficies of the plain , but may be made to move either horizontall , or at any other inclination : only the fiduciall edge must necessarily be made to move in the very meridian line , and must also be ( in all motions ) alwayes upright , or in the zenith line of the place ; and therefore must slide just under the meridian line . note also , that the zodiac though it be not necessarily confined to be set upon the superficies of the plain , yet it will be most conveniently limited thereunto . 6. what the meridians inclination meaneth , and how to finde the quantity of it . by the meridians inclination is meant what angle the same line maketh with the plain of the horizon , and in the former figure pag. 70 it is expressed by the complement of the angle d a p , or the excesse of it . to finde it , you may either apply the edge of an inclinatorie square to the said meridian line , and then the threed ( being drawn perpendicularly down by the plummet ) will shew the said elevation ( or inclination ) of the meridian line above the horizon , in the degrees of the limbe , if the same degrees be taken to the threed from that side of the square that stands perpendicularly upon the meridian line . or else another way may be by protracting or measuring the angle d a p , which is the complement of the former required inclination . it may be done thus . set one foot of your compasses at a , and let the other foot be extended to any point of the index , suppose to 8 , then measure the same length a 8 upon the meridian line , which let be a 7 , and to this length fit your line of chords ( upon the sector , or some like opening scale of chords . ) then take the length from 7 to 8 , and measure the same upon your scale of chords , so shall you finde the quantity of the angle d a p , whose complement ( or excesse ) is the inclination , or elevation , or depression of the meridian , which is here required . 7. other things to be noted concerning the zodiacall scale . 1. if the diall be described upon an irregular superficies , such as is not flat but writhen ( as by this course it may very well be ) then it is most convenient that the former scale be set in some other place , and not upon the superficies whereon the ellipsis is described . it may be contrived two wayes . first , if the diall superficies ( how irregular soever it be on that face whereon the houres are inserted ) be flat at the bottome , and be made to move ( the index standing still ) upon some other plain below it ) then the best way will be to make the zodiacall index upon the moving bottome of it , and to describe the zodiac upon the other plain upon which the motion is made . and to do this aright , you must project the meridian line upon that nether plain , and finde out the inclination of it , and so finde the scale by the secant of that inclination , just as you were before directed . and so this work will be compleat . the motion of dials unequall superficies must be in or parallel to this last projected meridian line , alwayes so as that the meridian of the diall must passe through the fiduc●all edge of the index . secondly , if the index be made to move ( the diall standing still ) and that the diall be upon some unequall super●icies , such as is unfit to receive the zodiac , then the foot of the index may have it inscribed upon it . now in this case of unequall superficies , it is supposed that this foot of the index cannot move parallel to that uneven superficies , but must move streight forward in some right line , just along with the meridian line . in this case you must finde the inclination of the foot of the index to the horizon , that is , what inclination a line drawn upon the foot of the index , either in or parallel to the meridian line , hath to the horizon , which must be done by some inclinatorie instrument , or some such way as is used in taking the reclination or inclination of plains . and when this is done , you are to finish the zodiac scale by the ●ecant of that inclination , in the same manner as was before shewed . so much for this . 2. for the other scales of the suns declination , amplitude , ascensionall difference , and graduall motion in the 12 signes , they are to be done in the selfe same manner , and by the same tables as before ( without any difference ) after that you have found your tangent scale or decimall scale out of which to describe them . 8. another observation . according to that generall observation pag. 36. note here ; that if to an index standing in the zenith line , a plain be set in the equinoctiall , the same plain shall have a circular diall upon it , equally divided . then whether the said equinoctiall plain or the index move , if they be made to move upon the horizontall plain , the zodiac for the horizontall plain must serve . but if the upright ( or zenith line ) index be made to move upon the equinoctiall plain , the former horizontall zodiac must be set upon the equinoctiall plain , and must there be enlarged above the horizontall zodiac , that is , every part of the horizontall zodiac must be made greater upon the equinoctiall plain in proportion as the radius to the co-secant of the latitude . that is , the scales that made the horizontall , being taken as radius , must now here be enlarged to be as co-secant of your latitude ; and from them ( so enlarged ) must the parts of the zodiac be inscribed upon the equinoctiall plain . note further . that for this equinoctiall plain which is to descend downwards from south towards north , you may insteed of it set another plain quite contrarily , that is , descending from north towards south ; which will be the most convenient of the two , because the upper face of this will give the houre all the seasons of the year , whereas the other will be only for sommer upon the upper face of it , and will require an under face for winter . sect . viii . hitherto of ellipticall dials to all superficies whether plain or curved , whose indexes stand upright in the zenith line of the place : there now followeth some other directions how the same thing may be done to any superficies , and to an index set casually in any position whatsoever . but first are premised some usefull propositions tending to the same purpose . 1. an index or streight line being set casually , how to finde the re / in-clination and declination thereof . if it stand upright , it is free from both those accidents , and falls to be the same case with the upright index before treated of . but if it lie leaning , then is it to be dealt withall in this place . and first for the re / in-clination , it is best to be taken by some inclinatory ( square or other ) instrument . the manner of the work is the very same that is performed in finding the re / in-clination of a plain : because the fiduciall edge of the index is ( here ) like the verticall line of a re / in-clining plain , and the application of the square must here be the same to this edge that was there to the verticall line of the plain , and the degrees of re / in-clination are to be numbered in both these cases alike . so that it will not be needfull here to make repetition of that which is so often elswhere declared . you are here also to observe whether the re / in-clination be towards the north or towards the south . to this new plain therefore , or to the two lines a b , c d , apply one of the streight edges of a quadrant , the limbe of it being turned alwayes towards the sun , and observe the horizontall distance of the sun , as is usually done in finding the declinations of plains . only here take this caution , that you count not this horizontall distance from that side of the quadrant which is perpendicular to the plain , but from that side which lies in the line e d , or upon the two lines a b , b c. and further , you are alwayes to account this horizontall distance from that end of the quadrants side which lookes the same way that the point b ( of the index , which is furthest remote from the plain a c ) doth look : as in the second figure is exprest , where the horizontall distance is more than a quadrant , it being there to be accounted from f to h. or for more easie conceit , you may alwayes suppose your quadrant in that posture to be continued to a semicircle , as is done in the second figure : and then count your horizontall distance from that term of the semicircles diameter which respecteth the same coast of the heavens with that point of the index which is most remote from the plain , which is b ; that is , you must count it from f , and so the horizontall distance will be f g h , more than a quadrant , in this particular case . the horizontall distance being thus accounted and observed , you are immediately to take the suns altitude and to finde the azimuth . then for the declination of the index , that is to know into what coast of the world , that is into what azimuth of the heavens ( it being continued from the plain at a infinitely forwards toward b ) it would point into : i say to finde this , you have only the same work to do which is usually done in finding the declinations of plains , the same work without any difference at all . and therefore i shall not here give any further directions in this particular , because i have done it often enough in other places , to which the reader may have recourse . and so i suppose the position of the index in respect of re / in-clination and declination to be fully found out , both for coast and quantity ; by which two things known , we are further to enquire what longitude from our meridian , and what latitude the said index pointeth unto , which will be the next proposition . ¶ note , that though i call a c a plain , yet it may be any curved superficies as well as a plain , for the diall will be described upon one as well as the other indifferently . 2. by having the re / in-clination and declination of any right line , to finde the longitude and latitude thereof . now in the second case before given where the index points downwards , or under the horizon , where the plain is inclining , and so looking downward , we may shape the projection a little otherwise , and the work then will be as easie as the former . let n e s w be the horizon , n the north , s the south , p the south pole n the nadir point opposite to the zenith , o the point or place to which the index respecteth . in the triangle n p o , are given , n p the complement of the latitude of the place , n o the inclination of the index ; and p n o the declination of the same index , by which we may finde p o , whose complement is the south latitude , or whose excesse is the north latitude required : and the angle at p will be found , which is the difference of longitude here sought after . the work is easie to them that understand the like work in re / in-clining plains , and may be performed either by calculation , or else instrumentally , as every one shall like best of . and this is all that needs to be said of this particular . 3. how to finde a meridian line , and to erect a true axis of the world from the foot of the index . the foot of the index i call that point where the index enters into the diall ●●perficies , whatsoever that superficies be , whether plain or curved . as in the two former figures may be understo●● by the point a. from the foot or po●●●at a , set some upright threed which may represent th●●●●enith-line of the place , as is to be seen in that figure pag. ●● . and is there represented by the line a d. then one way will be to make use of some other meridian line , observing when the sun comes to it , and at the same moment to note out the shadow cast by the line a d , for that is the meridian line . but this is for such superficies upon which the sun comes at noon : and it ties a man to the meridian moment . but without this we may ( according to the four first precepts ) set off the meridian line p e upon the pastboard , and by the zenith-line a d you may project it upon all objects that shall stand or be set in the way . for which purpose you must place some object upon that coast of a d upon which any one of the poles of the world may be projected , and also elevated above your diall superficies . and upon this obiect project the meridian neer that place whereabout you conjecture the axis of the world will passe through . then with your semicircle ( or projecting quadrant rather ) project the axis of the world , as is usually done in my wayes of dialling , and by that means you shall finde a point in the formerly projected meridian , which shall represent the pole of the world. and then further , if from the foot of the index at a , to this pole , you fasten a threed , the same threed will represent the true axis of the world comming from the foot of the index . 4. having an axis raised from the foot of the index , how to finde in what longitude and latitude the index it selfe lyeth , by a way easier and differing from the former , without looking after any re / in-clination or declination of the said index . this way will not prove ( perhaps ) so good as i expected , especially in the longitude , and therefore the former may be used . the former way that i used was troublesome enough for effecting of what was by it intended , which is the cause that i shall here endeavour to give another more easie . let the diall superficies be supposed to have upon it an index , an axis truly placed , issuing from the foot of the index , with a meridian line projected also to , or directly under the same axis , according as was done in the work of the preceding proposition . all the difficulty in this work is , that there is no stable place whereon to fix the compasses for measuring of the distance between the two knots m and n : but you may hold somthing underneath close to one of the points ( without disturbing the threeds place ) on which setting one foot of your compasses , you may measure the distance of the other point . or else , because it is supposed that the threed axis a x is fixed upon some solid thing at x , you may open your compasses to the radius a x ( to which also you are to have a radius of chords equall ) and set the same extent from a to r upon the index ( prolonging the index if need be by a threed till it come to be of a competent length ) and there fasten a slipping knot . then because x is a stable point , you may take the extent from x ( setting one foot of your compasses in that point ) to r , and measure the same upon your scale of chords , where it will give the former angle x a r , which is to be used as is shewed before . there will be found other wayes to do this last work . so much therefore for the latitude . secondly , for the longitude of the index , it will not be so easily had as the other was . the way that best likes me ( amongst many others ) for the present , is this . by the precepts , pag. 70. you must make some observation of the sun's azimuth ( and so must you also do by the last precedent proposition ) and you finde what azimuth that is too . you may also ( further ) project it upon your diall superficies , and then you have two azimuths upon the said superficies , namely , the meridian , and the observed azimuth . you are to know yet further , what altitudes the equinoctiall hath upon these two azimuths . and for the meridian it is certain enough that the altitude there , is equall to the complement of your latitude . for the other azimuth say thus . as the radius , is to the co-tangent of your latitude ; so is the co-sine of your azimuth , to the tangent of the equinoctials altitude upon that azimuth circle . after this you must make some mark or knot upon your axis a x , which suppose to be at m , and then with your projecting quadrant ( upon your meridian and azimuth ) from the point m , set on the respective equinoctiall altitudes , as the manner of projecting by that instrument useth to be . next , you must project this equinoctiall circle all over the diall-superficies ( and upon some other objects where need shall be , ) suppose it here to be the line s v. and furthermore , you are to proiect the axis a x , and the index a b , one upon the other , and to observe the line that they both ( in that position ) do make upon the diall-superficies , and other objects if need be . i say you must , first , principally observe this line , for it is the proper meridian line to the index or zenith line a b , because it passeth ( quoad superficiem ) through both the axis of the world and the zenith line . suppose it in the former plain to be a ☉ . and secondly you must note the intersection that this line makes with the equinoctiall line which was now projected , which we may imagine to be at v , upon some object laid in the way of purpose . and so also you must ( thirdly ) take notice of the intersection of the equinoctiall line with the meridian line , which suppose at s. for these two equinoctiall intersections and the point m , all three together , must do what we now intend . the thing here now intended is , to know what angle is contained between the two superficies x a b , and x a t , which are two meridian circles , or ( which is the same ) the angle v m s in the plain of the equinoctiall , which angle measures the said inclination of the two forenamed meridians , as it doth of all meridians . now to measure this angle v m s , the best way will be first to extend two threeds , one from v to m , and the other from s to m , crossing each other at m , and they may be continued further till they meet with some object ( in the way standing , or else set for that purpose ) where they may be both fixed as at f and g. then measure from s to m , and from v to m , which is the shortest , suppose s m to be the shortest , and v m the longest . first , take v m , and lay it down in a line , as there is done . then take s m , and measure it on the same line from m to s or d. thirdly , take the distance from v to d out of this line , and set it from v to d , upon the line v m , so shall m s and m d be equall one to the other . to this distance therefore m s or m d as a radius , open some line of chords , then take the chord or subtense from s to d ( s is a stable point upon which you may firmly set your compasses ) and measure the same length upon your line of chords , where you shall see how many degrees the angle s m d or s m v containeth . now all this businesse is to finde how great an angle is contained between the south part of the meridian of the place , and the meridian belonging properly to the zenith line . but here ( if it be observed well ) the angle now measured is made between the proper meridian and the north part of the meridian of the place , so that this angle must be the supplement ( to a semicircle ) of what is required . the supplement therefore of the angle now found , is ( in this case ) the longitude or meridian into which the zenith line a b pointeth . i say the supplement of that angle is the proper meridian of the zenith line ; or the difference of longitude proper to that line , from the meridian of the place . but forasmuch as the angle v m s doth measure the inclination or angle of this proper meridian from the north part of the meridian of the place , and that the shadow of the index a b doth first come to the proper meridian a o before it comes to a t the meridian of the place : we may therefore in such case say , that the angle v m s gives the difference of longitude ; and that the said difference of longitude lies eastward from the south part of the meridian of the place , as there in the figure it appears to lie westward from the north part . suppose that the difference of longitude were 39 gr . 25 min. and the proper latitude 40 gr . and so much for this also . if this way be thought not feasible enough , the former in the second proposition may be used ; yet variety in all kindes is delightfull , and not to be rejected in this . it is done , you see , without having respect to any declination or re / in-clination , which the former was founded upon . 5. how to forme the angles at the pole. this proposition makes way for computing tables of horizontall spaces , and equinoctiall altitudes to any longitude or meridian differing from the longitude of the east hour generall table west hours east hou generall table west hour 13 0 00 12 6 90 00 6   3 45 — a   93 45     7 30 — b   97 30   11 15 — c   101 15   11 15 00 1-d 5 105 00 7   18 45     108 45     22 30     112 30     26 15     116 15   10 30 00 2 4 120 00 8   33 45     123 45     37 30     127 30   a 1 55     131 15   a 39 25   3 135 00 9 a 1 50     138 45     41 15     142 30   9 45 00 3   146 15     48 45   2 150 00 10   52 30     153 45     56 15     157 30   8 60 00 4   161 15     63 45   1 165 00 11   67 30     168 45     71 15     172 30   7 75 00 5   176 15     78 45   12 180 00 12   82 30             86 15           angles at the pole. angles at the pole.   s 88 10 n proper merid. 00 00 3 t 84 25 n p 01 50 r   v 80 40 n 9q 05 35 e     76 55 n r 09 20 f     73 10 n   13 05 g 2   69 25 n   16 50 h     65 40 n 8 20 35 i     61 55 m   24 20 k     58 10 m   28 05 l 1   54 25 m   31 50 m     50 40 m 7 35 35 m     46 55 m   39 20 m     43 10 m   43 05 m 12   39 25 m   46 50 m     35 40 m 6 50 35 m     31 55 m   54 20 m     28 10 l   58 05 m 11   24 25 k   61 50 n     20 40 i 5 65 35 n     16 55 h   69 20 n     13 10 g   73 05 n 10 y 09 25 f   76 50 n   x 05 40 e 4 st 80 35 n   w 01 55 r u 84 20 n           z 88 05 n place , and to any latitude differing from the latitude of your place . after that the difference of longitude is known , you are first to frame a table of angles at the pole , which will not be hard to perform , and the manner of it will be seen best in an example , being altogether like the computation of the said angles at the pole for the houres of any place to the same declining plain . i have here first of all in a generall table set down the graduall distances of houres and quarters from 12 at noon or mid-day , which will be some help in performing the work . the houres that are on the left side of the degrees are the east or forenoon houres , and those on the right hand are the west or afternoon houres . suppose ( as before prop. 4. ) a difference of longitude were given 39 gr . 25 min. towards the east , and it were required to know what angles our hours ( namely those 12 that stand neerest to it , six upon one side , and six upon the other ) with their intermediate quarters do make with this longitude or proper meridian , for so i intend to call it . i first take the given longitude or proper meridian , 39 gr . 25 min and enter it among the east houres of the generall table , and i since it to fall in at a ; that is , between a quarter and halfe an houre past 9 before noon , and i finde also the circumstant numbers to be 37.30 the lesser , and 41.15 the greater . the differences between these two numbers and the longitude given , are 1 gr . 55 min. and 1 gr . 50 min. as you see them set down in the said generall table right against the letter a. these two differences are to be first placed in this second table of angles at the pole , as you see in the example , where , first of all is written proper meridian 00.00 separated from the rest as signifying only the place where it is to stand , and above it is placed ( the difference between 37.30 and the proper meridian 39.25 , namely ) 1 gr . 55 min. so again , below it is placed 1 gr . 50 min. ( which is the difference of 39 gr . 25 min. the proper meridian from the next greater number 41 gr . 15 min. ) now to these two radicall numbers , i adde ¼ ½ ¾ and one whole houre , namely a , b , c , and thereby produce e , f , g , on both sides the proper meridian . then to r , e , f , g , i adde 15 gr . d , namely , the number of one houre , and thereby do further produce h , i , k , l , on both sides the proper meridian , as before . and afterwards for more ease , to the numbers r , e , f , g , h , i , k , l , i adde 30 gr . or two houres , and do thereby produce on both sides the eight numbers noted with the letter m. and so still adding 30 gr . to the last eight numbers noted with m , or else adding 60 gr . to the first eight numbers noted with r , e , f , g , h , i , k , l , i shall make the eight remaining numbers noted with the letter n , both above and below , on both sides the proper meridian . and so this table of angles at the pole is compleated , for if it should ( in the same manner ) be further continued , the next numbers ( above ) would be 91 gr . 55 min. and ( below ) 91 gr . 50 min. both greater than 90 gr . beyond which there will be neither need nor expedience to go . this way of forming the angles i thought best to take , because it is more plain and easie than any other , which else might have been in this work used . you are afterward to place the houres ( in this last table ) about the proper meridian , just as you see their order to be about the letter a in the general table , and as you see they are in this particular table of angles at the pole. the like work serves for west longitudes . thus you have the angles for 12 houres and their quarters , the same angles serve for the other 12 houres : and whatsoever is hereafter computed for any of these 12 houres , must be understood to serve also for the 12 opposites , and so the work will be done for the whole circle , or 24 houres of the naturall day . 6. by knowing the angles at the pole , and the latitude of the place or horizon , how to finde the horizontall spaces thereto belonging . we must suppose that this longitude or meridian which we have before mentioned , is proper to some horizon or other , above which also the elevation of the pole must be imagined known , which we may suppose to be 40 gr . as before , prop. 4. in respect of this plain or horizon it is , that the spaces now mentioned are called horizontall spaces . the way to finde them for this ellipticall work is converse to that which is usuall in declining plains . namely thus , as the sine of the poles elevation above this horizon or plain , is to the radius ; so is the tangent of each of these angles at the pole , to the tangent of the horizontall space belonging to each of those foresaid angles . so having found all these horizontall spaces severally by this form of calculation , you may set them into a table , as here you see it done in the first of the two main columns . they signifie the distances of every of the houres and quarters from the proper meridian , and are to be accounted as numbered in the degrees of the said plain or horizon . 7. by knowing the angles at the pole , and the latitude of the plain or horizon , how to finde the equinoctiall altitudes or depressions ( above or under the same horizon ) due to the said angles at the pole or points of the equinoctiall . the proportion by which this is to be effected is this . as the radius , is to the co-sine of the poles elevation above the plain or horizon ; so is the co-sine of any of the angles specified in the former table of angles at the pole ; to the sine of the altitude required , and due to that angle , or houre ( rather ) in the equinoctiall circle . so in the former example . the latitude or elevation of the pole above the horizon was 40 gr . the complement of that is 50 gr . which is the meridian altitude and depression of the equinoctiall circle above and under the said horizon or plain , and is therefore to be set for the altitude of the proper meridian . now , as the radius , is to the altitude of 50 gr . so is the co-sine of every angle at the pole , or arke of the equinoctiall in the former table , to the altitude due thereunto . hours horizon spaces altitudes or profu . hours   88 49 1 24   3 86 24 4 16 3   83 58 7 08     81 30 9 59     79 00 12 49   3 76 26 15 37 2   73 48 18 24     71 04 21 08     68 15 23 50   1 65 18 26 28 1   62 13 29 03     59 00 31 33     55 35 33 58   12 51 58 36 17 12   48 09 38 29     44 06 40 33     39 48 42 29   11 35 14 44 14 11   30 24 45 47     25 19 47 8     20 00 48 14   10 14 28 49 05 10   8 47 49 40     2 59 49 58   hours horizon spaces . altitudes or profun hours proper merid. 00 00 50 00     2 51 49 58   9 8 39 49 40 9   14 21 49 06     19 53 48 16     25 13 47 09   8 30 18 45 49 8   35 08 44 16     39 42 42 31     44 00 40 36   7 48 04 38 32 7   51 53 36 20     55 30 34 01     58 55 31 36   6 62 09 29 06 6   65 14 26 32     68 11 23 53     71 00 21 12   5 73 44 18 28 5   76 22 15 41     78 56 12 53     81 27 10 03   4 83 55 7 12 4   86 21 4 20     88 46 1 28   and note that r e f g h , &c. below the propet meridian in the table of angles , are complements to n n n n n , &c. at the top of the table of angles : and again , those r e f g h , &c. above the proper meridian , are complements to the lowest n n n n n , &c. successively rising in order one after the other . so that , as the radius , is to the sine of 50 gr . so is the sine of p q r , &c. to the sines of the altitudes belonging to s t v , &c. and so again , the sines of w x y , &c. to the sines of the altitudes belonging to z u st , &c. and accordingly this table of altitudes and depressions is calculated . if the pole be elevated upon the plain or horizon , above 50 gr . it will be sufficient to compute these two tables to halfe houres only , and so to save halfe your labour . but in lesser elevations it is best to do it to quarters , as here is done n this example . 8. how to finde the proper meridian line duly belonging to any zenith line casually placed , and to draw it upon the plain . though the 4. proposition be not made use of for finding the longitude and latitude ( which of due pertaineth to any zenith line ) but instead of it the second precedent be thought most fit to be used ; yet so much of the fourth will be best to be used , as shall concern the finding out of the proper meridian line . that is , you must raise an axis from the foot of the index , as is a x , and then projecting this axis a x and the index a b one upon the other , as if they were both one , you shall thereby also project ( by them both together ) the proper meridian belonging to the zenith line a b , such as ( in the figure of the fourth ) is a o , which must be drawn upon the plain accordingly . the manner , of all the particular workings that do hereunto tend , is set down in the fourth precedent proposition , and therefore will not here again need to be repeated . 9. how to draw and divide the elli●sis into houres and quarters , to an index casually set , whose latitude and difference of longitude is discovered by the former works . when you know the position of your index in respect of longitude and latitude , you may then compute two tables to the same meridian or difference of longitude , and to the latitude of your index , as is done before in the 7 proposition , one of which tables is of horizontall spaces , the other of equinoctiall altitudes or depressions above and under that horizon which is proper to the index or zenith line casually placed . by these two tables the work will be done in such manner as was shewed before , prop. 2. the manner of the work is this . 1. you are to assume some point in your index a b , let the point be c , where you may fasten some knot of threed that it may not be lost again . 2. from this point you must draw an horizontall line , not in the levell of your own horizon , but in that horizon which is proper to that index or zenith line a b : that is , it must lie perpendicular to a b , making right angles ( in every part of it ) to that line , and must have respect ( in this perpendicularity ) to the point c. the meaning is , you must imagine a plain to passe through the point c , and the same plain to be perpendicular to the line a b ; or that the line a b is a perpendicular surgent line to the said plain passing through the point at c. [ and so it may be noted , that if other dials be described by the equinoctiall circle and not by the horizontall , the work of drawing the equinoctiall perpendicular to the axis will be difficult , but though no great accuratenesse be used , yet the work will be perfect enough , and no way defective for the losse of a degree or two in the perpendicularity required , which i thought good here also to note , because i have omitted it in all my other precepts of projecting dials . ] wherefore you may do it by some pastboard , applying one edge of it to any line projected from a b the index , as to a d , and keeping the edge there , you may turn the flat of it to the index a b , and draw a line by it , or make two pricks through it into the pastboard , whereby a line may be drawn , but above all note the point c upon it . then to this line of the index thus drawn , and from the noted point of it at c , erect a perpendicular : so applying your pastboard to its former place again ( the edge of it lying upon a d , and the flat of it applyed to the index a b , and the point in it , noted for c , being again fitted to c in the index , i say thus doing ) you may note where the last drawn perpendicular doth cut the line a d ( which must be extended by help of some threed if need be ) suppose at e : at the point e ( then ) you must say that one point of the proper horizontall line is to be taken . then in like manner you must seeke another horizontall point : first by projecting a line from the index ( any where ) such as is a f , and by applying one edge of a pastboard to that line , and the plain of that pastboard to the index a b , and so noting the point c , and drawing or marking the line a b upon it , &c. as was done before . so you shall finde another point of the same horizontal line or plain ( rather ) which suppose to fall at f. now having three points of the horizontall plain at c , e , and f , ( which i suppose not to lie in one and the same right line , for that must with carefulnesse be avoided here ) you may project some part of that line upon the plain , as e f , the rest of it ( so much as shall be found usefull ) may be made up with returns of threed , and regulated or kept in the same plain by projection , as the manner of working that way useth to be , and as here you see exprest , by the line h e f g , lying in the same plain with the point c , and that whole plain lying perpendicular to the index a b. 3. the next thing to be done is the drawing of the houre lines upon the plain . and the first thing h●re presupposed to be done , is the drawing of the proper meridian , performed by the 8 prop. that ( i say ) is supposed as already done before any of this work is begun . let the proper meridian be a v. having then made tables ( by the 7 prop. ) for the horizontall spaces of your hours from this proper meridian a b ; you must first apply a pastboard to the horizontall line e f , and fit the center of it to c : and upon the pastboard , project the proper meridian from c to a v , or from c to p. and then by the table of horizontall spaces in the 7 prop. ( for we now here suppose that this is the diall for which those tables were computed ) you may ( upon that pastboard ) set off all the houres and quarters from the proper meridian upon this pastboard : and applying the pastboard into its proper place , namely to the horizontall line e p f , and to the point c , you may project the houres and parts of houres from the pastboard to the horizontall line h e p f g , as the manner in this way of dialling is well known . 4. having transferred the houre-points into the horizontall line , you may ( by help of your index a b ) project and draw the houre lines upon the plain , which we will suppose done , because the manner of doing it is the same with that which was done before for upright indexes . 5. to know where the ellipticall line must come , or to finde the points in those houre lines , through which it must passe , we must work in the same manner as before ( in the 7 § of pag. 7● . ) is exprest , namely thus . we must make a scale of right sines of a fit length , and number them versedly , and out of that scale we must take such altitudes or profundities ( which you will , one or both ) as the table pag 97. giveth , which table we here suppose to be computed for this example whereabout we now are . and taking these altitudes from that scale ( that is , from the end at 90 , to the altitude numbered upon the parts of the scale ) we must insert them into their respective houres to which they belong . they must be inserted in this manner . having taken any altitude out of the scale , and found the houre upon which it must be placed , you must set one foot of that extent upon the houre line ( keeping it alwayes thereon , but ) removing it untill the other foot being turned about , may only touch the fiduciall edge of the index : and when the feet of the compasses are thus fitted , you must note upon what point of the houre line the foot that is thereon doth stand , for through that point of that houre must the ellipticall line passe . the same manner of work you must perform upon every houre , untill you have gone through 12 of them , which do make up halfe the houres of the whole diall . and if you strike the lines through the center , you shall have all the 24. and looke what is done upon any one houre line , the same is to be done upon the opposite . that is , looke what distance ( upon a plain ) the ellipsis hath from the center upon any one line , the selfe same distance from the center must the opposite line have . but if the description be made upon an uneven superficies , then this rule may not , perhaps , hold : yet this will ; namely , looke what distance from the index ( the least or perpendicular index i mean ) any point of the ellipsis hath , the same perpendicular distance is to be given for the ellipsis upon the opposite houre line . and by this means you may put in as many houres and ellipticall points as you please . and through these points you are to draw the ellipticall line . 10. concerning the motion that is to be made , either by the index , or by the ellipsis it selfe . the motion must be either in , or else parallel to , the proper meridian , and not elsewhere . now if the diall superficies be made to slide , and the index stand still ; then ( though the diall be upon an unequall superficies , yet ) the motion or sliding must be upon a plain . and it must be upon the proper meridian , projected by the index from the diall plate , unto or upon that said plain , and the motion must be in or parallel to it . and it must be noted , that this proper meridian thus projected never falls directly under any index that is ( not direct but ) declining , but it falls aside from it , according as the index stands aside from the proper meridian upon the diall superficies , and as it shall be laid by projecting it . but if the index move ( the diall standing still ) then the fiduciall edge of it must be made to move alwayes according to , and in the very proper meridian line it selfe , and not any where else , which may be contrived sundry wayes , as every man shall invent to his own liking . great care must be had that the fiduciall edge of the index ( in the motion of it ) keepe alwayes one justly parallel situation . 11. of the place of the suns annuall course or zodiac . the best place for it is to be considered of according to that which moveth . if the diall superficies move , it must ( though it be rough ( as i said before ) yet it must ) move upon a plain , this plain therefore in this case is fittest to receive the zodiac . and then there may a peculiar point or threed index ( serving only for the zodiac ) be set upon the diall-moving-plate , whereby it may be rectified according to the time of the year . if the index be made to move , then either the zodiac may be set upon the foot of the index which guides the motion of it , and the peculiar or zodiacall index may be set somewhere where it may stand to point at the zodiac . or else the zodiac it selfe may have its place upon some part of the standing plate or body , and the zodiacall index may be placed upon the foot of the diall index ; from whence it may be made to shew the zodiacall parts , as occasion shall be . 12. how the zodiac is to be limited in regard of length , and how to be described and set in its true place . the limitation of it must be according to the scale of sines by which the ellipsis is described , which scale is mentioned pag 102 § 5. and pag. 73 § 7 ▪ the index or diall plate ( that is the mover ) must move according to the proper meridian of the diall superficies , and precisely so , as that fiducial edge of the index must move ( not elsewhere , but ) in the very proper meridian it self . this is , if the index move . but if the diall superficies move , then the proper meridian thereof ( or the superficies of the proper meridian , namely the superficies made by the proper meridian cutting through the diall plate , which is of any thicknesse ) must move directly upon , or through , the fiduciall edge of the index . these things are intimated before . and that the zodiac is to be described upon a plain , though the diall it selfe be not so . and further , that if the diall plate move , ( though it be not a plain it selfe , yet ) it must move upon a plain , and that the proper meridian is to be described upon the same plain , and the motion to be directed according thereunto . these things are often inculcated , because they are hard to be conceived , and had need of the better consideration for that reason . now further . 1. if the diall plate be supposed thus to move upon a plain , and on it the proper meridian be drawn , then first of all , the angle is to be inquired that is made between the index and that part of the proper meridian which is projected upon the plain whereon the motion is made , which how to measure will be a hard matter to give rules for , because the variety of cases and positions of one to the other will be so various . it is first to be supposed , that it makes a just right angle with it , and consequently that the zodiac is described upon the proper horizon of the index . and if upon this supposition the zodiac be to be limited , then the rule will be the same with the former given in the like case , namely thus . upon your scale of sines ( by which you described your ellipticall line ) take from 90 , to the latitude of the index , and count that length for a new radius , and keepe it . then when you have found the forenamed angle ( of finding which more is said prop. 13. following ) to this new radius finde the secant of the complement or excesse of that angle : this length or secant will be the tangent of 45 gr . or the decimall scale by which you are to describe the zodiac on both sides from the equinoctiall point or line , according to the numbers in the generall tables made for this purpose , pag. 4 , 5 , 6 , and 7. [ for placing the equinoctiall point in the zodiac ( upon which all the other parts of that scale do depend ) you must set all in the same posture that they had when the diall was described : and when you have made fit place for the zodiac , and set on some peculiar zodiacall index , you must note the place upon which the said zodiacall index pointeth , for that place is the peculiar place for the equinoctiall , from whence all the other parts of the suns annuall course must be set on . ] 2. if the index be made to move in any depth under the diall superficies , then a slit must be made in the same superficies that the proper meridian would cut , and the index continued therein , down into one streight line till it meet with the foot whereon the index is fixed , and which being moved , carryeth the index along with it . and the zodiac must be described either upon this foot ( the zodiacall index being made to stand still while that said foot with its divisions moveth ) and that also parallel to ( or in ) the proper meridian projected upon it : or else the foot must carry the zodiacall index , and the zodiac must be described upon some other convenient place , as the sides ( &c. ) of that forementioned flat . and here , for limitation of the zodiacs length , you must finde the inclination of this moving foot to the index or zenith line ( which angle of inclination is mentioned before , and spoken of afterwards in the 13 prop. following ) and work directly as you did before by finding the new radius , and the secant of the complement ( or excesse ) of the angle . by which you shall finde the tangent of 45 gr . or the length of your decimall scale , out of which the zodiac may be described by the tables ▪ pag. 4 , 5 , &c. as is mentioned before in this proposition . thus ( in these cases ) is the zodiac to be set in its true place , thus it is to be limited and d●scribed . for occurrences of other sorts of cases than are here mentioned ( whereof there will not be many ) he that can understand to do these things aright , will be able to grapple with them ; and for such as do not understand what is here said , their best course is to let these difficulties alone . 13. how to finde what angle is made between the index and that part of the proper meridian which is projected upon the plain whereon the motion is made , or which drawn upon the foot of the index which maketh the same motion . the way that for the present i thinke of is this . apply a streight edge of pastboard to the projected line , and the plain of the same pastboard to the fiduciall edge of the index , and make two points upon the pastboard , by or through which the same fiduciall edge passeth . then taking away the pastboard , draw a streight line through those two points or marks , and so measure the angle made between the forenamed edge of the pastboard and this right line . if it be so that the line crosseth not the edge of the pastboar● , then draw such a parallel line to the edge as may crosse the former line , and then with a scale of chords you may measure the angle . 14. further observations concerning the motion and daily fitting of the diall and index , for setting them true . the best way is to make the diall plate move , and the index to stand still ( in these obliquely situated indexes ) for the zodiac will ( in such cases ) be most easily described and made usefull . and in this case the index may be also set fast first , and quite finished before the diall be drawn at all . then also the diall will be drawn more easily , and the motion of the diall plate may sooner ( this way ) be contrived , then can the motion of the index be contrived when the said index is to move and the diall stand still . the motion ( as hath been often said ) must be according ( that is parallel ) to the proper meridian : and the slit ( for the fiduciall edge of the index ) may be so contrived that the fiduciall edge it selfe ( which is best to be a fine threed ) may be also the proper index , and the zodiac may be described upon the diall-moving-plate , closely contrived to the threed . or thus at least , the lenght of the whole zodiac may be so limited as is before mentioned , and then if you desire to have it drawn upon some other place of the plain upon which the diall superficies moveth , and the same diall plate to carry the peculiar index for the zodiac ; then ( i say ) by these two terms prefixed to the length of the zodiac ( by the fiduciall edge of the index ) having reference to the diall-moving-plate ( as before ) you may determine the length of the same zodiac upon the standing plate whereon the diall plate moveth , for the new zodiacall index set fast into the moving plate , will ( upon the standing plate ) shew the terms of the same zodiac suting with the terms upon the moving plate fitted to the standing index . this will be direction enough because one way may serve as well as many . yet i doubt not to set down many more which would be without end , if they were all put together , for inventions in this kinde would be too too many . 2. if the index be thought best to move , then must a pattern of it ( that is some threed ) be first set up . and to this vice-index must the diall be described upon the plain ( or whatever curved superficies it be . ) and the zodiac is best to be fitted either upon the plain whereon the foot of the index is to move ( for a plain it must move upon , whatever be the superficies whereon the diall is described ) and that same foot to carry the zodiacall index upon it ; or else upon the foot of the index , and the zodiacall or peculiar index to stand fixt upon the plain ( or diall superficies ) and then when all this is done , the vice-index must be taken away and a true substantiall index put in the place of it . and for that purpose , you were best , before you take it away , to finde at what elevation it stood from the plain , and likewise to draw the perpendicular just under it . for by these two helps you may set up a true substantiall index , regulating it thereby into the same position ; which to do , must be left to the judgement of every man to contrive as he shall see requisite . but in both these wayes care must be had what inclination the zodiacall scale hath to the index , according to what is said in the 13 prop. 3. it must here again be taken for a rule , thot a plain , if it lie parallel to the index is not possibly capable of these kindes of ellipticall houres , but the index must have some inclination to the plain , and make some angle with the same . 4. all these precepts do serve to make such dials as are upon fixed walls or superficies that cannot be removed . but if it be to be done for moveable bodies , if they be regularly cut , then the declinations of every of the plains are known by their regularity ; so that such plains of them as are capable of any diall before mentioned ( for the index and the plain must not be parallel in any case , they may be perpendicular ) may have such dials upon them as they are capable of , either to indexes perpendicular to their plains , or to indexes lying in the zenith line of the place . but for these casuall indexes upon them , you will be put to it , to know what declinations they have from the meridian upon the regular body ( for i suppose that there is a meridian drawn upon some one of the plains or other . ) the way that for the present i think upon is this . set the body upon the foot true in respect of horizontall or upright position , that is , let the foot of it stand upon some just horizontall plain , though in respect of declination it stand at all adventures . then from the fiduciall edge of your casuall index , let fall a perpendicular ( i mean perpendicular to the horizon of the place , not to the plain ) and to that perpendicular point , from the point of the indexes concourse with the plain , draw a right line ; this line shall represent the azimuth wherein the said index lyeth . then again , with your eye repose a perpendicular hanging threed upon the two points ( namely , the center or concourse of the index with the plain , and the late found perpendicular point ) both together , and project the umbrage of the threed so hanging upon the horizontall plain . so also you must repose the shadow of some one of the cocks or axes with your eye , upon the meridian or line of 12 , and then also the perpendicular threed must be projected upon them both joyned before into one , so that threed , stile , and meridian must now be all as one line ; and the threed so hanging project the shadow of it upon the same horizontall plain . now these two lines thus projected either do concur and make an angle , or else ( by drawing a parallel to one of them through the other ) they must be made to concur , and looke what angle they make at their concurrence , the same being counted the right way is the angle of the casuall indexes declination from the meridian line . thus the declination is found . then the reclination may be found as before is declared , and by help of them the longitude and latitude of the index may be had , and so the diall made by the precedent precepts . a briefe demonstration of the 7th . and 8th . sections . in the 7 § all are made to the zenith line of the place . if therefore you imagine an horizontall ellipsis to be described to that zenith line , and upon that ellipsis a kinde of compressed cylinder to rise upright , parallel to the said zenith line , with houre lines raised from the severall points of the horizontall ellipsis , then will the zenith line shew the houre upon those surgent lines : and the same index ( or zenith line ) must shew the houre among tho●e upright lines in the compressed cylinder , by the same reason that it doth upon the ellipsis it selfe , upon which the said cylinder is raised , and if so , then it matters not what part of these lines is taken in for this use , since any one point of them will serve to do the work . the way therefore that is used for the projecting of these houres ( by help of the table in 29 pag. ) as it will serve to finde the ellipticall points upon the horizontall plain it selfe , so must it serve to finde some one point of these surgent houre lines upon any other superficies , because they keepe the same distances alwayes from the zenith line ( from whence the projection of them is made , and whereon they depend altogether ) and therefore it findes ( upon any superficies ) the points of those lines which passe through , or do intersect the diall superficies . therefore it is that these ellipticall points ( upon all superficies ) are points of those surgent lines rising from the horizontall plain , and that the zenith line must have the same relation and situation to them , that it had to the points upon the horizontall plain it selfe , and consequently , that this way must be of the same truth upon any superficies that it is upon the horizontall . so much for any upright index , set upon all sorts of superficies , with the houres thereon depending . in the 8 § . are handled casuall indexes ; which if well considered will fall to be the same with them in the 7 § . for the table of angles and altitudes by which they are made , are calculated to that horizon which is proper to the index , considered as a zenith line . and so from that horizon we may imagine the like surgent lines to rise all parallel to the index or zenith line , and the very same reason to hold in these , that held in the former . to the reader that shall have the view of this first draught of precepts . these rules here given may seeme to be stuffed with many impertinencies , and some need lesse difficulties , which the author acknowledgeth wil●ingly , and excuseth , by reason that they were his first med●tat●ons in th●s kinde ; and so much the more undigested by how much the lesse pra●●●se hath been by him used therein . the truth is , he never described any thing sutable to the cases of these two last sections . and if the reader be any way able to discern what it is to write upon a mathematicall subject wherein hath preceded no reall ●epresentation , he will not only excused difficulties and impert●nencies in the trad●tion , but will wonder if there be not some miscarriages in point of truth : of which ( notwithstanding ) the author is confident this treatise is clear . s. foster . 15. having placed a diall plate to move ( let the coast of the motion be casual● ) how to fit a stedfast index to it , and to describe an ellipticall diall upon the said moving plain . this case was forgotten before , but now here supplyed . you must observe the line ( upon the immoveable plate ) which the midst of the moving plate doth describe , and suppose that line to be the proper meridian . then from some convenient point of that proper meridian , raise a true axis . project the axis upon the proper meridian , and from any point of the said meridian raise a threed ( or index ) any where , only so as that both this threed , the axis , and the proper meridian may all three appear in one line . there six the threed , and then finde the longitude and latitude of it , and afterwards describe the diall to it , according to the rules given before . 16. if an index should be set up and made moveable upon a standing plain , there can no diall be described thereto , unlesse the index be made of wire or some such bending substance , but to such there may . for if you observe what streight line the foot makes in its motion , you must count that as the proper meridian , and so setting up an axis to some point of it , you may put in an index into that foot , so as the axis , and fiduciall edge of the index , and the proper meridian , may all three appear in one line . and then finish your work as is before directed . finis . circular horologiography . shewing , how to make an horizontall diall in a circle equally divided , to shew the hour of the day , and azimuth of the sun. invented and written by mr. samvel foster , late professor of astronomie in gresham-colledge . london , printed for nicholas bourn . 1654. circular horologiography . how to make an horizontall diall in a circle equally divided , to shew the houre of the day , and azimuth of the sun. here before we have had elliptical horologiography , now shall follow circular horologiography , which sheweth how to make a diall in a perfect circle equally divided into houres , ( whereby to finde the houre ) upon any plain whatsoever . divide a circle into 24 equall parts , and take so many of them as your horizon hath houres in the longest day , or rather so many as the degrees of your greatest amplitude ( east and west ) from the south do arise unto ; which here at london will be neer 18 of them . then divide each of these parts into 15 , which for the azimuth will signifie degrees , for the houre will stand for four minutes of time apiece . the altitude of the horary index may thus be found . adde your latitude to 90 gr . halfe that summe is the elevation of the horary index above the horizon . thus at london , 45 min. which is the altitude or elevation required . or , adde halfe the complement of your latitude ( which is 19¼ ) to your latitude ( 51½ ) the sum ( 70¾ ) is the elevation of the index . the standing and looking of it . it must stand right over the line of 12 , elevated above the said line 70¼ gr . and must looke toward ( but not into ) the north pole. the motion of the horary index . it must move to and fro , directly over the line of 12. or else the houres must move to and from it , according to the line of 12 , so as that the same line may alwayes lie under the foot of the said index . the motion of the one or the other is necessary , because else the circle of equall parts can never shew the true houre all the year long . how the zodiac is to be limited , and laid , and charactred . the motion of the index must be regulated by the zodiac . the zodiac therefore must lie either in , or else parallel to the meridian line . the length of it is thus to be limited . count the semidiameter of your circle ( viz. a b ) for the radius : to that radius , either 1. make the sine of 70¾ ( taken to the radius of your circle ) a tangent of 70¾ ; and to the radius of that tangent finde the secant of 19¼ ( the complement of 70¾ ) that length shall be the radius of the degrees , or the decimall of the tangents of the zodiac , to be inserted by the tables pag. 4 , 5 , 6 , and 7. 2. or else , make the sine of 19¼ ( estimated to the semidiameter of the circle a b as radius ) a radius , and to that radius finde the secant of 19 1 , this last length or secant shall be the quantity of the tangent of 45 gr . or of the decimall scale by which the numbers pag. 4 , 5 , &c. are to be inserted . and in these northern horizons ♋ and ♈ must be placed so that 12 may be neerest to the index in summer , and furthest off in winter . the manner how to fashion the cock which holdeth the index . the fashion may be seen by the figure a c d. at a is the place of the fiduciall point of the foot of the index to be assigned : and in that point a hole must be pierced , and a threed fixed . then the cock must have two holes more pierced , one at c , the other at d , both to stand perpendicularly over the line of 12. that at c must be so placed , that the angle c a b may be 70¾ gr . that at d must be placed so , that d a b may be an angle of 90 gr . how to place this diall for use . you must either fix your diall plate in the meridian line , and truly horizontall ▪ or else upon an horizontall or levell flat you must draw a meridan line , whereby to place it upon any occasion . then , to finde the houre . make use of the index a c , and rectifie the foot of the index to the requisite place in the zodiac , ( either to the day of the moneth , or the degree of the signe . ) when it is thus rectified and set , the shadow of the threed a c will shew the houre of the day . to finde the azimuth . put the threed from a to d , and let a d be your index . then ( alwayes ) place the foot of the threed in the center of the circle at a , so shall the threed a d give the azimuth . ¶ note , that the scales of declination , amplitude , ascensionall difference , may be placed by the zodiac , and used as is before shewed in the ellipticall dials . another way to make the same horizontall diall equally divided , to finde the houre and azimuth . the former way maketh the index to point up neer to the elevated pole. this other will require the index to point towards the contrary coast , namely , that of noon or mid day , or toward ( but not into ) that coast of the heavens where is the depressed pole under the horizon , but still the index is to be above the horizon . the altitude of the horary index above the horizon , must be halfe the complement of the latitude , or halfe the height of the equinoctiall . here at london 19¼ gr . which is the complement of the former wayes elevation 70¾ t●e standing and looking of it , must be right over the line of 12 , drawn out at competent length , and must justly point into the coast of 12 mid day : or towards ( but not into ) the south ( or depressed ) pole. it must move as the former did . the zodiac is to be laid as before , and to be limited and charactered thus . the semidiameter of your circle being taken for radius , you must use the two former rules in this manner . 1. either make the sine of 19¼ a tangent of 19¼ , and to the radius of that tangent finde the secant of ( the complement of 19¼ , namely ) 70¾ , that length shall be the radius of the degrees , or else the decimall scale of the tangents of the zodiac , and they must be inserted by the tables pag. 4 , 5 , &c. 2. or else , make the sine of 70¾ ( estimated to the semidiameter of the circle a b as to a radius ) to be a radius ; and to that radius finde the secant of 70¾ , this last length or secant , shall be the length of the tangent of 45 gr . or of the decimall scale , by which the numbers in the tables pag. 4 , 5 , &c. are to be inserted . by this instance for the horizon of london , where the numbers 19¼ and 70¾ are proper , may the like work be performed in other places of different latitudes from london , only by altering the numbers , according as the proper latitudes of such places shall require . and in all northern horizons ♋ ( or the longest dayes ) must be so placed that the point of 12 in the circle may then be neerest to the index , and in ♑ ( or the shortest dayes ) it may be furthest of . but in southern latitudes , the places of ♋ and ♑ are quite contrary , because there the sun being in ♑ makes the longest day , and in ♋ the shortest . the manner of fashioning the index , of placing the diall , and using it , is the same as was shewed before . how to make the like houres and azimuths by an equally divided circle , upon all plains whatsoever . you are first to finde these three things . 1. how much , and which pole is elevated above your plain , which is equall to the latitude of your plain . 2. the position of the ( usuall substylar or ) proper meridian of the plain . 3. the difference of the plains longitude from your own , or the angle at the pole made between yours and the plains meridian . and when these things are had , the work will be like that which was before . for the way will be twofold ▪ as there it also was . the first way . take halfe the complement of the plains latitude , and adde it to the plains latitude it selfe , the summe will give the elevation of the index above the plain . this index is to point towards ( but not into ) that pole which is elevated above the plain . the other circumstantiall things about the index are as those before , if insteed of the line of 12 there mentioned , we here take the proper meridian ( or substilar drawn in its right position ) of the plain . the limitation of the zodiac must be somewhat as before . that is , you must count the semidiameter of your diall circle to be as the radius . and to that radius , 1. either finde the sine of the elevation of the index , and make that sine a tangent of the same elevation of the index ; then to the radius of this tangent finde the co-secant of the former elevation , the same co-secant shall be the radius or decimall scale whereby to insert those dayes or ellipticall degrees set down in the tables pag. 4 , 5 , &c. 2. or else to that same radius of the circle , finde the co-sine of the elevation of the index , and to this being made a radius , finde the co-secant of the indexes elevation , and so shall this co-secant be the radius or decimall scale as before . the second way . the altitude of the index above the plain is the halfe complement of the plains latitude . the index must ( in this way ) point towards ( but not into ) that pole which is depressed under the plain ( i mean toward that coast of the world , but must still remain above the plain . ) the standing and motion of the index , with the charactering of the zodiac , and other circumstantiall matters , as of the cock for one index , and the usage for the houre , must be agreeable to the former rules given before . the limitation of the zodiac must be done by the two former rules given ( in generall terms ) in this page . how to divide the circle of houres . it must be divided into 24 equall parts , and those of the 24 taken into use as will be serviceable , the rest left out . every of these 24 parts may be divided into 15 gr . or into quarters and halfe quarters ( for here in such plains there will be no azimuth shewed for the place , nor well for the plain it selfe , and therefore it is best to omit it in these altogether . ) all the doubt will be where you must begin your division . from the intersection of the proper meridian ( of the plain ) with the circle , set off ( upon the same circle , and to the true coast ) an arke equall to the plains difference of longitude . and from that point so inserted , ( which you must suppose to be the point of 12 ) you are to begin the division of your circle . the numbring of the houres on both sides 12 will not be hard to finde , in respect of their course , for the course of the shadow of the index will give that . note 1. in all polar plains ( such as our upright east and west , &c. ) the index in both these cases or wayes ( here before given ) will be elevated 45 gr . and the radius of the diall circle will be the radius or decimall scale by which the zodiac scale is to be inserted out of the tables , pag. 4 , 5 , &c. 2. if at any time the zodiac prove longer than the diameter of the horologicall circle , and so the index do go without the circle ; at such time as it is without , the shadow of the index will go one way till it touch the circle , and then back again the same way it formerly came . a briefe demonstration of these circular wayes of making dials . the circular wayes have dependance , and are deduced out of the precedent ellipticall wayes ; and the cases are but two , wherein the same may be done upon any horizon or plain , as may be perceived by the former precepts , wherein the altitudes of the indexes above the plain are ever made to be either halfe the complement of the plains latitude , or else ( the summe of the latitude and that halfe complement , which summe is equall to ) the complement of that forenamed halfe complement . according to these two cases here are two schemes fitted . the projections are made upon the plain of the meridian circle p z h x. p is the pole , z the zenith of the place , h a the horizon of the place , ae a the equinoctiall circle , d a and r a are drawn in the middle of z p and ae h or ae o. d b c x represents one azimuth or verticall circle proper to the plain r a , and by that one all the rest of them may be understood . d a represents the index in both the former cases , and r a the horizontall plain ( not of the place but that horizontall plai● which is ) properly belonging to the index or zenith line d a. the first scheme shews the first of the two former cases , the second shews the second of them . their demonstrations will be both one . for the question in them both is , how to these inclinations of the index , the diall , or horary line , that falls upon the horizontall plain of the place , comes to be a perfect circle . first , therefore consider , that in all the ellipticall dials , the horary line hath relation to the equinoctiall circle , and to the index or line that is to give the shadow ( whatsoever the superficies be upon which the projecture is made , whether plain or otherwise it matters not , as appears before , but here we deale only with plains , because ellipses and circles are plain figures . secondly , consider , that through the severall houre points of the equinoctiall circle , right lines are to be supposed to passe infinitely extended , till they meet with the plain whereon the projecture is to be designed . which issuing out of these infinite lines ( if they be regularly cloathed about with an inflected superficies must comprehend a cylindricall concave , either round or compressed according as the forenamed infinite lines are either perpendicular to the plain of the equinoctiall , or not perpendicular to it . thirdly , these parallel lines so drawn through the houre points of the equinoctiall , parallel to the index , must all fall perpendicular to the horizontall plain which is properly belonging to the index as a zenith line . and the same lines upon that proper horizon do alwayes make an ellipsis , except only these two cases . first , if the horizon be the same with the equinoctiall circle , and the zenith line or index the same with the axis of the world , for then those lines do there make a perfect circle . secondly , if the horizon be a polar or meridian horizon , and the index or zenith line fall into the equinoctiall ( having no latitude from it ) for then those lines do all fall into ( or coincide with ) the plain of the equinoctiall , and consequently do make ( upon their proper horizon ) a direct right line : ●ea more , because the said lines do lie all upon the plain of the equinoctiall , and are all drawn out of the equall divisions of the equinoctiall circle , and besides are all parallel one to the other ; therefore they make ( in this case ) a double line of sines , as is said before in the fifth note pag. 37. now to come to my intended purpose , which is , to prove 〈◊〉 indexes so laid ( as is mentioned before in the two precedent cases ) will require ( upon the horizontall plain of the place ) a true circle and not an ellipsis . suppose therefore d a to be the index ( according to one of the two precedent conditions ) then must r a be the proper horizon to that index . a●d according to the former doctrine , if ae , and b , and a , be taken for three of the houre points in the equinoctiall , and azimuths drawn through them from d the verticall point of the index , then i say ( in these two cases and not otherwise ) looke how the equinoctiall is divided by those circles , in the same manner is h o the horizon of the place divided . but because the equinoctiall is ever equally divided in respect of houres , therefore the circles from d to x ( from the zenith of the proper horizon r a to the nadir , which are azimuthall circles ) shall cut the horizon of the place ( h o ) into equall parts . this , i say , is true , because d ae b and x h c are equall sphericall triangles in three of their quantities [ for d ae is equall to x h , and the angles at d and x are equall , as also the right angles at ae and h ] and therefore equall in all the rest , if like be compared to like : that is , h c is equall to ae b ; and consequently a c equal to a b : so also c x equall to b d. and if so , then the chord b c must be parallel to the index or zenith line d x , and so ae h is parallel to d x , and so contrarily ( in these two cases ) right lines drawn from the houre points in the equinoctiall circle , parallel to the index , must cut ( upon the horizon of the place ) equall parts , equall to one another , and equall to the houres upon the equinoctiall circle . that is to say , they cut the horizontall circle it selfe into 24 equall parts or hours . and consequently ( in these two cases ) those forementioned right lines must designe ( the horizontall circle it selfe , that is ) a circle upon the horizontall plain of the place , though upon the proper horizon it selfe ( r a ) they must designe an ellipsis ( falling perpendicularly thereon ) according to the third observation in the 128 pag. and from that ellipsis these lines b c , h ae , may be conceived as surgent lines ( such as are mentioned before pag. 113 ) containing a compressed cylinder , which cylinder being cut two wayes , ( that is subcontrarily ) will again reveile the originall circle ( at least in one of the two sections ) from whence it selfe and the surgent lines tooke their forme and places . note . 1. the same reason holds in all plains , that in them also there may be found such a circular horologiography . for they are horizons to some place of the world or other , and are therefore capable of the same accidents that the forementioned horizon of a particular place is . 2. that these cases are the same ( in a manner ) with those § . 5. for here a new zenith line ( as d a ) is found to h o the horizon of the place , such a zenith line ( or index ) as may lie equidistant from the horizon of the place h o , and from the aequinoctiall circle ae a : and there there is a plain set up which may stand so , as that the zenith line of the place may stand equally distant from the aequinoctiall circle and this plain . in both , the index shands equally distant from the equinoctiall , and from the plain whereon the houre points are to be inserted . for the limitation of the zodiac . the first work in the limitation of it is for the ellipsis that should be described upon the proper horizon , and the ground of that is the same with what was before delivered pag. 75. the reason of both will be plain enough out of the foregoing diagrams . for the zenith line d a , if it were to move upon the proper horizon r a , it must move in the proper horizontall zodiac , as formerly it did , pag. 17 , 18. which zodiac is to be made as before , and here in the first work is shewed : the reason of the division of it is given in pag. 61 , 62. here is only to be shewed the reason of the enlargement . if the index were set upon the proper horizon r a , then it must move thereon ( perpendicularly ) from a to s suppose . but it moves not upon that but upon h o the horizontall plain of the place , and yet to keepe the same distances in motion from the first position of it in d a x , that is , it must move parallel alwayes to d a , but not upon a s but upon a h , which to do , it must not have the parts of the zodiac in a s ( to guide upon h o ) but proportionall ones to them , that is , instead of a s must be taken a m ; which is all one as to say , as a s the radius , to a m the cosecant of s m a , or d a o , the indexes elevation ( above the horizon h o ; ) so are the parts of a s the prime or fundamentall zodiac , to the parts of a m , the secondary or inlarged zodiac . thus much of circular dialling . appendix . see the figure in page 120. to that zodiacs length you may finde such a radius as shall be thereto justly competent so as to make the ellipticall horizontall diall r t s ; and having made that diall , the upright index d a will give the hour upon it , as the slope index a c will give the houre upon the circle . and these two being severall indexes and dialls must set themselves as the other do , and wil not stand right till they agree . onely this cannot be so punctuall , because the two indexes a c and a d are no● so far distan●●s in the other horizontall di●ll . but the azimuth will here be still shewed , as was mentioned before . having the zodiac limited , how to finde the two extream diameters of the ellipsis . the length of halfe the zodiac is alwayes to be esteemed the tangent of 23½ gr . now because we here suppose this zodiac already made , and to be intended for an horizontall diall ; therefore , either thus , 1 as the tangent of 23½ , is to the tangent of your latitude ; so the length of the semizodiac upon your diall , to the sine of your latitude , which is the shorter semidiameter a t. or else thus , 2 as the tangent of 23½ , is to the radius or tangent of 45 gr . so is your known and limited zodiac , to the cosine of your latitude . the radius , to which cosine is the larger semidiameter a w. thus having fitted your ellipsis ( to the zodiac ) you may describe and divide it in such manner , as was formerly shewed , pag. 1. finis . rectilineal or diametral horologiography . shewing the manner how to describe the houres upon a finite right line , or the diameter of a circle , and to fit a moveable index unto the diall so described . invented and written by mr. samvel foster , late professor of astronomie in gresham-colledge . rectilineal or diametral horologiography . concerning dials made upon a finite streight line , with a moveable index . this may seem no difficult nor new thing , since the most vulgar known way of pro●racting dials is either by one contingent line ( which is a right line ; ) or else by two tangent lines , as by the sector is usually done . but it must be considered , that the contingent line is infinite , and this line ( here mentioned ) is finite . again , that other way by the sector , is not by one finite line , but by two at the least ; whereas this is by one finite line ▪ some difference there is . and besides ; that other by tangents may be as well done upon a single crooked line , this cannot . this streight line is mentioned before , that in right horizons or polar plains the ellipsis shutreth up into a streight line . and it is true , that this falls out to be so , onely in right horizons ( if the index stand perpendicular to the plain , as is before intended ) which cases happen but seldome , and to some plains onely . but here my purpose is to shew how it may be done to all plains whatsoever : observing only this one condition ; alwayes to set the index so , as to point up into the equinoctiall circle : for so it must do , and will not be done any other way , according to this forme of casting the houres . there are two cases which will offer themselves . the first is , when the index pointeth up both into the equinoctiall circle , and likewise into the plains proper meridian circle . the second is , when it points up into the equinoctiall circle of the heavens ( for so it must ever do ) and into some other meridian circle , such as is not proper to the plain . ¶ 1. how to make a diall upon a finite streight line , drawn upon any plain , the index being so ordered that it point up into the equinoctiall circle , and into the plains proper meridian . in this case , the index is to lie just over the proper meridian of the plain , and is to be elevated so much above that line as the height of the equinoctiall above that plain comes unto , and must point up into the equinoctiall circle it selfe . it must therefore here be supposed that the situation of the plain , with such things as are thereon depending , are known . such are these that follow . 1 the plains reclination or inclination and declination . 2 the position of the substylar , or plains proper meridian 3. the elevation of the pole above the plain . 4. the plains difference of longitude from your meridian . these things are to be found by the common wayes of making of dials upon plains , which need not here be repeated , but shall be supposed to be already known , and placed in their true positions upon the plain . let a b be the proper meridian of the plain truly placed , and a d the index standing over it , and pointing up into the equinoctiall . let the degrees of the difference of the plains longitude be c l , and let the point l stand from c upon the right coast according to the exigence of the plain . then for drawing and dividing the line that is to give the houres , do thus . 1. supposing a to be the point of the equinoctiall ( in the zodiacall motion of the index ) through that point draw r a s perpendicular to b a c the proper meridian ; and upon a as the center , and to any convenient extent , describe the circle r c s , in which , c l is to be set according to the difference of longitude , as was before mentioned . so that l is the point from whence you must begin to divide your circle , for from l is the point of the houre 12 deduced . therefore , 2. divide your circle from l into 24 houres at a , b , c , d , e , f , &c. l , m , n , o , &c. and from each of these points let perpendiculars fall upon the diameter r s , as l m , an , bp , &c. the same perpendiculars shall divide the diameter into the houres required : m shall be the point of 12 , because it falls from l , the point of the difference of longitude : n and o shall be the next houres to 12 on both sides : p and q shall be the second houre points on both sides 12 , and so the rest in their orders . but then in these dials , when the difference of longitude falls not just upon an hour ( as very seldome ( and then but casually ) it doth ) the line r s will be crosse divided as you see it in this example , for e r u is one hour , and unequally broken and set upon the diameter r s : so t s o is the space of 5 houres ; and the points t and o stand crosse or distant one from the other , and are not coincident ; which i say must alwayes happen when the difference of longitude is not equall to one or more just full houres . 3. in these crosse houres ( for so i may call them ) you may divide each side of the line severally , as thus : or else you may make a double line in this manner , supposing the fiduciall line to be the white that lies intire and undivided betwixt them . this kinde of diall will be somwhat difficult in the use , because of this crosse division of the houres . yet i thought best not to omit it , because it is the ground of the next case which will be more usefull . concerning the zodiac and the index . for the limitation of the zodiac , let a r or a s be as radius . to that radius let a f be the tangent of 23½ . this tangent is to be augmented according to the proportion that the radius bears to the secant of the complement of the angle d a b , which is the elevation of the equinoctiall above the plain : that is , as the radius , to the secant of the poles elevation above the plain ( which is the same with the said complement ; ) so is the tangent of a f , to the line a b suppose . therefore now a b is the length of halfe vour zodiac , and must be taken as a tangent of 23½ gr . ( or as a scale of 4348 / 10000 equall parts ) by which you may insert the moneths or signes as is before shewed , pag. 4 , 5 , &c. these must be set on and charactered as the nature of the plain shall require them to be ordered . the index though it move , yet must alwayes lie just over the proper meridian a b , and must also keepe the angle d a b unchanged , and must be moved from b to a , and so to b , as the time of the year shall require . or the index may stand and the diall move upon the line b b , as the manner of these kindes of dials may be , and which have been too often mentioned . the second case follows . ¶ 2. how to make a diall upon a finite streight line , drawn upon any plain , the index being so ordered , that it point up into the equinoctiall circle , and into a just hour point in the same circle . as before , so here , it must be supposed that the situation of the plain with the other things ( before-mentioned ) which depend thereupon are known and placed upon the plain . and the matter must be so ordered , that the index must both looke into the equinoctiall circle , and also it must point to some just hour therein , that so one division of the diall line may very well serve for accounting the houres both wayes , the whole houres and parts upon one side justly corresponding to the just houres and parts upon the other side . this will be for more expedite use than the former way could be , and no such trouble nor confusion in the numeration of houres and parts of houres , as will appear in the next example . in this case that we are now going upon , we must consider these things . 1. how the index is to be ordered , so as that it may look up into some just houre point of the equinoctiall ? 2. how the line of the diall or houres ( the horologicall line ) is to be placed , and limited and divided ? 3. how the zodiac is to be placed and limited ? ¶ first , for the index : the best way will be to finde what declination and reclination ( in respect of the meridian and zenith of the plain ) the houre point ( into which the index must point ) hath . and that is done by these two rules . 1. as the radius , to the sine of the equinoctials altitude above the plain : so is the co-sine of the distance of that houre point from the meridian of the plain , to the co-sine of the reclination ; or to the sine of the elevation of the index above the plain . 2. as the radius , to the sine of the poles altitude above the plain : so is the co-tangent of the former distance ( of the intended houre point from the meridian of the plain , ) to the co-tangent of the declination of the index from the meridian of the plain . or thus , 2. as the sine of the poles altitude above the plain , to the radius , so is the tangent of the former distance , to the tangent of the declination of the index from the plains meridian . thus , ] suppose a b were the proper meridian of the plain , make the angle b a c on the right coast , and equall to the declination of the index ( now found ) and elevate a d just perpendicularly over a c , to the angle d a c equall to the elevation of the index above the plain , and so you shall place the index in a just position , that it shall looke up into a just houre point of the equinoctiall . that is the first thing . ¶ secondly . the horologicall line is to be set for position as it was before in the precedent case , namely , perpendicular to the proper meridian , as in the former and this scheme you see r a s. then again , this line which in the former figure was equall to r s the diameter of the circle ( or two assumed radiusses ) must now in this case be enlarged , according as you have ordered your index to wrye from the proper meridian of the plain to any other just houre ; as in this figure it is made to be distant from that proper meridian unto the houre of 12 : that is , it is supposed to be turned so , as to looke into our 12 a clock point in the equinoctiall : which point of 12 in the former figure was noted at l , and so in this figure let it be supposed again . i say , let it be supposed that ( because the former horologicall line was crossely divided , and therefore untoward for use ) this way that is now spoken of shall be to take away that inconvenience in use which was in the former . and therefore suppose the. style or index to be wryed from the proper meridian of the plain to the meridian of the place , or to the point of 12. it may as well be put to any other just houre , as to 12 , but be it so in this example , and let the point be l selected from all the rest . from the selected point l , draw the diameter l a m , and from the selected point l divide the circle into 24 equall parts or houres . then from each two houre points correspondently distant on both sides the diameter l m draw right lines , as g h and n o , and g o , which will be all parallel to the diameter l m. draw these lines ( i say ) and continue them untill they meet with the horologicall line r a s , sufficiently prolonged both wayes . by this draught thus made , the horologicall line r a s shall be both prolonged to h o , and also divided just as a line of sines , so the parts of it are in the same proportion that the parts of the diameter g a g are , which are right sines both wayes from the center , or versed sines from g through a to g. ¶ note that here ( as formerly ) you may assume the circle of what extent you list , and shall finde most fitting your purpose . it is plain then by this example , first , that the radius of the circle a r , is to be enlarged , according to the secant of the equinoctiall distance between the proper meridian of the plain , and that just hour ( wha●ever it be ) inro which the index is made to point . as you see that arke to be here l c , or g r , or g s , whose secant is a h , or a o , in respect of the assumed radius a r. and secondly , that h o thus enlarged is divided like a double line of sines , as it ought to be . and for numbering the houres , if a be 12 , and the index look toward the south , then must o be 2 , and the other o must be 6 ; and the first o , 10 at night . and h must be 9 in the morning ; the other h , 6 in the morning ; and the first h 3 in the morning . but if the index had not pointed up into 12 at l , but into some other hour as p , then must a diameter have been drawn at first from p to the center a quite through , and all the houres parallel to p a , and then the horologicall line would have been thus numbered with 7 at the two extreams , and 1 in the midst at a. and here is one speciall thing to be noted ( as formerly pag. 125 ) but this here is alwayes so , and in all oblique horizons . that in our climates where the dayes are in summer longer than 12 houres , the shadow of the index doth really go forward and backward . as to make it plain . if the diall were described upon an horizontall plain ( for instance ) and it were our longest day , then the shadow of the index would begin at 4 a clock ( figured below the line ) and would go thence towards the left hand to 5 , 6 , and 7. at 7 it would return back towards the right hand through 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , 5 , 6 , untill 7 , and then being arrived at 7 , the utmost point on the right hand , it will come back again towards 8 and 9 to the left hand . this for the placing , limiting , and dividing of the horologicall line . ¶ thirdly , concerning the zodiac how it must be placed and limited , consider of these things . the zodiac ( in this case of wrying the index from the plains proper meridian ) must be augmented in length , and changed in situation from what it was in the former case where the index looked up into the plains proper meridian . yet i will here first begin with that way so far as it goes , and then proceed to what is further needfull . let the figure pag. 137 be here repeated . to the radius a r , the tangent of 23 and a halfe is a f , and the same is there augmented thus . as the radius , to the secant of the poles elevation above the plain ; so let af be supposed to be , to ab . so that if the index were pointing directly to the meridian point ( in the equinoctiall ) which is proper to the plain , then the businesse was at an end ; for a b must be the length of the zodiac , and standing square to r a s. but now we here suppose the index to be wryed from that point , as much as the arke c l comes to ; which turning aside will both lengthen the zodiac again , and turn it out of its perpendicular standing to r a s. which thing is thus to be effected . to a f as a radius , finde the tangent of the poles elevation above the plain ; suppose that tangent were g h. then say , as the radius , to the tangent of c l the quantity of wrying , so g h to x z. make b k equall to x z , and parallel to a r , and draw a k , so shall a k be the length , and shall also give ( in this position ) the situation of the zodiac . the manner how to divide it need not be repeated , because it hath been often set down heretofore , you must only remember to place this line upon that coast of a b which is contrary to the coast of the indexes deviation or wrying from the proper meridian point of the plain : this work may either be protracted , or calculated , or wrought instrumentally , as shall best be liked of . another way i will here adde to do the same work , which may perhaps be not inferior to the former . let all things preparatorie in the former figure , be here again repeated , r a the radius , and a f the tangent of 23 degrees and a halfe to that radius : then to a f as radius , finde the sine and co-sine of the poles elevation above the plain : let f p be the sine , and a p the co-sine . make a m equall to a p ; and from m draw the line m n , making the angle n m c equall to the angle b a c pag. 142 , and standing on the contrary coast , which is the angle of the declination of the index from a c in this figure , the proper meridian of the plain . then say , as the radius , is to the tangent of the reclination of the index ( which is the complement of d a c ( pag. 142 ) the elevation of the index above the plain , ) so is the sine f p to a fourth quantity , suppose the same to be d g : make m n equall to d g , and draw the streight line a ● , so shall a n in this figure be equall to a k in the precedent , and the angle n a c be ( here ) equall to the angle k a b in the former . so that a n must be the length of the halfe zodiac , and will lie in its true position by this work . this may be done either by calculation , protraction , or instrument , as all other works may also be . care must alwayes be had to place these zodiacs on the right coast from f a c , and that the zodiac be rightly divided and charactred according to the seasons of the year , as they shall have respect to the nature of the plain . or thirdly , it may be done thus . first , 1. as the radius , to the bine of the poles elevation above the plain ; so the tangent of the equinoctiall discession or wrying , to the tangent of the declination of the zodiac ; which is the angle n a c in this figure , o● b a k in the former . 2. as the co-sine of the discession , or equinoctiall wrying , is to the radius ; so is the tangent of the poles elevation above the plain , to the tangent of another arke . 3. and , as the radius , is to the secant of that arke ( or as the co-sine of that arke to the radius ; ) so is the tangent of 23½ gr . to a number . if now you make the radius of the circle first assumed , namely r a or a s to be a decimall scale , and out of that decimall scale do take this number , the same shall give the length of the zodiac a n in this figure , or a k in the former . the division of this zodiac now it is thus placed and limited , must be done by the tables and directions in the 4 and 5 pages . for a n must be taken as a tangent of 23½ gr . or as the number 4348 of a decimall scale of 10000 parts . thus much for this kinde of diall upon a streight line , both crossely and evenly divided . a demonstration of the former way . in pag. 28 , note 5 , there is mention made of polar plains or right horizons , that in them , the ellipsis closeth up into a right line . that right line is here the line that we have to do withall . for we here suppose alwayes that the index pointeth up into the equinoctiall circle ; and the index we alwayes suppose to be a zenith line , and therefore the horizon proper to that index as a zenith line , must needs make right angles w●th the equinoctiall circle , and must therefore be a right horizon or polar plain . when the index is pitched in a true posture , all the houre points of the equinoctiall circle must be supposed to have lines issuing from them all parallel to the index . this is a generall truth in the whole course of the ellipticall dialling , wherefore , first of all , these lines will ( in this case ) fall all upon the plain of the equinoctiall ; and because the common section of any two plains must be a right line , therefore it is that all these lines so issuing , will fall upon one and the same streight line , namely , whensoever the equinoctiall meets with or cuts any other plain . hence comes it to be rectilineal horologiography , being performed upon a plain . secondly , because they are all parallel to the index , therefore they must all be parallel to one another ; and because they come originally from a circle , and are parallel , they will all also keep distance one frō another as the sines in a circle do . thirdly , because the index moves alwayes in a parallel position to it selfe , and would shew the houre upon the circular divisions of the equinoctiall circle ( at all times of the year , that is , when this equinoctiall is made to represent all other parallels of declination ) it must follow that the index when ( the shadow of ) it falleth upon any one such point , it must also fall upon the whole line that issueth through that point . the reason is , because the line is alwayes parallel to the index , therfore is it touch one point it must shadow the whol . fourthly , hence will follow , that any line drawn upon the plain of the equinoctiall , crossing those forementioned parallel lines , shall be a line of sines ( or a double line of sines , if you will rather expresse it properly ) and if it passe through the center o● the equinoctiall circle ( where the index also doth alwayes passe ) then is such a line one of the dials that we here are to deal with . because every plain is taken for a great circle cutting the equinoctiall in the center . and the divisions of this line ( the sinicall divisions ) must be the houre points , as is said before . now all that we have here to say is the making good of the former rules of limiting out and placing the houre line , the zodiacall line , and the index . if we therefore ( in all cases ) suppose the index to be a zenith line , and then an horizontal plain to be placed perpendicular to it , this plain must be the proper horizon to that index ( or zenith line ) and must ( in these cases here handled ) go through the axis of the world ( a part of which i must call the primary zodiac ) and will cut the equinoctiall plain in a right line placed perpendicularly to the index , which i call the primary horizontall line . and again , all that is done in other plains ( not perpendicular to the index ) must be imagined primarily ( or originally ) to be deduced from this proper horizontall plain . first therefore , for the houre line . if the index lie in the proper me●idian of the plain ( as is supposed in the first of the former cases ) then the diameter of the circle that is at first assumed , serves for the horologicall line without any alteration , because the proper horizon , and the plain upon which you work , do ( both of them ) cut the equinoctiall in one and the same line , which is to be the horologicall line . and therefore in plain reason , so much of the equinoctiall as the index is wryed from this proper meridian of the plain , so much of the same equinoctiall must the proper horizon be wryed with it , and consequently , so much will be between the proper horizontall line and the horologicall line of the plain , but still both these lines will continue upon the same plain of the equinoctiall , and consequently they must in both cases be one and the same line , namely that which is made by the projection of the equinoctiall circle upon the plain , on●y it must b● augmented according to the proper horizontall lines departure from the section of the plain with the equinoctiall , that is , according to the wrying of the index , as is plain enough by what is said in the 142 pag ▪ and so for the division of that line in the same page . for g g and h o , do both lie in one and the same plain of the equinoctiall : and all the projecture must be made parallel to the index , whence h a is the secant ( to g a radius ) of r g or c l , the wrying of the index . then for the zodiac , you are to deduce it from the axis which is perpendicular to the zenith line or index : and from so much of it as containeth the tangent of 23½ gr . to the radius g a. now though the index do keep the proper meridian of the plain , yet the axis is elevated above the plain . and because the projecture of that tangent or part of the axis must be parallel to the index , therefore look what elevation the axis or pole of the world hath , to such a secant of the radius equall to the tangent of 23½ before mentioned must the zodiac be extended . by which means the index keeps the same situation and distance from the equinoctiall plain that it would do if it moved according to the proper and primary tangents upon the axis , and no otherwise . and so doth the former enlarging of the horologicall line produce the like effect , namely , that the index doth upon that enlarged line shew the same houres that it would do upon the proper horizontall line . then for the zodiac to a wryed index , it is both displaced and enlarged too , according as right lines drawn through the divisions of the axis , and parallel to the index , would make a projection of it . according to which , there are three wayes given . the first makes a f equall to the tangent of 23½ to the radius r a , to be radius , and to that radius findes f h , and to that again as radius , finde h z the tangent of wrying . and so a z is the semizodiac placed and enlarged . as f p radius , to p z the tangent of the angle p f z ; so is f p in the former known length ( found to a f tangent of 23½ gr . ) to p z. therefore z is the point terminating the extremity of the projected zodiac , and so a z must be the semizodiac , according to what was delivered in the second way pag. 147. the third way resolves a sphericall rectangled triangle , made between the proper meridian of the plain p a , the plain it selfe a b , and the meridian in which the index lyeth , or into which it pointeth , p b. in which three quantities are known . p a the altitude of the pole above the plain . the angle at a a right angle . and the angle at p equall to the arke of the equinoctiall or wrying of the index from the proper meridian of the plain . and there is first required a b. as the radius p n , to the tangent of ( n m = ) p ; so the sine of p a , to the tangent of a b , = n a c , pag. 147 , then secondly , the arke p b is required , and that 〈◊〉 wrought by the second work , pag. 148 , which in the second figure pag. 153 , may be understood to be the angle f a z , then because a f z is a plain triangle rectangled at f , and the angle f a z is now known , and the side a f ( equall to the tangent of 23½ ) it will be required to finde a z in the same parts . therefore in the third work ( pag. 148 ) it is said , as a f , radius , to a z , secant of f a z ; so is a f in parts , ( as a tangent of 23½ ) to a z in the same parts . and so the zodiac is both placed and limited . appendix . it hath been forwerly shewed ( in the viii . section ) how to make the ellipticall diall to any index however placed , and to any superficies flat or curved . that doctrine will shew how to project this kinde of rectilineall diall ( and the former circular diall too ) onely i shall adde a word or two to shew that that way will produce a single line in this case , and such a line as must answer to a double line of sines . 1. see the 6 and 7 propositions pag. 95 , 96. the first of them is to finde the horizontall spaces . and in this case ( because the index lookes alwayes up into the equinoctiall circle ) the latitude of the horizon proper to this index line , is nothing . so that , as the sine of the poles elevation , which is nothing , is to the radius ; so is the tangent of each houre from the proper meridian of the index , to the tangent of an infinite length : [ for so it must be because the first term is 00. ] which infinite tangent is belonging to 90 gr . therefore it will follow that all the houres and each of them are 90 gr . from the proper meridian of the index : that is , that they are all coincident into one line , which is perpendicular to the proper meridian of the index . this shews that this dial will be a single line , and that by the former generall doctrine , section viii . 2. then for the second thing , that it will be a line of sines , or ( though a curved line , yet ) to be thence deduced or divided , it will appear by the seventh proposition pag. 96. because , as the radius , to the co-sine of the poles elevation ; which here is the radius ; so the co-sine , &c. to the same term again . for if the two first terms be the same , then the two latter must be equall too . and since the third terms are a line of sines , the fourth terms ( which are to divide the diall line ) being the same with the third , will be also such as will make a line of sines . so this is made good likewise . and there may other wayes be found to effect this same thing upon a curved superficies . as first to project the equinoctiall line or section with the uneven supersicies , upon the said superficies : and when that is done , to lay the edge of some boa●d or ruler , in the superficies of the equinoctiall , with the index lying in the plain of the equinoctiall , will ( by the former equinoctiall line drawn ) direct you in , and afterwards to proceed answerably to the nature of the diall , &c. but what need i to mention any such thing thus obscurely . they that understand the former wayes will be able to invent many more , and so i ce●se from further tro●bling my selfe or the reader . one proposition more must here be added , which conduceth to this rectilineal horologiography , namely : how to lay a line that shall point up to any assigned place or point in the heavens . this is usefull for the laying of an index to any condition that shall be required : namely , so as to cut through any just houre , either in the equinoctiall circle , or any other circle in the heavens . the first work must be to finde what reclination and declination the said point shall have in respect of our own zenith and meridian . this is an easie sphericall work , and therefore i shall omit it . only note , that the reclination here mentioned is the distance of it from our zenith ; and the declination is , what azimuth from the meridian the said point fals into . the next work is , how to lay or stretch a line to any given declination and reclination . the way that for the present i think on shall be this . for the houre points . you are here left to your choise to take which ( of three wayes ) you like best . either , 1. these points in the circle may serve for the houre points without any more draughts , and then this will be another circular diall , differing from those that were handled before in circular dialling . 2. the perpendicular diameter d e may have the hours inserted by drawing lines from each pare of houres equally distant from d and e houre points of the circle , parallel to the index c a , and so you shall have the diameter d e divided like a double line of sines ( unequally ) at 12 , 11 , 1 , 2 , 3 , and 10 , 9 , 8 , &c. 3. you may likewise ( if you allow not the two other wayes ) draw any line as f g , and by lines drawn through the houre points of the circle , all parallel to the index c a , the same line f g shall be divided as a double line of sines , at 12 , 1 , 2 , &c. and 11 , 10 , 9 , &c. and will shew the houre as the other do , and the two extream parallels 4 f , and 4 g , shall limit the length of the line f g. ☞ upon this circular way note thus much . that though the index a c should not lie upon a just houre , but otherwise accidentally , yet because the houre points are upon a circle , the houres will be distant from each other , and not breed that distraction in computing that was before noted necessarily to fall upon the streight horologicall line . and so you are left at liberty to set your index how you will , only so as ever to looke into one and the same meridian circle . but you must then be carefull to place some one houre line true , so as to answer to some just houre circle in the heavens , which upon this equinoctiall plain ( when the axis is once truly placed ) will be easie . for if the axis give a shadow , and you at the same instant time ( noting the shadow ) do observe the houre , that shadow will represent the same houre . if therefore upon that shadow line ( as a diameter ) you describe a circle , you may ( from the intersections of the said line , with the circle ) set off any one full and just houre , and then from that just houre you may divide the circle into the 24 houres . this equall setting of the index is allowable upon the circle , but not upon the right lines d e or f g , for in them there will be confusion of parts , which in the circle are distant . note further , that you need not make the center of the horologicall circle just in the point a , where the extremity of the index which came down upon the plain , but you may make it in any point where the index a c will touch the plain , as at h , or in other like places . concerning the zodiac . you must account the semidiameter d a of the horologicall circle to be as radius . and to that length ( and none other ) you must make the length of the semizodiac a b upon the axis to be the tangent of 23 and a halfe , and by that scale of tangents you must put in the sines or moneths as hath been often shewed before . and whereas i have hitherto said that the index must move upon the axis , and agree to a zodiac thereon inscribed , it must be understood that in some sense this is necessary , but in some sense not so . as thus , the index d a or d e , must alwayes move perpendicular a x the axis . a c the zodiac . a d the index , a b another supporter differing from the axis . to the axis a x. but it is not tied to move upon the axis it selfe . it may move according to any line that lies in the same plain with the index d e and the axis a x , that is , it may move according to the line a b , which we here suppose to be in the same plain or meridian with a d and a x. this condition is alwayes requisite , and it must ever keep one & the same angle with the ●ight line a b , and such an angle that may keepe it alwayes perpendicular to the true axis , that is , parallel to the equinoctiall plain . and this being observed , you may insert the zodiac upon the line a b , but it must be proportioned from the axis a x , thus ; as a c radius , to a e the secant of the angle c a e , so is a c the tangent of 23 and a halfe ( in respect of the radius of the horologicall circle , ) to a fourth quantity ; the length whereof gives a e. and now this line a e must be esteemed as a tangent of 23 degrees and a half , and by it must the semizodiac or halfe year be put in , as hath been shewed formerly . so this is done also , for the index will move upon a e , according to the parts upon a c , as it is bound to do . only note that these dials upon the equinoctiall plains can serve but halfe a year upon one side ▪ & therefore for the other half year you must do the like work upon the under side of the equinoctiall plain . the work i say is like without any reall difference , and therefore there will need no more words concerning it . for demonstration of these things . there needs nothing to be added more than what hath formerly been done for rectilineal dials . for what is done there in the proper horizon ( proper to the index i mean ) by a line of sines , is here expressed by the line of sines d a e , pag. 162 : only there , all the lines issuing from the equinoctiall houre points did meet with the proper horizontal plain , and did thereon create a single line of ●80 sines , and so the equinoctiall circle standing towards the eye ( being infinitely removed from it ) edgewise is projected upon his own diameter , and appears nothing else but as a diameter ( or streight line ) to the eye . but here the same forementioned parallel lines issuing from the houre points of the equinoctiall circle , and flowing out upon the plain of that circle , do so appear in full view to the eye , because the eye stands so as fully to view the said plain and not looking edgewise upon it . when the plain stood edgewise to the eye , the eye did project it into the diameter , but now we conceive the eye to stand only to looke what was effected upon the plain by the former projection . and further . looke what projection was formerly made ( in this case of the indexes pointing into the equinoctiall circle ) upon the indexes proper horizon , and was there received by a streight line , the same is here done by drawing the diameter d a e , pag. 162 , which line must be supposed to be the common section of the indexes proper horizon with the equinoctiall plain . and so in effect both these cases are nothing different . then for the line f g , the parts of it , and the points of the circle , and the points of d e must necessarily all fall into one houre line , and all shew the true houre ( whichsoever of them you take for your horologicall line ) because as the lines issuing from the houre points of the equinoctiall are all parallel one to the other , and all again to the index , therefore if any part of the shadow of the index do fall upon any part of one of those lines , the whole shadow must fall upon the whole line , and consequently the correspondent points upon d e and f g and the circle , will all joyntly , or each of them severally , give the houre of the day . other things are plain enough . of the parallelisme of the index and plain . it must further be noted , that an index may be made to point up into the equinoctiall circle , and yet also be parallel to any plain whatsoever . for every plain cuts the equinoctiall circle somwhere : now if an index be made to lie in that common section of these two circles or plains , then will it lie parallel both to the equinoctiall plain and to the other plain , and it may be made to move alwayes parallel to them both , if it be moved at right angles to any line that shall stand perpendicular to the said common section . but for all this , there can no such diall ( as we have now spoken of ) be made upon any other plain besides the equinoctiall . i say there can no parallelisme of the index to the plain be ( in this kinde of dialling ) allowed , more than what is said may be done upon the equinoctiall plain . for though the index may be parallel to the equinoctial and plain both , yet the equinoctiall it selfe must make angles with all such plains . and because the index is parallel to the common section , therefore all the houre points projected by parallel lines to the index , must be upon the equinoctiall plain it selfe , and must run parallel to the line of common section : that is , they will all run so , as never to meet so much as with the line of common section ( which line is upon the equinoctiall plain ) much lesse ever to meet with any part of the plain it selfe , which is every where distant more or lesse from the equinoctiall , except only in the line of common section . so that there is no more to be said of this subject . finis . elliptical horologiography . shewing how upon any plain to draw an elliptical diall , to an index set any way , by sphericall ( and not projective ) work . invented and written by mr. samvel foster , late professor of astronomie in gresham-colledge . london , printed for nicholas bourn . 1654. elliptical horologiography . hitherto we have had the whole businesse of elliptical horologiography , so far as that more cannot seem to be thought of , or required . yet , because to any index set eith●r on purpose or casually , the houres have been ( formerly ) found out by projection , i thought fit here to adde another way ( but for plains only ) that shall not need that projective manner of working . and so the probleme will be this : how upon any plain to draw an ellipticall diall , to an index set any 〈◊〉 , by sphericall ( and not projective ) work . the substance of what i shall here say is most of it delivered before , only i must now shew what propositions a●e to be referred hither , and how they must be used . things that are pre-requisite to this work . 1. you must know the situation of your plain , that is , what declination and reclination it hath in respect of your own horizon . this is to be done the ordinary way , as in all plains is usuall . 2. you must know the declination and reclination of your index . that is , if the index be set casually you must then finde out the declination and reclination thereof , which is shewed , pag. 80. how it must be done , namely , by observation from the sun , and cannot be otherwise performed . or if you choose a longitude and latitude wherein to lay your index , then must you ( by that longitude and latitude given ) finde what declination and reclination is due thereunto , in respect of your own horizon : and according to this declination and reclination you are to lay your index . see the schemes pag. 83 , 84 , for by those or the like you are to conceive of your sphericall work in this kinde . 3. having thus found the situation both of plain and index , in respect of your own horizon , you must then finde what situation the index hath in respect of the plain ; that is , what declination it hath from the proper meridian of the plain , and what reclination it hath from the plains zenith . this proposition hath not been handled before , but must be put here in this place . the next work is to finde what situation an index ( set up upon the plain ) shall have to the plain , and the fundamental lines thereon described , viz. the verticall line , and proper meridian . i suppose first , that the declination and reclination of the index ( in respect of your own horizon ) is known by the former things already spoken of , and from thence you may gather what is here required . thus let the index point up to x : and let z x ( the reclination from your zenith ) be known , and s z x ( the declination from your meridian ) be also known . now therefore in the triangle t z x , there are three things known . 1 t z the complement of the plains reclination in respect of your horizon . 2 z x the complement of the indexes reclination in respect of your horizon . 3 thirdly , the angle t z x , which is the difference of the two declinations , namely of the plain and of the index . by these all the rest will be found ; as namely , 1 t x , the reclination of the index x from t the pole of the plain , which is called the proper reclination of the index . 2 z t x , or v y , or rather v m , which shews how much the * subindicall line lies from the verticall line of the plain ( in degrees of a circle described upon the plain ) and consequently , because p t z was known before , therefore p t x is now also become known , which shewes the declination of the * subindicall line from the plains proper meridian , which may therefore be called the proper declination of the index . 3 the angle z x t ( if it be of any use here ) shews how much ( in azimuthal position ) your zenith z lies from t , the zenith of the plain upon that hotizon whose zenith line is the same with the index , which is therefore the proper horizon to the index . or if you would set an index so as to fit it to any assigned difference of longitude , and to any assigned latitude also , you must work thus . let x p s be the assigned difference of longitude , and p x the complement of the latitude , into both which the index shall point up in x. because the plains difference of longitude z p t. is known before , therefore the angle t p x is known . and p t the complement of the plains latitude is also before known . therefore in the triangle t p x these three things are known , viz. p x , p t , and the interjacent angles t p x. by which three you may finde all the other quantities : namely , p t x , or b y , the departure of the subindicall line from the plains proper meridian : t x the proper reclination of the index x from t the pole of the plain , or the complement of it ( x y ) which is the elevation of the index above the plain just over ( perpendicularly over in respect of the plain ) the subindicall line : with the angle p x t ( if there were any use of it ) which shewes how much the pole of the plain declines from the proper meridian of the index . i say , it shewes this declination upon that horizon ( whatever it should be ) which shall be properly belonging or perpendicularly placed to the said index as the proper zenith-line . thus the situation of the index ( whether casually , or by election set ) is found to the true coasts of the plain it self . what remains to be done . the rest of the work for making the elliptical diall it self to the index assigned , and for limiting and dividing the zodiacall scale depends upon this one rule . namely , to project the houre points of the equinoctiall , and the zodiacall scale upon the axis ( which zodiacall scale upon the axis it self is made by the tangents of 23 gr . and an half , to the radius of that sphere wherein you shall imagine your circles to be ) i say . to project ( all such points ) upon the plain , by lines passing through each one of them , and all and each of the same lines to be drawn parallel to the index given . this is the summe of what is here now to be done , and of what hath been done formerly quite through this book . for all that is before effected upon any plain or curved superficies , is nothing more then what i have now said ; namely , a determining of the true places of every such ( houre and zodiacall ) point , as they shall be projected upon the dialls superficies , by lines going through them , and going parallel also to the index that is to give the houre . for so , the equinoctiall circle will very well supply every parallel of the sun , the index and horologicall line being rectified by the zodiac in a true site one to the other , as hath been often mentioned before ; and then the index , casting its shadow upon the houre point , either in the equinoctiall or in the parallel circle , must also cast it upon the whole projecting line , which passeth th●ough the hour point , ( because that line is made alwayes parallel to the index ) and consequently upon that point of the houre , which ( by the said line ) is carried to the diall-superficies . and so for the points of the zodiac , they are also projected in such manner , as that the index moving in them ( however they may seem to stand awry and out of order ) shall alwayes cut the axis of the world ( if it were there placed ) and cut it according to the tangents of declination thereon expr●ssed . and this will serve for a brief demonstration of that work which is ●ow to come afterwards . in which we are first to bring down the houre points and zodiac to the plain considered as an horizon , and the projection of those points to be made into an ellipsis , by lines all perpendicular to the said plain , that is , parallel to the plains proper zenith line . secondly , we are to project or transpose these first ( both houre and zodiacall ) points from the primary ellipsis into a secondary ellipsis , such as the position of the true intended index will require : and so also the points of the zodiac must be dealt withall . how to make an ellipticall diall upon any plain , to an index perpendicular to the plain . the first work is done before , partly by calculation , and partly by protraction , pag. 25 , 26 , but more fully and better , proposition 5 , 6 , 7 , pag. 92 , 95 , 96 , by which you make two tables , one of horizontal spaces , the other of altitudes . so that i shall not need to say any more of it . by that work ( i say ) you may make two tables which may serve to prick down the houre points , which points you shall see to winde about the center in forme of an ellipsis . 1. it is first requisite that the pr●per meridian of the plain be rightly situated upon the plain ( as is there also expressely required . and it is again requisite that the proper line of the two sixes ( proper i mean to the plain , considered as an horizon by it selfe ) be also drawn : which line ( as in all horizontal dials is usuall ) must be perpendicular ( upon the plain ) to the proper meridian . and the points also of the two sixes must be carefully set on , whose distance from the center is the longer radius of the ellipsis . the reason why these points of the proper sines must be had , is , because the ●ew ellipsis ( which we now look for ) must passe through them both . so also the just point of the proper 12 ( or the section of the ellipsis with the plains proper meridian ) must be with the same care designed upon the plain . and the subindicall line must be drawn in its true position upon the plain , according to the proper declination of the index . 2. after this is done , you must call to minde what the declination and reclination of your index is ( i speak now of the proper declination and reclination before mentioned ) as pag. 172 is mentioned . 3. then from every houre point lately made upon the plain , you are to draw right lines , parallel to the subindicall line , and draw them both wayes or on both sides of each houre point , because you know not which of them ( as yet ) you are to make use of for the pricking down of your new or distorted ellipsis . 4. take the larger radius of your primary ellipsis ( that is the length from the center of it , to the point of the proper six a clock ) and ( upon the sector , for that instrument is most fit for this use or practise ) entering it upon the tangent of 45 degrees , open your sector to tha● length : then take over from the reclination of the index ( the proper reclination i now speak of ) counted on the same scale of tangents , to the other leg , and reserve this length . that is , as the radius , to the tangent of the proper reclination of the index ; so is the larger semidiameter of the primary ellipsis , to a fourth quantity , which is equall to the ●●rmer reserved length , and to that large semidiameter or radius , is the tangent of the indexes proper reclination . 5. this reserved length must again be entered from 90 in the line of sines , and the sector set according to it , and being so set it is fit for the rest of the work that is to ensue . because this proportion we are to go upon : as the radius , to the tangent of the indexes proper reclination ( which two terms are they that now do set the sector , ) so is the sine of every altitude of the horary points in the equinoctiall , ( which altitudes your table of altitudes computed by the forementioned proposition , pag. 96 , 97 , will help you unto , ) so is the sine of every ( horary-equinoctiall-points ) altitude , to the tangent of the reclination of every one of those severall houre points of the equinoctiall circle . note that the reclinations here mentioned are , in respect , and from the proper zenith line of the plain . and note also that these tangents thus found , are to be estimated to the larger semidiameter of the primary ellipsis as to their proper radius . 6. by these tangents so found , you are to set on the new houre points , by which both the new ellipsis is to be described and also divided . i say the new houre points are to be set on upon the lines drawn through each of the old houre-points parallel to the subindicall line , and they are to be s●t on from each of the said old houre points in those lines . but which way , or on which side of them , will be still doubted ▪ as was before mentioned § 3. and to take away this doubt , it will be best to consider the point of the proper 12 upon the primary ellipsis , when the sun is ( any day ) in that proper 12 point of the heavens , then must the upright ( or proper index of the plain ) cast its shadow into the proper 12 point of the primary ellipsis . then further , supposing the sun to stand still there , imagine how your slope index is to lie from the proper index , and by that you shall easily discern which way the inclination of that index will wry the proper point of 12 from the primary ellipsis , and consequently you shall be able ( thereby ) to judge upon which part of that line which is drawn through the point of proper 12 ( parallel to the subindicall line ) the houre ( of proper 12 ) shall be cast into the secondary ellipsis from the primary . this point being set right , you may place all the houre points from that proper 12 to the two proper sixes , the same way with this first point : but all the other points on the remoter side of the proper 12 , must go the contrary way , as you will easily ●erceive , the very figure of the ellipsis will require it should so be . as here . r s is the subindical line , over which the index leaneth . o the point of proper 12 upon p o q the primary ellipsis . o a , 9 b , ● c , &c. parallel lines through the houre points in the primary ellipsis , parallel i say to the subindicall line r s. p q the two proper fixes , through which alwayes the distorted ellipsis must passe . then because the index leaneth from the upright index ( or zenith line of the plain ) towards s , therefore the proper 12 at o must go towards s , viz. to a : and therefore also 9 b , 8 c , &c. 11 g , &c. must go the same way till you come to the two proper sixes at p and q , and then the houre points d e f must be set from 3 2 1 , the contrary way to that wherein the first houres were placed ; as you may plainly see the form of the new ellipsis will require . for it cannot possibly be made to turn again from p towards m , if it must make up an ellipsis , as in this case it is necessary it should do . another way followeth . a second way to make the ellipsis and houres to a slope index upon a plain : and without trouble of making that table before men●ioned . this way will be more expedite than the former . i shall here suppose these particulars to be again done by the former precepts . 1. that the plains proper meridian r o , is rightly placed . 2. that p r q is drawn perpendicular thereunto . 3. that the subindicall line r s is also rightly placed from r o the proper meridian . 4. that r p or r q being determined for the radius , r o and r k are each of them the sines of the plains propet latitude . 5. from the points o and k ( which are the two proper points of 12 to an index standing perpendicularly to the plain ) let o a and k l be drawn parallel to r s the subindicall line . 6. then to the radius r p , finde the sine of the plains difference of longitude from your meridian [ because i purpose here to shew how to work all from your 12 a clock line , though there is no obligation to that houre , but all may be done just in the same manner from any other houre point , if you account the plains difference of longitude from some one of them , which is easie to do : as if in this example the difference of longitude from 12 be 36 gr . then from 11 it must be 21 gr . from 10 it must be 6 gr . from 1 a clock it must be 51 gr . &c. to all or any of the other houres ] which here for example sake we will say to be 36 gr . with that sine set off r t , and draw t i parallel to r o the proper meridian . then again to the radius r o , finde out the co-sine of the difference of longitude , and set it upon the line t i , from t to i. all this work is only to finde the point of 12 in the primary ellipsis , as you see it expressed in the former figure at 12. the like work you must do to finde the point of one of the houres of six in the primary ellipsis , because six is 90 gr . from 12. [ if you had chosen the point of 11 to work by , you must have lookt for the point of 5 a clock , because that point is the 90th gr . from 11 , if 10 , then 4 , &c. ] that is , to the radius r o finde the sine of the difference of longitude , and set it from r to v , and then draw v x parallel to p r the line of the plains proper six . then to the radius r p finde the co-sine of the said difference of longitude . and set it upon v x from v to x : so shall x be the point of your own six a clock upon the proper ellipsis ; as you may see in the figure is expressed . now through these two points i and x draw right lines parallel to the subindical line r s. for if you divide r 12 ( both wayes ) as a line of sines ( r e and r f being sines of 15 gr . r d and r g sines of 30 gr . &c. r 12 and r 12 sines of 90 gr . ) at a , b , c , d , e , r , f , g , h , i , k ; you must then draw right lines through each of those points ( each one parallel to the conjugate diameter 6 r 6. ) and when that is done , make r 6 a radius , and to that radius make e 7 , e 5 , f 5 , f 7 , to be the sine of 75 gr . or one houre under 90. make d 8 , d 4 , g 4 , g 8 , the sine of 60 gr . or two houres lesse than 90. so c 9 , c 3 , is the sine of three houres under 90 : b 10 , b 2 , of four houres under 90 : a 11 , a 1 , is the sine of five houres under 90 , or of one houre . the like you may do for halves and quarters , or what other parts you desire . thus much likewise for this second way . alwayes you must remember to put all things upon their right coasts . concerning the zodiac , how it is to be limited and placed . to omit other wayes , i shall here pitch upon that which is most suitable to this manner of working upon which we now are . to the radius p r ( which is the larger semidiameter of the primary ellipsis ) first , finde the tangent of 23 degrees and a halfe ; which suppose to be r ξ : and then to this line , being taken as a new radius , finde the co-sine of the plains latitude , or the sine of the equinoctials altitude , which suppose to be the length r w , which is also to be set from the center r upon the proper meridian of the plain , as here it is from r to w. then through the point w , draw a line , as w z , parallel to the subindical line r s. when this is done you are to finde the sine of the poles elevation above the plain , to the same radius r ξ , which sine you may suppose to be the length of the line φ ψ. then say again ; as the radius , to the tangent of the indexes reclination ; so φ ψ ( the sine of the poles elevation above the plain ) to a fourth line ; which line let be represented by λ ω. take this line λ ω therefore , and set it upon w z , the line formerly drawn , from w to z. and lastly , draw the line r z , and make r y equall to it , so shall z y be the zodiac in its true length and position : and r z or r y shall give the halfe of it , which halfe is to be estimated as a tangent of 23 degrees and a halfe , and so the zodiac to be set on as hath formerly been often shewed . you are here to take care that all things be set in their right position . in this example , z turns from w towards the left hand , because it is here supposed that the north pole is elevated upon the plain , and so the axis riseth out of r above the plain : and therefore that 23½ tangent degree upon that axis a●e above the plain . from which extream point of 23½ above the plain , you may suppose a line to issue out parallel to the index , which inclines from r towards s , and therefore that line inclining so too , must project that extream point from w towards z. all the reason of the work depends upon that which hath been formerly said : namely , that both houre points of the equinoctiall , and declination points in the axis , are all projected from their true positions in the sphere ( upon or unto a plain ) by right lines passing through each of them , parallel to the assigned index . i shall need to say no more , because i think enough ( if not too much ) hath been said already . this shall serve for the effection and demonstration also of this proposition or probleme . coronidis loco : i shall here further adde ; how upon an horizontal plain , to describe an ellipticall diall to an index lying aslope , and pointing into some assigned longitude and latitude . this i shall adde as a corollary to that which hath last of all been delivered in general for all plains and indexes . out of which doctrine this ( as a particular branch ) is deduced . it containeth not any new thing , but because it may be more frequently made use of ( in practice ) than other cases , therefore i adde it , that it may also be the more plain and easie in operation . it is ( for more ease ) intended that the index should point up into some just houre circle , and to have some just degrees of reclination from the zenith of the horizon . i purpose not to give variety of wayes , for they are shewed in the last precepts now given which are generall , and may easily be brought and applyed to this particular case , as a branch of that generall before . suppose in our horizon 51 gr . 30 min. in latitude , and to two indexes set upon an horizontall plain , the one pointing south-east into the morning circle of 7 a clock , the other pointing south-west into the evening circle of 5 a clock , and each of them reclining 45 gr . it were required to make two ellipticall dials : how must they be done ? it must be noted that the making of one will make the other , because they are to be set upon one plain , and their differences of the indexes longitudes , is in each 5 houres , and the reclination of the index in each 45 degrees . only the coasts in respect of situation must be changed , because the positions of the indexes are not parallel , but contrary one to the other , and so will the ellipses be , as you may see in the next page . the first thing to be done , is , by having the difference of the indexes longitude ( 75 gr . ) and the reclination of the index ( 45 gr . ) to finde what declination the index must have : or to finde the declination of the subindicall line from our meridi●n . when this work is done ; draw a r b for the meridian line of north and south , and 6 r 6 for the line of east and west , crossing each other at r , which must be the center of the diall , and draw the subindicall line r s 60 gr . 5 min. from a r upon the right coast , that is , north-east in the first plain , and north-west in the other plain . then making 6 r for radius , finde thereto the sine of the latitude of the place 51 gr . 30 min. which let be the line r c , and set it upon the north part of the meridian line a b ( i call it the meridian line because it is the said line upon the horizontal plain whereupon these descriptions are now made ) from the center r to c. then from the point c , draw c d parallel , and the same way with r s the subindicall line . so shall c be the point of 12 upon the primary ellipsis which should be drawn , if an ellipticall diall were to be made upon the horizontall plain to the upright index , which is the proper zenith line of the horizontall plain . after this , to the same radius r 6 , finde the co-sine of your latitude , or the sine of the equinoctials height , or rather ( in this particular work ) the depth of it under the horizon , which sine let be the line e f. then say , as the radius , is to the tangent of 45 gr . the indexes reclination ; so is the sine e f , to a fourth ; which fourth will be ( in this example ) the same line with e f , because the tangent of 45 ( the reclination ) is the same length with the radius , yet i will call it the fourth quantity . take this fourth quantity and set it off upon the line c d , from c to d , so shall d give the point of 12 upon the distorted ellipsis , which is the 12 that fitteth with the reclining index with which we have now to deal . then draw ( from r to d ) the line r d , both wayes , as d r g is done . by these meanes we have now got two conjugate diameters . and now the rest of the work will be easie . for you are onely to divide the semidiameters r d and r g ( from r towards the extremities ) each of them as a line of sines . that is , r m , r o , make the sine of 1 hour or 15 gr . so r l , r p , the sine of 30 , or 2 hours : r k , r q , the sine of 45 gr . or 3 houres : r i the sine of 60 gr . or 4 hours : r h the sine of 5 houres : r d or r 12 , the whole sine or sine of 90 gr . then through each of these points at h , i , k , l , m , r , o , p , q , draw a line parallel to the other conjugate diameter , namely , to 6 r 6. this done , make r 6 a radius , and to that radius make m7 , m5 , 07 , 05 ( each of them ) a sine of 5 hours or 75 degrees . make l 8 , l 4 , p 4 , p 8 , a sine of 4 hours or 60 gr . make k9 , k3 ; q3 ; q 9 , ( each of them ) a sine of 3 houres or 45 gr . : so i 10 , i● , must be the sine of 2 houres : and h 11 , h 1 , the sine of 1 houre or 15 gr . by this work you shall ( upon the former parallel lines drawn through h , i , k , l , m , o , p , q , ) finde the houre points of 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 ; in which you are to signe out the houre points , and through which you are to draw the ellipsis , as you see the figure for a pattern . you might also have divided each of the radiusses , noted with r 6 into parts correspondent to a line of sines , as you see partly done at the letters s t u , in the last of the two former scheams ) as at s t u is expressed , and through those points you must draw paralle●s to the other diameter d r g. then to the radius d r make s 7 , s 5 , the sine of 15 gr . or 1 hour : t 8 , t 4 , the sine of 30 gr . or ●wo hours : u 9 , u 3 , the sine of 45 gr . or 3 houres ; and so forward : by which work you shall finde the same houre points ( 3 , 4 , 5 , 6 , 7 , 8 , 9 , &c. ) that you did before , and so you may finde out and draw the ellipsis by them . the same manner of work is to be performed for halves and quarters of the severall houres . how the zodiac is to be placed and limited . it is the same work with that which went before , pag. 185. yet i will here insert it again . first , work thus . as the radius , is to the tangent of 23½ gr . so let e f ( which was formerly found to be the sine of the equinoctials altitude above the horizon 38 gr . 30 mn . in respect of the radius r 6 , and must now again be here used ) be to a fourth quantity , which suppose to be f w. take therefore that length f w , and set it from r to x on the north side ( in this example : ) and through x draw x γ parallel to the subindicall line r s. then say again . as the radius , is to the tangent of 23 gr . and an half ; so let r c ( which was before found to be the sine of the latitude of the place 51 gr . and an half in respect of the radius r 6 , and comes now to be here used again ) be to a fourth quantity , which suppose to be f z. take therefore f z , and set it upon the line formerly drawn through x , from x , contrary to the going of the subindical line , namely southward ( in this example ) i say from x to γ then from γ through r , draw γ r π , and make r π equall to r γ , so shall γ π be the zodiac both placed rightly and justly limited . for division of it , you must conceive that r γ or r π is a tangent of 23 and an half gr . the suns greatest declination , and consequently you may by it set on either 12 moneths , or 12 signes , as you will ; the manner whereof hath been often shewed in this book , and especially pag. 10 , 11. and pag. 14 , 15 , &c. so that now all this work is come to an end . the index must slide to and fro upon the line γ π ( or the index standing , the ellipsis must move after the same rate ) γ π is the proper meridian of the index . and if your work ( when you make this diall , according to this case here before put ) be exactly true , you shall finde the zodiacall line π γ to fall just into the houre lines of 5 and 7 , as they ought to do , because the difference of the indexes longitude was assumed to be 75 gr . from the south meridian of the place , which is the same with those houres . the signe ♋ must be placed at γ , and ♈ at π in this diall . the index must lie ( when it is in the center r ) just over the line r s , and must alwayes move in a parallel position to it self . and it must in this example recline 45 gr . or rise above the horizon ( or above r s making an angle above that line 45 gr . also . but though it lie thus , yet it must move in the line γ π , and it must be so set on , that when you project the fiduciall edge of it into a point ( as workmen use to do , when with their eye they tr●e the straightnesse of a line ) all that edge may at once justly appear to run into the line γ π , wherever the index be set in the zodiac . this is to be done carefully . i have here set two of them , whose proper meridians , ( i mean proper to the index , which are the lines of their zodiacs , viz. γ π ) do not lie in one and the same right line , or parallelism ( which is the samenesse of po●ition ) but make angles one to the other . this is done , because they should not lie in one meridian : for by that meanes they will alwayes set one another , which the horizontal diall with the single ellipsis to an upright index will not do at all times . these two , i say , will set themselves : and if you adde a common horizontall diall more , to stand between them , as you see done there ▪ they will be the more specious and usefull for setting each other . i have set that whose index looks north-east upon the left hand , you may set it on the right hand if you will , and the other on the left . for this makes no materiall change . the index is best to be a threed . other circumstances i remember none but what the workman will be well able to go through . another way for the description of an ellipticall diall upon the horizontall plain of 51 gr . 30 min. latitude , whose zodiac , and the motion of the index is performed upon the ●oures of 7 in the morning , and of 5 in the afternoon , as the same houres are drawn upon the common horizontall diall . this way was formerly written , and though it need not to have been here placed , in respect the way last given is the best , yet because it doth somwhat differ from what was set down before , i thought best not to omit it : because it is delightfull to see differ●●t way ●● meet both in one effect . 1. first , you are to get ( this way ) the altitudes of each houre point in the equino●tiall , above the horizon , by this r●le . as the radius , to the sine of 38 gr . 30 min. so ●he sines of 75 , 60 , 45 , 30 , 15 gr . to so many 4th sines . the co-sines of these will limit out the points ( upon the houres of the horizontall diall formerly drawn ) through which the ellipsis upon the horizontall plain ( made for an upright index ) is to passe : as is shewed before in the 28 , 29 , 30 , and 31 , pages , which points in the figure of the next page may be represented by 8 , 9 , 10 , 11 , 12 , 1 , 2 , 3 , &c. 2. as the radius , to the tangent of the reclination of the index ; so are the 4th sines before found , to so many 7th terms . 3. describe the regular horizontall ellipsis , as you see p o q which you must know to be for the latitude of 51 gr . 30 min. and that p q are the points of the two sixes , and o k the points of the two twelves , viz. at noon and midnight : and also that all the houres are regularly set down to o p q k , &c. suppose here an index to   gr . m. recline northwards 45 00 differ in longit. s e / w 75 00 consequently it must   decline north east / west 60 05 have north latitude 50 37 4. from each point there set down , draw lines parallel to the subindicall line r s , as o a , 8 c , 9 b , &c. the angle o r s must be 60 gr . 5 min. 6. through these new points b f d a e g , &c. you are to draw a new ellipsis , which will be distorted from the former ellipsis first drawn . thus the diall it selfe for the houre points is limited , you may adde halves and quarters . for the zodiac . in this example remember , that the index declineth north east / west 60 gr . 5 min. reclineth 45 gr . differeth in longitude south east / west 75 gr . hath latitude 50 gr . 37 min. 1. because the difference of longitude is 75 gr . from the south , therefore our two houres of 7 in the morning , and 5 at evening , are the proper meridians upon which the indexes must slide . 2. for that cause you must consider how much of those two meridians is intercepted between the equinoctiall and our horizon . you may finde it by this rule . as the radius , is to the sine of 15 gr . ( the conpl . of 75 , ) so is the tangent of 38 gr . 30 min. to the tangent of 11 gr . 38 min. which is the arke required . 3. because the latitude of the index is 50 gr . 37 min. north , the complement of it must be 39 gr . 23 min. and so much under the equinoctiall points of 7 and 5 ( and upon the houre circles of 7 and 5 , which are the proper meridians to the two indexes ) doth the proper horizon of each of those indexes cut the said houres of 7 and 5 upon the south-east and south-west ; and so much do they cut the same meridians above the equinoctiall upon the south-west and south-east . so that upon the southerly coast , that portion of these meridians which is comprehended between the equinoctiall and our horizon , namely 11 gr . 38 min. must be taken from 39 gr . 23 min. and the remainder will be 27 gr . 45 min. which is the arke comprehended between the proper horizon of the index and our own horizon , which upon the south part ( in this particular example ) is below our horizon , on the north part above . 4. first , therefore the limitation of the semi-zodiacs length must be to that latitude of the index or proper horizon 50 gr . 37 min. and then secondly , the said zodiac first found and limited must be augmented ( in this example ) in proportion of the radius to the secant of 27 gr . 45 min. in this manner , by the second rule given at the beginning of this book pag. 11. to the radius p r finde the cofine of the latitude of the index , namely the sine of 39 gr . 23 min. which suppose to be n t. to this length n t as a radius , finde out the secant of the forementioned arke 27 gr . 45 min. which suppose to be n v. this length n v shall give the radius or tangent of 45 gr . which is the tangent or decimall scale , out of which ( either of them ) you are to make your zodiac by the numbers of those tables in the 6 and 7 pages . to this radius or decimall scale , or tangent of 45 gr . n w is the tangent of 23 gr . and a halfe , wherefore the semizodiac , r l or r m is to be made equall in length to n w. thus the zodiacs length is limited . for the position of the zodiac , it was before said , that it must lie in the line of 7 in the morning , or 5 in the afternoon . so in this last figure you see it points to z , which is 7 in the morning , ( and consequently also to x , which is 7 in the evening ) in the distorted ellipsis : because this is that diall whose index looks north-east : you see it do so likewise in the first figure pag. 189 , as in the second figure there it lies to 5 in the evening , and consequently also to 5 in the morning . observe in the last figure ( which is as true and just in the two other pag. 189. ) that the angle p r z , and so q r x , is equall to the angle made ( upon our common horizontall diall ) between 6 and 7 a clock , namely , 18 gr . 54 min. and consequently that 1 r 1 ( a line drawn from 1 to 1 in the primary ellipsis ) is perpendicular to the zodiac l m : but p r c is not equall to the angle thereon between 6 and 8 a clock ; nor is it so in any other of the houres . observe again , that if a line be drawn parallel to o k. through a the point of 12 in the distorted ellipsis , till it meet with the zodiac ( extended ) at y ; observed ( i say ) that if r m be taken as a tangent of 23½ gr . ( as it is taken in this work ) then will r y be the tangent of ( the complement of your latitude , namely ) 38 gr . 30 min. and this must be so , because if the sun should decline 38 gr . 30 min. southward , then the sun must only peep up upon the horizon at 12 , and consequently the shadow of the index ( namely the point of it that intersects with your horizon , for the sun hath no altitude at that place and time ) must run full from south to north to shew the point of 12 at a. other observations might be made , but i go no further . the reason of the precedent work will be demonstrated out of the former things delivered in this book : especially out of this last treatise , which begins pag. 171 , and ends pag. 186. and therefore i wholly omit to adde any thing more to this particular subject . they that understand the things before , will not be to seeke in the necessary demonstrations of these particulars . a note concerning the framing of dials to finde the azimuth . as the axis of the world is to the equinoctial circle , so is the zenith line to the horizon of any place . whence it will follow : that whatsoever is in this book declared concerning the projection of the houre points of the equinoctiall circle upon any plain , &c. the same may be applied to the projection of azimuths or points of the horizon upon any plain , &c. onely mutatis mutandis , ( i.e. ) instead of the equinoctiall and poles of the world , you are to use the horizon and the poles of it , ( i.e. ) zenith and nadir . then for the zodiac ( which depends upon the scale or tangents of declination , which exceed not 23 and a half gr . you are to use ( in the azimuthal work ) a scale of tangents ( rightly limited ) going up from oo to so much altitude as you mean to use ( if it be for the suns course , to 62 gr . here at london . ) as there you rectifie the index to the declination of the sun : so here you must ( answerably ) rectifie it to the suns altitude taken by observation , and then the index being set to the altitude gives the azimuth for that moment and no more . corollary . therefore , to finde the azimuth by this way , you must take the suns altitude , and so finde the azimuth for one moment onely . quest. but why is there so little said here of this ? answ. because it is a thing of no use , but of a great deal of trouble : for the altitude is to be observed , and the index set thereto every moment , quest. is it not so too in the dials before described ? answ. no ; for there the index set once , may serve in that place one whole day very well , because the suns declination doth not alter much in one day . quest. why then is this azimuthal businesse mentioned at all ? answ. because the reader might see , that the authour of this treatise was not ignorant of it : yet it is mentioned also that if any delight in such a curiosity he may to his liking effect it . ¶ and for a full conclusion note , that , all the former dials may be made in a craticular way . finis . an index of the chief particulars : of the elliptical diall , with an index perpendicular to the plain . upon an horizontall , or any other direct plain , see from page 8 to page 22 upon a declining plain , from 22 to 29 another way to prick down and divide the ellipsis , 29 to 34 some uses and varieties of this elliptical diall , 34 to 37 some varieties of the structure of it , 37 to 47 an advertisement concerning some other uses of the last description , 47 to 53 a demonstration of what soever went before , 53 to 69 of the elliptical diall with an index standing in the zenith-line . upon any plain , or other curved superficies : done by projection , from page 69 to 80 to any superficies , and to an index casually set : done also by projection , 80 to 112 a demonstrations of all , 112 to 115 circular horologiographie , performed and demonstrated , 115 to 133 rectilineal horologiography , how done , and demonstrated , 133 to 169 elliptical horologiography , 169 upon all plain ( not curved ) superficies to an index ( casually , or by election ) placed any way : performed by spherical operations , ( not by projection , and demonstrated , 169 to 201 a note concerning azimuths how they may be found by such kinde of dials made in analogie to the former , 201 all the former works may be craticularly performed , 202 finis . courteous reader , be pleased to take notice of these books following , which are very usefull and necessary for all merchants , tradesmen , accomptants , they are sold by nicholas bourn at the south entrance of the royall exchange . introduction to merchants accompts , compiled by iohn gollines , student in the mathematicks , and professour of merchants accompts . tabula foeneratoria , or tables for the forbearance and discompts of money ; likewise tables for the forbearance , discompt and purchase of annuities to 31 years , at the rate of 6 per centum , per annum , calculated by roger clavell , gent. student in the mathematicks . manuel of millions or accompts ready cast up , whereby you may both suddenly and truly know the value of any commoditie at any price whatsoever : the second edition : to which is aded a second part of very great use , by richard hodgetts . posthuma fosteri , the description of a ruler , upon which is inscribed divers scales , and the uses thereof ; with divers propositions in astronomie , navigation and dialling , with the delineating of horizontal dialls , by samuel foster , late professour of astronomie in gresham-colle●ge . these following are books of other sorts : bread for the poor , an advancement of the english nation ; promised by the inclosure of the wastes and common grounds of england , by adam moore , gent. bills of lading in all languages , for shipping forth of merchants goods to all countreys of trading . indentures for virginia , bermoodas , new-england , and all other new plantations . a confutation of athisme , by dr. dove . bishop downame , on the lords prayer . popular errours in physick , by dr. primrose . sweedish wars compleat in four volumns , 4o. youths guide , or the fathers legacie to his son. dr. preston , of faith and love. merchants mirrour , or directions for the perfect booking or surveying of his estate ; framed by way of debitor and creditor , after the so termed italian manner ; and compiled by richard dafforne , of london accomptant . certain sermons , preached by dr. preston at cambridge and london upon these texts following , 2d to the chronicles 7 , 14.5 of the ephesians , 32. and the 5 to the ephesians , 22 , 23 , 24. called the golden scepter , 4o. dr. featly , handmaid to devotion . surgions mate , by mr. woodall , folio . notes, typically marginal, from the original text notes for div a40031-e51860 * subindicall line i call that line upon the plain which lies perpendicularly under the index , as the substilar doth under the stile . i mean perpendicularly in respect of the plain , not of the horizon . if this way of finding the two points i and x , seeme not so expedite , you must remember that all the work here is done without the two forementioned calculated tables . mathematicall recreations. or, a collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. not vulgarly manifest till now. written first in greeke and latin, lately compi'ld in french, by henry van etten, and now in english, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. invented and written by william oughtred. récréation mathématique. english. 1653 approx. 447 kb of xml-encoded text transcribed from 177 1-bit group-iv tiff page images. text creation partnership, ann arbor, mi ; oxford (uk) : 2005-12 (eebo-tcp phase 1). a48262 wing l1790 estc r217635 99829293 99829293 33730 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. a48262) transcribed from: (early english books online ; image set 33730) images scanned from microfilm: (early english books, 1641-1700 ; 1992:7) mathematicall recreations. or, a collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. not vulgarly manifest till now. written first in greeke and latin, lately compi'ld in french, by henry van etten, and now in english, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. invented and written by william oughtred. récréation mathématique. english. oughtred, william, 1575-1660. aut [40], 286, [17] p. : ill. printed for william leake, at the signe of the crown in fleetstreet, between the two temple gates, london : m d c liii. [1653] translation of: jean leurechon. recreation mathematique. henry van etten is a pseudonym of jean leurechon. with an added engraved title page reading: mathematicall recreations or a collection of sundrie excellent problemes out of ancient and moderne phylosophers. "the description and use of the double horizontall dyall" has separate title page dated 1652; register is continuous. running title reads: mathematicall recreation. 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 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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 science -problems, exercises, etx. -early works to 1800. mathematics -problems, exercises, etc. -early works to 1800. fireworks -early works to 1800. scientific recreations -early works to 1800. sundials -early works to 1800. 2005-02 tcp assigned for keying and markup 2005-05 spi global keyed and coded from proquest page images 2005-06 jonathan blaney sampled and proofread 2005-06 jonathan blaney text and markup reviewed and edited 2005-10 pfs batch review (qc) and xml conversion mathematicall recreations . or , a collection of many problemes , extracted out of the ancient and modern philosophers , as secrets and experiments in arithmetick , geometry , cosmographie , horologiographie , astronomie , navigation , musick , opticks , architecture , stati●k , mechanicks , chemistry , water-works , fire-works , &c. not vulgarly manifest till now . written first in greeke and latin , lately compi'ld in french , by henry van etten , and now in english , with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring : and the double horizontall diall . invented and written by william oughtred . london : printed for william leake , at the signe of the crown in fleetstreet , between the two temple gates , mdcliii . on the frontispice and booke . all recreations do delight the minde , but these are best being of a learned kinde : here art and nature strive to give content , in shewing many a rare experiment , which you may read , & on their schemes here look both in the frontispice , and in the book . upon whose table new conceits are set , like dainty dishes , thereby for to whet and winne your judgement , with your appetite to taste them , and therein to taka delight . the senses objects are but dull at best , but art doth give the intellect a feast . come hither then , and here i will describe , what this same table doth for you provide . here questions of arithmetick are wrought , and hidden secrets unto light are brought , the like it in geometrie doth unfold , and some too in cosmographie are told : it divers pretty dyals doth descrie , with strange experiments in astronomie , and navigation , with each severall picture , in musick , opticks , and in architecture : in statick , machanicks , and chymistrie , in water-works , and to ascend more hie , in fire-works , like to joves artillerie . all this i know thou in this book shalt finde , and here 's enough for to content thy minde . for from good authors , this our author drew these recreations , which are strange , and true so that this book 's a centre , and t is fit , that in this centre ; lines of praise should meet w. mathematicall recreations or a collection of sundrie excellent problemes out of ancient & moderne phylosophers both vsefull and recreatiue printed for william leake and are to be solde at the crowne in fleet streete betweene the two temple gates . to the thrice noble and most generous lo. the lo. lambert verreyken , lo. of hinden , wolverthem , &c. my honourable lo. amongst the rare and curious propositions which i have learned out of the studies of the mathematicks in the famous university of pont a mousson , i have taken singular pleasure in certaine problemes no lesse ingenious than recreative , which drew me unto the search of demonstrations more difficult and serious ; some of which i have amassed and caused to passe the presse , and here dedicate them now unto your honour ; not that i account them worthy of your view , but in part to testifie my affectionate desires to serve you , and to satisfie the curious , who delight themselves in these pleasant studies , knowing well that the nobilitie , and gentrie rather studie the mathematicall arts , to content and satisfie their affections , in the speculation of such admirable experiments as are extracted from them , than in hope of gaine to fill their purses . all which studies , and others , with my whole indevours , i shall alwayes dedicate unto your honour , with an ardent desire to be accounted ever , your most humble and obedient nephew and servant , h. van etten . by vvay of advertisement . five or six things i have thought worthy to declare before i passe further . first , that i place not the speculative demonstrations with all these problems , but content my self to shew them as at the fingers end : which was my plot and intention , because those which understand the mathematicks can conceive them easily ; others for the most part will content themselves onely with the knowledge of them , without seeking the reason . secondly , to give a greater grace to the practice of these things , they ought to be concealed as much as they may , in the subtiltie of the way ; for that which doth ravish the spirits is , an admirable effect , whose cause is unknowne : which if it were discovered , halfe the pleasure is l●st ; therefore all the finenesse consists in the dexterity of the act , concealing the meanes , and changing often the streame . thirdly , great care ought to be had that one deceive not himselfe , that would declare by way of art to deceive another : this will make the matter contemptible to ignorant persons , which will rather cast the fault upon the science , than upon him that shewes it : when the cause is not in the mathematicall principles , but in him that failes in the acting of it . fourthly , in certaine arithmeticall propositions they have onely their answers as i found them in sundry authors , which any one being studious of mathematicall learning , may finde their originall , and also the way of their operation . fifthly , because the number of these problemes , and their dependances are many , and intermixed , i thought it convenient to gather them into a table : that so each one according to his fancie , might make best choise of that which might best please his palate , the matter being not of one nature , nor of like subtiltie : but whosoever will have patience to read on , shall finde the end better than the beginning . to the reader . it hath been observed by many , that sundry fine wits as well amongst the ancient as moderne , have sported and delighted themselves upon severall things of small consequence , as upon the foot of a fly , upon a straw , upon a point , nay upon nothing ; striving as it were to shew the greatnesse of their glory in the smalnesse of the subject : and have amongst most solid and artificiall conclusions , composed and produced sundry inventions both philosophicall and mathematicall , to solace the minde , and recreate the spirits , which the succeeding ages have imbraced , and from them gleaned and extracted many admirable , and rare conclusions ; judging that borrowed matter often-times yeelds praise to the industry of its author . hence for thy use ( courteous reader ) i have with great search and labour collected also , and heaped up together in a body of these pleasant and fine experiments to stirre up and delight the affectionate , ( out of the writings of socrates , plato , aristotle , demosthenes , pythagoras , democrates , plinie , hyparchus , euclides , vitruvius , diaphantus , pergaeus , archimedes , papus alexandrinus , vitellius , ptolomaeus , copernicus , proclus , mauralicus , cardanus , valalpandus , kepleirus , gilbertus , tychonius , dureirus , josephus , clavius , gallileus maginus , euphanus tyberill , and others ) knowing art imitating nature that glories alwayes in the variety of things , which she produceth to satisfie the minde of curious inquisitors . and though perhaps these labours to some humourous persons may seeme vaine , and ridiculous , for such it was not undertaken : but for those which intentively have desired and ●ought after the knowledge of those things , it being an invitation and motive to the search of greater matters , and to imploy the minde in usefull knowledge , rather than to be busied in vaine pamphlets , play-books , fruitlesse legends , and prodigious histories that are invented out of fancie , which abuse many noble spirits , dull their wits , & alienate their thoughts from laudable and honourable studies . in this tractate thou maist therefore make choise of such mathematicall problemes and conclusions as may delight thee , which kinde of learning doth excellently adorne a man ; seeing the usefulnesse thereof , and the manly accomplishments it doth produce , is profitable and delightfull for all sorts of people , who may furnish and adorne themselves with abundance of matter in that kinde , to help them by way of use , and discourse . and to this we have also added our pyrotechnie , knowing that beasts have for their object only the surface of the earth ; but hoping that thy spirit which followeth the motion of fire , will abandon the lower elements , and cause thee to lift up thine eyes to soare in an higher contemplation , having so glittering a canopie to behold , and these pleasant and recreative fires ascending may cause thy affections also to ascend . the whole whereof we send forth to thee , that desirest the scrutability of things ; nature having furnished us with matter , thy spirit may easily digest them , and put them finely in order , though now in disorder . a table of the particular heads of this book , contracted according to the severall arts specified in the title-page . experiments of arithmetick . page 1 , 2 , 3 , 16 , 19 , 22 , 28 , 33 , 39 , 40 , 44 , 45 , 51 , 52 , 53 , 59 , 60 , 69 , 71 , 77 , 83 , 85 , 86 , 89 , 90 , 91 , 124 , 134 , 135 , 136 ▪ 137 , 138 , 139 , 140 , 178 179 , 181 , 182 , 183 , 184 , 185 , 188 , 208 , 210 , 213. experiments ●n geometrie . pag. 12 , 15 , 24 , 26 , 27 , 30 , 35 , 37 , 41 , 42 , 47 , 48 , 49 , 62 , 65 , 72 , 79 , 82 , 113 , 117 , 118 , 119 , 214 , 215 , 217 , 218 , 234 , 235 , 236 , 239 , 240. experiments in cosmographie . pag. 14 , 43 , 75 , 106 , 107 , 219 , 220 , 225 , 227 , 228 , 229 , 230 , 232. experiments in horologiographie . pag. 137 , 166 , 167 , 168 , 169 , 171 , 234. experiments in astronomie . pag. 220 , 221 , 222 , 223 , 224. experiments in navigation . pag. 105 , 233 , 234 , 237 , 238. experiments in musick . pag. 78 , 87 , 126. experiments in opticks . pag. 6 , 66 , 98 , 99 , 100 , 102 , 129 , 131 , 141 , 142 , 143 , 144 , 146 , 149 , 151 , 152 , 153 , 155 , 156 , 157 , 158 , 160 , 161 , 162 , 163 , 164 , 165. experiments in architecture . pag. 16 , 242 , 243. experiments in staticke . pag. 27 , 30 , 32 , 71 , 199 , 200 , 201 , 283 , 204 , 205 , 207. experiments in machanicks . pag. 56 , 58 , 68 , 88 , 95 , 108 , 110 , 128 , 173 , 174 , 176 , 246 , 248 , 258 , 259. experiments in chymistrie . pag. 198 , 255 , 256 , 257 , 260 , 262 , 263 , 264. experiments in water-workes . pag. 190 , 191 , 192 , 193 , 194 , 196 , 247 , 249 , 250 , 252 , 253. experiments in fireworkes . from page , 265. to the end . finis . a table of the contents , and chiefe points conteined in this book . problem . ii. how visible objects that are without , and things that passe by , are most lively represented to those that are within . page 6 prob. 1 of finding of numbers conceived in the minde . 1 , 2 , 3 prob. 5 of a geographicall garden-plot fit for a prince or some great personage . 14 prob. 37 any liquid substance , as water or wine , placed in a glasse , may be made to boile by the motion of the finger , and yet not touching it . 54 prob. 3 how to weigh the blow of ones fist , of a mallet , a hatchet or such like . 9. prob. 30 two severall numbers being taken by two sundry persons , how subtilly to discover which of those numbers each of them took . 46 prob. 4 that a staffe may be broken ▪ placed upon two glasses , without hurting of the glasses . 12 prob. 7 how to dispose lots that the 5 , 6 , 9 , &c. of any number of persons may escape . 16 prob. 13 how the weight of smoke of a combustible body , which is exhaled , may be weighed . 27 prob. 12 of three knives which may be so disposed to hang in the aire , and move upon the point of a needle . 27 prob. 17 of a deceitfull bowle , to bowle withall . 32 prob. 16 a ponderous or heavy body may be supported in the aire without any one touching it . 30 prob. 18 how a peare , or apple , may be parted into any parts , without breaking the rinde thereof . 33 prob. 15 of a fine kinde of dore which opens and shuts on both sides . 30 prob. 9 how the halfe of a vessell which containes 8 measures may be taken , being but onely two other measures , the one being 3 , and the other 8 measures . 22 prob. 8 three persons having taken each of them severall things , to finde which each of them hath taken . 19 prob. 6 how to dispose three staves which may support each other in the aire . 15 prob. 14 many things being disposed circular ( or otherwise ) to finde which of them any one thinks upon . 28 prob. 19 to finde a number thought upon without asking questions . 33 prob. 11 how a milstone or other ponderosity may hang upon the point of a needle without bowing , or any wise breaking of it . 26 prob. 20 and 21 how a body that is uniforme and inflexible may passe through a hole which is round , square and triangular ; or round , square and ovall-wise , and exactly fill those severall holes . 35 , 37 prob. 10 how a stick may stand upon ones finger , or a pike in the middle of a court without falling . 24 prob. 22 to finde a number thought upon after another manner than those which are formerly delivered . 39 prob. 23 to finde out many numbers that sundry persons or any one hath thought upon . 40 prob. 24 how is it that a man in one & the same time may have his head upward , and his feet upward , being in one and the same place ? 4● prob. 25 of a ladder by which two men ascending at one time , the more they ascend , the more they shal be asunder , notwith standing the one be as high as the other . 42 prob. 26 how is it that a man having but a rod or pole of land , doth brag that he may in a right line passe from place to place 3000 miles . 42 prob. 27 how is it that a man standing upright , and looking which way he will , he looketh true north or south . 43 prob. 28 to tell any one what number remaines after certaine operations being ended , without asking any question . 44 prob. 29 of the play with two severall things . 45 prob. 31 how to describe a circle that shall touch 3 points placed howsoever upon a plaine , if they be not in a right line . 47 prob. 32 how to change a circle into a square forme . 48 prob. 33 with one and the same compasses , and at one and the same extent or opening , how to describe many circles concentricall , that is , greater or lesser one than another . 49 prob. 34 any number under 10. being thought upon , to finde what numbers they were . 51 prob , 35 of the play with the ring . 52 prob. 36 the play of 3 , 4 , or more dice . 53 prob. 38 of a fine vessell which holds wine or water being cast into it at a certain height , but being filled higher it will runne all out of its owne accord . 56 prob. 39 of a glasse very pleasant . 58 prob. 40. if any one should hold in each hand as many pieces of money as in the other , how to finde how much there is . 59 prob. 41 many dice being cast , how artificially to discover the number of the points that may arise . 60 prob. 42 two metals as gold and silver or of other kinde , weighing alike , being privately placed into two like boxes , to finde in which of them the gold or silver is . 62 prob. 43 two globes of divers metals ( as one gold the other copper ) yet of equall weight , being put in a box as b.g. to finde in which end the gold or copper is . 65 prob. 44 how to represent divers sorts of rainbowes here below . 66 prob. 45 how that if all the powder in the world were inclosed in a bowle of paper or glasse , and being fired on all parts , it could not break that bowle . 68 prob. 46 to finde a number which being divided by 2. there will remaine 1. being divided by 3. there will remaine 1. and so likewise being divided by 4 , 5 , or 6. there will still remaine one , but being divided by 7 will remaine nothing . 69 prob. 47 one had a certaine number of crownes , and counting them by 2 and 2 , there rested 1. counting them by 3 , and 3 , there rested 2. counting them by 4 , and 4 , there rested 3. counting them by 5 , and 5 , there rested 4. counting them by 6 , and 6 , there rested 5. but counting them by 7 and 7 , there rested nothing , how many crownes might he have ? 71 prob. 48 how many sorts of weights in the least manner must there be to weigh all sorts of things betweene one pound and 121 pound , and so unto 364 pound ? 71 prob. 49 of a deceitfull balance which being empty seems to be just , because it hangs in aequilibrio , notwithstanding putting 12 pound in one ballance , and 11 in the other , it will remaine in aequilibrio . 72 prob. 50 to heave or lift up a bottle with a straw . 74 prob. 51 how in the middle of a wood or desert , without the sight of the sun , starres , shadow , or compasse , to finde out the north , or south , or the 4 cardinal points of the world , east , west , &c. 75 prob. 52 three persons having taken counters , cards , or other things , to finde how much each one hath taken . 7● prob. 53 how to make a consort of musick of many parts with one voice or one instrument onely . 78 prob. 54 to make or describe an oval form , or that which is neare resembled unto it at one turning , with a paire of common compasses . 79 prob. 55 of a purse difficult to be opened . 80 prob. 56 whether is it more hard and admirable without compasses to make a perfect circle , or being made to finde out the centre of it ? 82 prob. 56 any one having taken 3 cards , to finde how many points they containe . 83 prob. 57 many cards placed in divers ranks , to finde which of those cards any one hath thought . 85 prob. 58 many cards being offered to sundry persons to finde which of those cards any one thinketh upon . 86 prob. 59 how to make an instrument that helps to heare , as gallileus made to help to see . 87 prob. 60 of a fine lamp which goeth not out , though one carries it in ones pocket , or being rolled on the ground will still burne . 88 prob. 61 any one having thought a card amongst many cards , how artificially to discover it out . 89 prob. 62 three women a , b , c. carried apples to a market to sell : a had 20. b had 30. c 40. they sold as many for a penny one as the other , and brought home one as much money as another , how could this be ? 90 prob. 63 of the properties of some numbers . 91 prob. 64 of an excellent lamp which serves or furnisheth it selfe with oile , and burnes a long time . 95 prob. 65 of the play at keyles or nine-pins . 97 prob. 66 of spectacles of pleasure of spectacles which give severall colours to the visage . 98 of spectacles which make a towne seeme to be a city , one armed man as a company , and a piece of gold as many pieces . 99 how out of a chamber to see the objects which passe by according to the lively perspective . 100 of gallileus admirable optick-glasse , which helps one to see the beginning and ending of eclipses , the spots in the sunne , the starres which move about the planets , and perspicuously things far remote . of the parts of gallileus his glasse . 102 prob. 67 of the magnes and needles touched therewith . how rings of iron may hang one by another in the aire . 103 of mahomets tombe which hangs in the aire by the touch of the magnes . 104 how by the magnes only to finde out north and south 105 of a secrecie in the magnes , for discovering things farre remote . 106 of finding the poles by the magnes 107 prob. 68 of the properties of aeolipiles or bowles to blow the fire . 108 prob. 69 of the thermometer , or that which measures the degrees of heat and cold by the aire . 110 of the proportion of humane bodies , of statues , of colosses , or huge jmages and monstrous giants . 113 of the commensuration of the parts of the bodie the one to the other in particular , by which the lion was measured by his claw , the giant by his thumbe , and hercules by his foot . 115 , 116 of statues or colosses , or huge images ; that mount athos metamorphosed by dynocrites into a statue , in whose hand was a towne able to receive ten thousand men . 117 of the famous colossus at rhodes which bad 70 cubits in height , and loaded 900. camels , which weighed 1080000 l. 118 of nero his great colossus which had a face of 12 foot large . 119 of monstrous giants of the giant og and goliah . 119 , 120 of the carkasse of a man found which was in length 49 foot ; and of that monster found in creet , which had 46. cubits of height . 120 of campesius his relation of a monster of 300 foot found in sicile , whose face according to the former proportion should be 30 foot in length . 121 prob. 71 of the game at the palme , at trap , at bowles , paile-maile , and others . 122 prob. 72 of the game of square formes . 124 prob. 73 how to make the string of a viol sensibly shake without any one touching it . 126 prob. 74 of a vessell which containes 3 severall kindes of liquor , all put in at one bung-hole , and drawne out at one tap severally without mixture . 128 prob. 75 of burning-glasses . archimedes his way of burning the ships of syracuse . 129 of proclus his way , and of concave and sphericall glasses which burne , the cause and demonstration of burning with glasses . 131 of maginus his way of setting fire to powder in a mine by glasses . 131 of the examination of burning by glasses . 133 prob. 76 of pleasant questions by way of arithmetick . of the asse and the mule. 134 of the number of souldiers that fought before old troy. 135 of the number of crownes that two men had . 136 about the houre of the day . 137 of pythagoras schollers . 137 of the number of apples given amongst the graces and the muses . 138 of the testament or last will of a dying father . 138 of the cups of croesus . 139 of cupids apples . 139 of a mans age. 140 of the lion of bronze placed upon a fountaine with his epigram . ibid. prob. 77 in opticks , excellent experiments . principles touching reflections . 141 experiments upon flat and plaine glasses . 142 how the images seeme to sink into a plaine glasse , and alwayes are seene perpendicular to the glasse , an● also inversed . 143 the things which passe by in a street may by help of a plaine glasse be seen in a chamber , and the height of a tower or tree observed . 143 how severall candles from one candle are represented in a plaine glasse , and glasses alternately may be seene one within another , as also the back-parts of the body as well as the fore-parts are evidently represented . 144 how an image may be seene to hang in the aire by help of a glasse : and writing read or easily understood . 146 experiments upon gibbous , or convex sphericall glasses . how lively to represent a whole city , fortification , or army , by a gibbous glasse . 147 how the images are seen in concave glasses . 149 how the images are transformed by approaching to the centre of the glasse , or point of concourse ; and of an exceeding light that a concave glasse gives by help of a candle . 151 how the images , as a man , a sword , or hand , doth come forth out of the glasse . 152 , 153 of strange apparitions of images in the aire , by help of sundry glasses . 152 , 154 of the wonderfull augmentation of the parts of mans body comming neare the point of inflammation , or centre of the glasse . 155 how writing may be reverberated from a glasse upon a vvall , and read. 156 how by help of a concave glasse to cast light into a campe , or to give a perspective light to pyoneers in a mine , by one candle only . 156 how excellently by help of a concave glasse and a candle placed in the centre , to give light to read by . 157 of other glasses of pleasure . 158 of strange deformed representations by glasses ; causing a man to have foure eyes , two mouthes , two noses , two heads . of glasses which give a colour to the visage , and make the face seeme faire and foule . 160 prob. 78 how to shew one that is suspicious , what is in another chamber or roome , notwithstanding the interposition of that wall . 160 corolary , 1. to see the besiegers of a place , upon the rampa●●t of a fortification 161 corolary 2. and 3. notwithstanding the interposition of vvalls and chambers , by help of a glasse things may be seen , which passe by . 162 prob. 79 how with a musket to strike a marke not looking towards it , as exactly as one aimed at it . 162 how exactly to shoot out of a mu●ket to a place which is not seene , being hindred by some obstacle or other interposition . 163 prob. 80 how to make an image to be seen hanging in the aire , having his head downward . 164 prob. 81. how to make a company of representative souldiers seeme to be as a regiment , or how few in number may be multiplyed to seem to be many in number . 165 corolarie . of an excellent delightfull cabinet made of plaine glasses . 165 prob. 82 of fine and pleasant dyalls in horologiographie . of a dyall of herbs for a garden . 166 of the dyall upon the finger and hand , to finde what of the clock it is . 167 of a dyall which was about an obelisk at rome . 168 of dyals with glasses . 168 of a dyall which hath a glasse in the place of the stile . 169 of dyals with water , which the ancients use● 171 prob. 83 of shooting out of cannons or great artillery . how to charge a cannon without powder . 173 to finde how much time the bullet of a cannon spends in the aire before it falls to the ground . 174 how it is that a cannon shooting upward , the bullet flies with more violence , than being shot point blanke , or shooting downeward . 174 vvhether is the discharge of a cannon so much the more violent , by how much it hath the more length ? 176 prob. 84 of prodigious progressions , and multiplications of creatures , plants , fruits , numbers , gold , silver , &c. of graines of mustardseed , and that one graine being sowne , with the increase thereof for 20 yeares will produce a heap greater than all the earth a hundred thousand times . 178 of pigges , and that the great turke with all his revenne , is not able to maintaine for one yeare , a sow with all her increase for 12 yeares . 179 of graines of corne , and that 1 graine with all its increase for 12 yeares , will amount to 244140625000000000000 graines , which exceeds in value all the treasures in the world. 183 of the wonderfull increase af sheepe . 182 of the increase of cod-fish . 182 of the progressive multiplication of soules ; that from one of noahs sonnes , from the flood unto nimrods monarchie , should be produced 111350 soules . 183 of the increase of numbers in double proportion , and that a pin being doubled as often as there are weekes in the yeare , the number of pinnes that should arise is able to load 45930 ships of a thousand tunne apiece , which are worth more than tenne hundred thousand pounds a day . 183 , 184 of a man that gathered apples , stones , or such like upon a condition . 185 of the changes in bells , in musicall instruments , transmutation of places , in numbers , letters , men and such like ▪ 185 of the wonderfull interchange of the letters in the alphabet : the exceeding number of men , and time to expresse the words that may be made with these letters , and the number of books to comprehend them . 187 , 188 of a servant hired upon certaine condition , that he might have land lent him to sowe one graine of corne with its increase for 8 yeares time , which amounted to more than four hundred thousand acres of land. 188 prob. 85 of fountaines , hydriatiques ; stepticks , machinecks , and other experiments upon water , or other liquor . first , how water at the foot of a mountaine may be made to ascend to the top of it , and so to descend on the other side of it 190 secondly , to finde how much liquor is in a vessell , onely by using the tap-hole . 191 thirdly , how is it , that a vessell is said to hold more water at the foot of a mountaine , then at the top of it 191 4 how to conduct water from the top of one mountaine to the top of another 192 5 of a fine fountaine which spouts water very high and with great violence , by turning of a cock 193 6 of archimedes screw which makes water ascend by descending . 194 7 of a fine fountaine of pleasure . 196 8 of a fine watering pot for gardens . 197 9 how easily to take wine out of a vessell at the bung hole without piercing a hole in the vessell . 198 10 how to measure irregular bodies by help of water . 198 11 to finde the weight of water . 199 12 to finde the charge that a vessell may carry , as ships , boats or such like . 200 13 how comes it that a ship having safely sailed in the vast ocean , and being come into the port or harbour , will sinke down right . 200 14 how a grosse body of metall may swim upon the water . 201 15 how to weigh the lightnesse of the aire . 203 16 being given a body , to mark it about , and shew how much of it will sink in the water , or swim above the water . 204 17 to finde how much severall metalls or other bodies do weigh lesse in the water than in the aire . 204 18 how is it that a ballance having like weight in each scale , and hanging in aequilibrio in the aire , being removed from that place ( without diminishing the weights in each balance , or adding to it ) it shall cease to hang in aequilibrio sensibly , yea by a great difference of weight . 205 19 to shew what waters are heavier one than another , and how much . 206 20 how to make a pound of water weigh as much as 10 , 20 , 30 , or a hundred pound of lead , nay as much as a thousand or ten thousand pound weight . 207 prob. 86. of sundry questions of arithmetick , and first of the number of sands calculated by archimedes and clavius . 208 2 divers metalls being melted together in one body , to finde the mixture of them . 210 3 a subtile question of three partners about equality of wine and vessels . 213 4 of a ladder which standing upright against a wall of 10 foot high , the foot of it is pulled out 6 foot from the wall upon the pavement , how much hath the top of the ladder descended . 214 prob. 87 witty suits or debates between caius and sempronius , upon the forme of figures , which geometricians call isoperimeter , or equall in circuit , or compasse . 214 1 incident : of changing a field of 6 measures square , for a long rectrangled fiel of 9 measures in length and 3 in breadth : both equall in circuit but not in quantity . 215 2 incident : about two sacks each of them ho●ding but a bushell , and yet were able to hold 4 bushels . 217 3 incident : sheweth the deceit of pipes which conveygh water , that a pipe of two inches diameter , doth cast out foure times as much water as a pipe of one such diameter . 218 7 heapes of corne of 10 foot every way , is not as much as one heap of corne of 20 foot every way . 218 prob. 88 of sundry questions in matter of cosmographie , and astronomy . in what place the middle of the earth is supposed to be . 219 of the depth of the earth , and height of the heavens , and the compasse of the world , how much . 219 how much the starry firmament , the sun , and the moone are distant from the centre of the earth . 220 how long a mill-stone would be in falling to the centre of the earth from the superficies , if it might have passage thither . 220 how long time a man or a bird may be in compassing the whole earth . 220 if a man should ascend by supposition 20 miles every day : how long it would be before he approach to the moone . 221 the sunne moves more in one day than the moone in 20 dayes . 221 if a milstone from the orbe of the sun should descend a thousand miles in an houre how long it would be before it come to the earth . 221 of the sunnes quick motion , of more than 7500 miles in one minute . 221 of the rapt and violent motion of the starry firmament , which if a horseman should ride every day 40 miles , he could not in a thousand yeares make such a distance as it moves every houre . 221 to finde the circle of the sunne by the fingers . 223 prob. 93 of finding the new and full moone in each moneth . 224 prob. 94 to finde the latitude of countreys . 225 prob. 95 of the climates of countreys , and how to finde them . 225 prob. 96 of longitude and latitude of the places of the earth , and of the starres of the heavens . 227 to finde the longitude of a countrey . 228 of the latitude of a countrey . 229 to finde the latitude of a countrey . 230 to finde the distance of places . 230 of the longitude , latitude , declination , and distance of the starres . 231 how is it that two horses or other creatures comming into the world at one time , and dying at one and the same instant , yet the one of them to be a day older than the other ? 232 certaine fine observations . in what places of the world is it that the needle hangs in aequilibrio , and verticall ? 233 in what place of the world is it the sun is east or west but twice in the yeare ? 233 in what place of the world is it that the sunnes longitude from the equinoctiall paints and altitude , being equall , the sunne is due east or west ? that the sunne comes twice to one point of the compasse in the forenoone or afternoone . 233 that in some place of the world there are but two kindes of winde all the yeare . 233 two ships may be two leagues asunder under the equinoctiall , and sayling north at a certaine parallell they will be but just halfe so much . 233 to what inhabitants , and at what time the sunne will touch the north-part of the horizon at midnight . 234 how a man may know in his navigation when he is under the equinoctiall . 234 at what day in the yeare the extremitie of the styles shadow in a dyall makes a right line . 234 what height the sunne is of , and how far from the zenith , or horizon , when a mans shadow is as long as his height . 234 prob. 97 to make a triangle that shall have three right angles . 234 prob. 98 to divide a line in as many parts as one will , without compasses or without seeing of it . 235 prob. 99 to draw a line which shall incline to another line , yet never meet against the axiome of parallells . 236 prob. 100 to finde the variation of the compasse by the sunne shining . 237 prob. 101 to know which way the winde is in ones chamber without going abroad . 238 prob. 102 how to draw a parallel sphaericall line with great ease . 239 prob. 103 to measure an height onely by help of ones hat. 240 prob. 104 to take an height with two strawes . 240 in architecture how statues or other things in high buildings shall beare a proportion to the eye below either equall , double , &c. 242 prob. 106 of deformed figures which have no exact proportion , where to place the eye to see them direct . 243 prob. 107 how a cannon that hath shot may be covered from the battery of the enemy . 244 prob. 108 of a fine lever , by which one man alone may place a cannon upon his carriage . 245 prob. 109 how to make a clock with one wheele 246 of water-workes . prob. 110 how a childe may draw up a hogshead of water with ease . 247 prob. 111 of a ladder of cords to cary in ones pocket , by which he may mount a wall or tower alone . 248 prob. 112 of a marvelous pump which drawes up great quantity of water . 249 prob. 113 how naturally to cause water to ascend out of a pit. 250 prob. 114 how to cast water out of a fountaine very high . 252 prob. 115 how to empty the water of a pit by help of a cisterne . 253 prob. 116 how to spout out water very high . 253 prob. 117 how to re-animate simples though brought a thousand miles . 255 prob. 118 how to make a perpetuall motion . 255 prob. 119 of the admirable invention of making the philosophers tree , which one may see to grow by little and little . 256 prob. 120 how to make the representation of the great world 257 prob. 121 of a cone , or pyramidall figure that moves upon a table 258 prob. 122 how an anvill may be cleaved by the blow of a pistoll . 258 prob. 123 how a capon may be rosted in a mans travells at his sa●●le-bowe . 259 prob. 124 how a candle may be made to burne three times longer than usually it doth 259 prob. 125 how to draw wine out of water 260 prob. 126 of two marmouzets , the one of which lights a candle , and the other blowes it out . 261 prob. 127 how to make wine fresh without ice or snow in the height of summer . 262 prob. 128 to make a cement which lastes as marble , resisting aire and water . 262 prob. 129 how to melt metall upon a shell with little fire . 263 prob. 130 of the hardning of iron and steele . 263 prob. 131 to preserve fire as long as you will , imitating the inextinguible fire of the vestales . 264 finis . ad authorem d.d. henricum van etenium , alumnum academiae ponta mousson . ardua walkeri sileant secreta profundi , desinat occultam carpere porta viam . itala cardani mirata est lampada docti terra , syracusium graecia tota senem : orbi terrarum , ptolemaei clepsydra toti , rara dioptra procli , mira fuêre duo , anglia te foveat doctus pont-mousson alumnum : quidquid naturae , qui legis , hortus habet . docta , coronet opus doctum , te sit tua docto digna , syracusii , arca , corona , viri . arca syracusiis utinam sit plumbea servis , aurea sed dominis , aurea tota suis. mathematical recreation . problem i. to finde a number thought upon . bid him that he quadruple the number thought upon , that is , multiply it by 4 , and unto it bid him to adde 6 , 8 , 10 , or any number at pleasure : and let him take the halfe of the sum , then ask how much it coms to , for then if you take away half the number from it which you willed him at first to add to it , there shall remain the double of the number thought upon . example the number thought upon 5 the quadruple of it 20 put 8 unto it , makes 28 the halfe of it is 14 take away halfe the number added from it , viz 4 , the rest is 10 the double of the number thought upon , viz. 10 another way to finde what number was thought upon . bid him which thinketh double his number , and unto that double adde 4 , and bid him multiply that same product by 5 , and unto that product bid him adde 12 , and multiply that last number by 10 ( which is done easily by setting a cypher at the end of the number ) then ask him the last number or product , and from it secretly subtract 320 , the remainder in the hundreth place , is the number thought upon . example . the number thought upon 7 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . his double 14 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . to it add 4 , makes 18 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . which multiplyed by 5 makes 90 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . to which add 12 makes 102 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . this multiplyed by 10 which is only by adding a cypher to it , makes 1020 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . from this subtract 320 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . rest 700 for which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon . to finde numbers conceived upon , otherwise than the former . bid the party which thinks the number , that he triple his thought , and cause him to take the half of it : ( if it be odde take the least half , and put one unto it : ) then will him to triple the half , and take half of it as before : lastly , ask him how many nines there is in the last half , and for every nine , account four in your memory , for that shall shew the number thought upon , if both the triples were even : but if it be odde at the first triple , and ev●n at the second , for the one added unto the least halfe keep one in memory : if the first triple be even , and the second odde , for the one added unto the least halfe keepe two in memory ; lastly , if at both times in tripling , the numbers be odde , for the two added unto the least halfes , keep three in memory , these cautions observed , and added unto as many fours as the party sayes there is nines contained in the last halfe , shall never fail you to declare or discern truly what number was thought upon . example . the number thought upon 4 or 7 the triple 12 or 21 the half thereof 6 or 10 , one put to it makes 11 the triple of the halfe 18 or 33 the halfe 9 or 1● , one put to it makes 17 the number of nines in the last halfe 1 or 1 the first 1. representeth the 4. number thought upon , and the last 1. with the caution makes 7. the other number thought upon . note . order your method so that you be not discovered , which to help , you may with dexterity and industry make additions ▪ substractions , multiplications , divisions , &c. and instead of asking how many nines there is , you may ask how many eights tens , &c. there is , or subtract 8.10 . &c. from the number which remains , for to finde out the number thought upon . now touching the demonstrations of the former directions , and others which follow , they depend upon the 2 , 7 , 8 , and 9 , books of the elements of euclide : upon which 2. book & 4. proposition this may bee extracted , for these which are more learned for the finding of any number that any one thinketh on . bid the party that thinks , that he break the number thought upon into any two parts , and unto the squares of the parts , let him adde the double product of the parts , then ask what it amounteth unto , so the root quadrat shall be the number thought upon . the number thought upon 5 , the parts suppose 3 and 2. the square of 3 makes 9 the sum of these three nūbers 25 , the squa●e root of which is 5 , the number thought upon the square of 2 makes 4 the sum of these three nūbers 25 , the squa●e root of which is 5 , the number thought upon the product of the parts . viz. 3 by 2 makes 6 , which 6 doubled makes 12 the sum of these three nūbers 25 , the squa●e root of which is 5 , the number thought upon or more compendiously it may be delivered thus . break the number into two parts , and to the product of the parts , adde the square of half the difference of the parts , then the root quadrat of the aggregate is halfe the number conceived . examination . the problems which concern arithmetick , we examine not , for these are easie to any one which hath read the grounds and principles of arithmetick , but we especially touch upon that , which tends to the speculations of physick , geometry , and optickes , and such others which are of more difficulty , and more principally to be examined and considered . problem ii. how to represent to those which are in a cham●er that which is without , or all that which passeth by , it is pleasant to see the beautifull and goodly representation of the heavens intermixed with clouds in the horizon , upon a woody scituation , the motion of birds in the aire , of men and other creatures upon the ground , with the trembling of plants , tops of trees , and such like : for every thing will be seen within even to the life , but inversed : notwithstanding , this beautifull paint will so naturally represent it self in such a lively perspective , that hardly the most accurate painter can represent the like . but here note , that they may be represented right two manner of wayes ; first , with a concave glasse : secondly , by help of another convex glasse , disposed or placed between the paper and the other glasse : as may be seen here by the figure . now i will add here only by passing by , for such which affect painting and portraiture , that this experiment may excellently help them in the lively painting of things perspectivewise , as topographicall cards , &c. and for philosophers , it is a fine secret to explain the organ of the sight , for the hollow of the eye is taken as the close chamber , the ball of the apple of the eye , for the hole of the chamber , the crystaline humor at the small of the glasse , and the bottome of the eye , for the wall or leafe of paper . examination . the species being pressed together or contracted doth not perform it upon a wall , for the species of any thing doth represent it selfe not only in one hole of a window , but in infinite holes , even unto the whole sphere , or at least unto a hemisphere ( intellectuall in a free medium ) if the beams or reflections be not interposed , and by how much the hole is made less to give passage to the species , by so much the more lively are the images formed . in convexe , or concave glasses the images will be disproportionable to the eye , by how much they are more concave , or convexe , & by how much the parts of the image comes neer to the axis , for these that are neer are better proportioned then these which are farther off . but to have them more lively and true , according to the imaginary conicall section , let the hole be no greater than a pins head made upon a piece of thin brasse , or such like , which hole represents the top of the cone , and the base thereof the term of the species : this practice is best when the sun shines upon the hole , for then the objects which are opposite to that plaine will make two like cones , and will lively represent the things without in a perfect inversed perspective , which drawn by the pensill of some artificiall painter , turn the paper upside down , and it will be direct and to the life . but the apparences may be direct , if you place another hole opposite unto the former , so that the spectator be under it ; or let the species reflect upon a concave glass , and let that glas reflect upon a paper or some white thing . problem iii. to tell how much waighs the blow of ones fist , of a mallet , hatchet , or such like , or resting without giving the blow scaliger in his 331 exercise against cardan , relates that the mathematicians of maximillian the emperour did propose upon a day this question , and promised to give the resolution ; notwithstanding ●caliger delivered it not , and i conceive it to be thus . take a balance , and let the fist , the mallet , or hatchet rest upon the scale , or upon the beam of the balance , and put into the other scale as much weight as may counterpoyse it ; then charging or laying more waight into the scale , and striking upon the other end , you may see how much one blow is heavier than another , and so consequently how much it may waigh for as aristotle saith , the motion that is made in striking adds great waight unto it , and so much the more , by how much it is quicker : therefore in effect , if there were placed a thousand mallets , or a thousand pounde waight upon a stone , nay , though it were exceedingly pressed down by way of a vice , by levers , or other mechanick engine , it would be nothing to the rigor and violence of a blow . is it not evident that the edge of a knife laid upon butter , and a hatchet upon a leafe of paper , without striking makes no impression , or at least enters not ; but striking upon the wood a little , you may presently see what effect it hath , which is from the quicknesse of the motion , which breaks and enters without resistance , if it be extream quick , as experience shews us in the blows of arrows , of cannons , thunder-boults , and such like . examination . this problem was extracted from scaliger , who had it from aristotle , but somwhat refractory compiled , & the strength of the effect he says depends only in the violence of the motion ; then would it follow that a little light hammer upon a piece of wood being quickly caused to smite , would give a greater blow , and do more hurt than a great sledge striking soft ; this is absurd , and contrary to experience : therefore it consists not totally in the motion , for if two severall hammers , the one being 20 times heavier than the other , should move with like quickness , the effect would be much different , there is then some thing else to be considered besides the motion which scaliger understood not , for if one should have asked him , what is the reason that a stone falling from a window to a place neer at hand , is not so forceable as if it fell farther 〈◊〉 when a bullet flying out of a peece and striking the mark neer at hand 〈◊〉 not make such an effect as striking 〈…〉 that scaliger and 〈…〉 this subiect ▪ would not be less troubled to resolve this , than they have been in that . problem iv. how to break a staffe which is laid upon two glasses full of water , without breaking the glasses , spilling the water , or upon two reeds or straws without breaking of them . in like manner may you doe upon two reeds , held with your hands in the aire without breaking them ▪ thence kitchin boyes often break bones of mutton upon their hand , or with a napkin without any hurt , in only striking upon the middle of the bone with a knife . now in this act , the two ends of the staffe in breaking slides away from the glasses , upon which they were placed ; hence it commeth that the glasses are no wise indangered , no more than the knee upon which a staffe is broken , forasmuch as in breaking it presseth not : as aristotle in his mechanick questions observeth . examination . it were necessary here to note , that this thing may be experimented , first , without glasses , in placing a small slender staffe upon two props , and then making tryall upon it , by which you may see how the staffe will either break , bow , or depart from his props , and that either directly or obliquely : but why by this violence , that one staffe striking another , ( which is supported by two glasses ) will be broken without offending the glasses , is as great a difficulty to be resolved as the former . problem v. how to make a faire ge●graphic●ll card in a garden plot , fit for a prince , or great personage . it is usuall amongst great men to have faire geographicall maps ▪ large cards , and great globes , that by them they may as at once have a view of any place of the world , and so furnish themselves with a generall knowledge , not only of their own kingdoms form , scituation , longitude , latitude , &c. but of all other places in the whole universe , with their magnitudes , positions , climats , and distances . now i esteem that it is not unworthy for the meditations of a prince , seeing it carries with it many profitable and pleasant contentmen●s : if such a card or map by the advice and direction of an able mathematician were geographically described in a garden plot form , or in some other convenient place , and instead of which generall description might particularly and artificially be prefigured his whole kingdoms and dominions , the mountains and hils being raised like small hillocks with turfs of earth , the valleys somwhat concave , which will be more agreeable and pleasing to the eye , than the description in plain maps and cards , within which may be presented the towns , villages , castles , or other remarkable edifices in small green mo●●e banks , or spring-work proportionall to the pl●tform , the forrests and woods represented according to their form and capacity , with herbs and stoubs , the great rivers , lakes , and ponds to dilate themselves according to their course from some artificiall fountain made in the garden to passe through chanels ; then may there be composed walks of pleasure , ascents , places of repose , adorned with all variety of delightfull herbs and flowers , both to please the eye or other senses . a garden thus accommodated shall farre exceed that of my lord of verulams specified in his ●ssayes ; that being only for delight and pleasure , this may have all the properties of that , and also for singular use , by which a prince may in little time personally visit his whole kingdom , and in short time know them distinctly : and so in like manner may any particular man geographically prefigure his own possession or heritage . problem vi. how three staves , knives , or like bodies , may be conceived to hang in the aire , without being supported by any thing but by themselves . take the first staffe ab , raise up in the aire the end b , and upon him cros-wise place the staffe cb , then lastly , in triangle wise place the third staffe ef ▪ in such manner that it may be under ab , and yet upon cd . i say that these staves so disposed cannot fall , and the space cbe is made the stronger , by how much the more it is pressed downe , if the staves break not , or sever themselves from the triangular forme : so that alwayes the center of gravitie be in the center of the triangle : for ab is supported by ef , and ef is held up by cd , and cd is kept up from falling by ab , therefore one of these staves cannot fall , and so by consequence none . problem vii . how to dispose as many men , or other things in such sort , that rejecting , or casting away the 6 , 9 , 10 part , unto a certain number , there shall remaine these which you would have . ordinarily the proposition is delivered in this wise : 15 christians and 15 turkes being at sea in one shippe , an extreame tempest being risen , the pilot of the shippe saith , it is necessary to cast over board halfe of the number of persons to disburthen the shippe , and to save the rest : now it was agreed to be done by lot , and therefore they consent to put themselves in rank , counting by nine and nine , the ninth person should alwayes be cast into the sea , untill there were halfe throwne over board ; now the pilote being a christian indeavoured to save the christians , how ought he therefore to dispose the christians , that the lot might fall alwayes upon the turkes , and that none of the christians be in the ninth place ? the resolution is ordinarily comprehended in this verse . populeam virgam mater regina ferebat . for having respect unto the vowels , making a one , e two , i three , o foure , and u five : o the first vowell in the first word sheweth that there must be placed 4. christians ; the next vowel u , signifieth that next unto the 4. christians must be placed 5 turkes , and so to place both christians and turkes according to the quantity and value of the vowels in the words of the verse , untill they be all placed : for then counting from the first christian that was placed , unto the ninth , the lot will fall upon a turk , and so proceed . and here may be further noted that this probleme is not to be limited , seeing it extends to any number and order whatsoever , and may many wayes be usefull for captaines , magistrates , or others which have divers persons to punish , and would chastise chiefely the unruliest of them , in taking the 10 , 20 , or 100. person , &c. as we reade was commonly practised amongst the ancient romans : herefore to apply a generall rule in counting the third , 4 , 9 , 10 , &c. amongst 30 , 40 , 50 , persons , and more or lesse ; this is to be observed , take as many units as there are persons , and dispose them in order privately : as for example , let 24 men be proposed to have committed some outrage , 6 of them especially are found accessary : and let it be agreed that counting by 8 and 8 the eight man should be alwayes punished . take therefore first 24 units , or upon a piece of paper write down 24 cyphers , and account from the beginning to the eighth , which eighth mark , and so continue counting alwayes marking the eighth , untill you have markt 6 , by which you may easily perceive how to place those 6 men that are to be punished , and so of others . it is supposed that josephus the author of the jewish history escaped the danger of death by help of this problem ; for a worthy author of beliefe reports in his eighth chapter of the third book of the destruction of jerusalem , that the town of jotapata being taken by main force by vespatian , josephus being governour of that town , accompanyed with a troop of forty souldiers , hid themselves in a cave , in which they resolved rather to famish than to fall into the hands of vespatian : and with a bloudy resolution in that great distresse would have butchered one another for sustenance , had not josephus perswaded them to die by lot and order , upon which it should fall : now seeing that josephus did save himselfe by this art , it is thought that his industry was exercised by the helpe of this problem , so that of the 40 persons which he had , the third was alwayes killed . now by putting himselfe in the 16 or 31 place he was saved , and one with him which he might kill , or easily perswade to yeild unto the romans . problem . viii . three things , and three persons proposed , to finde which of them hath either of these three things . let the three things be a ring , a piece of gold , and a piece of silver , or any other such like , and let them be known privately to your self by these three vowels a , e , i , or let there be three persons that have different names , as ambrose , edmond , and john , which privately you may note or account to your selfe once known by the aforesaid vowels , which signifie for the first vowel 1 , for the second vowell 2 , for the third vowell 3. now if the said three persons should by the mutuall consent of each other privately change their names , it is most facill by the course and excellencie of numbers , distinctly to declare each ones name so interchanged , or if three persons in private , the one should take a ring , the other a piece of gold , and the third should take a piece of silver ; it is easie to finde which hath the gold , the silver , or the ring , and it is thus done . take 30 or 40 counters ( of which there is but 24 necessary ) that so you may conceale the way the better , and lay them down before the parties , and as they sit or stand , give to the first 1. counter , which signifieth a , the first vowell ; to the second 2. counters , which represent e , the second vowel ; and to the third 3. counters , which stand for i , the third vowell : then leaving the other counters upon the table , retire apart , and bid him which hath the ring , take as many counters as you gave him , and he that hath the gold , for every one that you gave him , let him take 2 , and he that hath the silver for every one that you gave him , let him take 4. this being done , consider to whom you gave one counter , to whom two , and to whom three ; and mark what number of counters you had at the first , for there are necessarily but 24. as was said before , the surpluse you may privately reject . and then there will be left either 1.2.3.5.6 or 7. and no other number can remaine , which if there be , then they have failed in taking according to the directions delivered : but if either of these numbers do remaine , the resolution will be discovered by one of these 6 words following , which ought to be had in memory , viz. salve , certa , anima , semita , vita , quies· 1. 2. 3. 5. 6. 7. as suppose 5. did remaine , the word belonging unto it is semita , the vowels in the first two syllables are e and i , vvhich shevveth according to the former directions , that to vvhom you gave 2 counters , he hath the ring ( seeing it is the second vovvell represented by tvvo as before ) and to vvhom you gave the 3. counters , he hath the gold , for that i represents the third vovvel , or 3. in the former direction , and to vvhom you gave one counter , he hath the silver , and so of the rest : the variety of changes , in vvhich exercise , is laid open in the table follovving . rest men hid rest men hid 1 1 a 5 1   2 e 2   3 i 3   2 1 e 6 1   2 a 2   3 i 3   3 1 a 7 1   2 i 2   3 e 3   this feat may be done also without the former words by help of the circle a. for having divided the circle into 6 parts , write 1. within and 1. vvithout , 2. vvithin and 5. vvithout , &c. the first 1.2.3 . vvhich are vvithin vvith the numbers over them , belongs to the upper semicircle ; the other numbers both vvithin and vvithout , to the under semicircle ; now if in the action there remaineth such a number which may be found in the upper semicircle without , then that which is opposite within shews the first , the next is the second , &c. as if 5 remains , it shews to whom he gave 2 , he hath the ring ; to whom you gave ● , he hath the gold , &c. but if the remainder be in the under semicircle , that which is opposite to it is the first ; the next backwards towards the right hand is the second ; as if 3 remains , to whom you gave 1 he hath the ring , he that had 3 he had the gold , &c. problem ix . how to part a vessel which is full of wine conteining eight pints into two equall parts , by two other vessels which conteine as much as the greater vessell ; as the one being 5 pints , and the other 3 pints . let the three vessels be represented by abc , a being full , the other two being empty ; first , poure out a into b until it be full , so there will be in b 5 pints , and in a but 3 pints : then poure out of b into c untill it be full : so in c shall be 3 pints , in b 2 pints , and in a 3 pints , then poure the wine which is in c into a , so in a will be 6 pints , in b 2 pints , and in c nothing : then poure out the wine which is in b into the pot c , so in c there is now 2 pints , in b nothing , and in a 6 pints , . lastly , poure out of a into b untill it be full , so there will be now in a only 1 pint in b 5 pints , and in c 2 pints . but it is now evident , that if from b you poure in unto the pot c untill it be full , there wil remain in b 4 pints , and if that which is in c , viz. 3 pints be poured into the vessell a , which before had 1 pint , there shall be in the vessel a , but halfe of its liquor that was in it at the first , viz. 4 pints as was required . otherwise poure out of a into c untill it be full , which pour into b , then poure out of a into c again untill it be full , so there is now in a onely 2 pints , in b 3 , and in c 3 , then pour from c into b untill it be full , so in c there is now but 1 pint , 5 in b , and 2 in a : poure all that is in b into a , then poure the wine which is in c into b , so there is in c nothing , in b onely 1 pint , and in 7 a 7 pints : lastly , out of a fill the pot c , so there will remain in a 4 pints , or be but halfe full : then if the liquor in c be poured into b , it will be the other half . in like manner might be taken the half of a vessell which conteins 12 pints , by having but the measures 5 and 7 , or 5 and 8. now such others might be proposed , but we omit many , in one and the same nature . problem . x. to make a stick stand upon the tip of ones finger , without falling . fasten the edges of tvvo knives or such like of equall poise , at the end of the stick , leaning out somevvhat from the stick , so that they may counterpoise one another ; the stick being sharp at the end , and held upon the top of the finger , vvill there rest vvithout supporting : if it fall , it must fall together , and that perpendicular or plumb-wise , or it must fall side-wise or before one another ; in the first manner it cannot : for the centre of gravitie is supported by the top of the finger : and seeing that each part by the knives is counterpoised , it cannot fall sidevvise , therefore it can fall no vvise . in like manner may great pieces of timber , as joists , &c be supported , if unto one of the ends be applied convenient proportionall counterpoises , yea a lance or pike , may stand perpendicular in the aire upon the top of ones finger : or placed in the midst of a court by help of his centre of gravitie . examination . this proposition seems doubtfull ; for to imagine absolutely , that a pike , or such like , armed with two knives , or other things , shall stand upright in the aire , and so remain without any other support , seeing that all the parts have an infinite difference of propensity to fall ; and it is without question that a staff so accommodated upon his centre of gravity , but that it may incline to some one part without some remedy be applied , and such as is here specified in the probleme will not warrant the thing , nor keep it from falling ; and if more knives should be placed about it , it should cause it to fall more swiftly , forasmuch as the superiour parts ( by reason of the centricall motion ) is made more ponderous , and therefore lesse in rest . to place therefore this prop really , let the two knives , or that which is for counterpoise , be longer always then the staffe , and so it will hang together as one body : and it will appear admirable if you place the centre of gravity , neer the side of the top of the finger or point ; for it will then hang horizontall , and seem to hang onely by a touch , yet more strange , if you turn the point or top of the finger upside down . problem xi . how a milstone or other ponderosity , may be supported by a small needle , without breaking o● any wise bowing the same . let a needle be set perpendicular to the horizon , and the center of gravitie of the stone be placed on the top of the needle : it is evident that the stone cannot fall , forasmuch as it hangs in aequilibra , or is counterpoysed in all parts alike ; and moreover it cannot bow the needle more on the one side then on the other , the needle will not therefore be either broken or bowed ; if otherwise then the parts of the needle must penetrate and sinke one with another : that which is absurd and impossible to nature ; therefore it shall be supported . the experiments which are made upon trencher plates , or such like lesser thing doth make it most credible in greater bodies . but here especially is to be noted , that the needle ought to be uniforme in matter and figure , and that it be erected perpendicular to the horizon , and lastly , that the center of gravity be exactly found . problem xii . to make three knives hang and move upon the point of a needle . fit the three knives in form of a ballance , and holding a needle in your hand , and place the back of that , knife which lyes cross-wise to the other two , upon the point of the needle : as the figure here sheweth you ; for then in blowing softly upon them , they will easily turne and move upon the point of the needle with ●ou falling . problem xiii . to finde the weight of smoak , which is exhaled of any combustible body whatsoever . let it be supposed that a great heape of fagots , or a load of straw weighing 500 pound should be fired , it is evident that this grosse substance will be all inverted into smoak and ashes : now it seems that the smoak weighs nothing ; seeing it is of a thin substance now dilated in the aire , notwithstanding if it were gathered and reduced into the thickest that it was at first , it would be sensibly weighty : weigh therefore the ashes which admit 50 pound , now seeing that the rest of the matter is not lost , but is exhaled into smoake , it must necessarily be , that the rest of the weight ( to wit ) 450 pound , must be the weight of the smoak required . examination . now although it be thus delivered , yet here may be noted , that a ponderosity in his own medium is not weighty : for things are said to be weighty , when they are out of their place , or medium , and the difference of such gravity , is according to the motion : the smoak therefore certainly is light being in its true medium ( the aire , ) if it should change his medium , then would we change our discourse . problem xvi . many things being disposed circular , ( or otherwise ) to finde which of them , any one thinks upon ▪ suppose that having ranked 10 things , as abcdefghik , circular ( as the figure sheweth ) and that one had touched or thought upon g , which is the 7 : ask the party at what letter he would begin to account ( for account he must , otherwise it cannot be done ) which suppose , at e which is the 5 place , then add secretly to this 5 , 10 ( which is the number of the circle ) and it makes 15 , bid him account 15 backward from e , beginning his account with that number hee thought upon , so at e he shal account to himself 7 , at d account 8 , at c account 9 , &c. so the account of 15 wil exactly fall upon g , the thing or number thought upon : and so of others : but to conceal it the more , you may will the party from e to account 25 , 35 , &c. and it will be the same . there are some that use this play at cards , turned upside downe , as the ten simple cards , with the king and queen , the king standing for 12 , and the queene for 11 , and so knowing the situation of the cards : and thinking a certain houre of the day : cause the party to account from what card he pleaseth : with this proviso , that when you see where he intends to account , set 12 to that number , so in counting as before , the end of the account shall fall upon the card : which shall denote or shew the houre thought upon , which being turned up will give grace to the action , and wonder to those that are ignorant in the cause . problem xv. how to make a door or gate , which shall open on both sides . all the skill and subtilty of this , rests in the artificiall disposer of foure plates of iron , two at the higher end , and two at the lower end of the gate : so that one side may move upon the hooks or hinges of the posts , and by the other end may be made fast to the gate , and so moving upon these hinges , the gate will open upon one side with the aforesaid plates , or hooks of iron : and by help of the other two plates , will open upon the other side . problem xvi . to shew how a ponderosity , or heavy thing , may be supported upon the end of a staffe ( or such like ) upon a table , and nothing holding or touching it . take a pale which hath a handle , and fill it full of water ( or at pleasure : ) then take a staffe or stick which may not rowle upon the table as ec , and place the handle of the pale upon the staffe ; then place another staffe , or stick , under the staffe ce , which may reach from the bottom of the pale unto the former staffe ce , perpendicular wise : which suppose fg , then shall the pale of water hang without falling , for if it fall it must fall perpendicularly , or plumbe wise : and that cannot be seeing the staffe ce supports it , it being parallel to the horizon and susteined by the table , and it is a thing admirable that if the staffe ce were alone from the table , and that end of the staffe which is upon the table were greater and heavier than the other : it would be constrained to hang in that nature . examination . now without some experience of this probleme , a man would acknowledge either a possibility or impossibity ; therefore it is that very touchstone of knowledge in any thing , to discourse first if a thing be possible in nature , and then if it can be brought to experience and under sence without seeing it done . at the first , this proposition seems to be absurd , and impossible . notwithstanding , being supported with two sticks , as the figure declareth , it is made facile : for the horizontall line to the edge of the table , is the centre of motion ; and passeth by the centre of gravity , which necessarily supporteth it . problem xvii . of a deceitfull bowle to play withall . make a hole in one side of the bowle , and cast molten lead therein , and then make up the hole close , that the knavery or deceit be not perceived : you will have pleasure to see , that notwithstanding the bowle is cast directly to the play , how it wil turn away side-wise : for that on that part of the bowle which is heavier upon the one side then on the other , it never will go truly right , if artificially it be not corrected ; which will hazard the game to those which know it not : but if it be known that the leady side in rolling be always under or above , it may go indifferently right ; if otherwise , the weight will carry it always side-wise . problem . xviii . to part an apple into 2.4 . or 8. like parts , without breaking the rinde . passe a needle and threed under the kinde of the apple , and then round it with divers turnings , untill you come to the place where you began : then draw out the threed gently , and part the apple into as many parts as you think convenient : and so the parts may be taken out between the parting of the rind , and the rind remaining alwayes whole . problem xix . to finde a number thought upon without asking of any question , certaine operations being done . bid him adde to the number thought ( as admit 15 ) halfe of it , if it may be , if not the greatest halfe that exceeds the other but by an unite , which is 8 ; and it makes 23. secondly , unto this 23. adde the halfe of it if it may be , if not , the greatest halfe , viz. 12. makes 35. in the meane time , note that if the number thought upon cannot be halfed at the first time , as here it cannot , then for it keep 3 in the memory , if at the second time it will not be equally halfed , reserve 2 in memory , but if at both times it could not be equally halved , then may you together reserve five in memory : this done , cause him from the last summe , viz. 35. to subtract the double of the number thought , viz. 30. rest 5. will him to take the halfe of that if he can , if not , reject 1. and then take the halfe of the rest , which keep in your memory : then will him to take the halfe againe if he can , if not , take one from it , which reserve in your memory , and so perpetually halveing untill 1. remaine : for then mark how many halfes there were taken , for the first halfe account 2 , for the second 4 , for the third 8 , &c. and adde unto those numbers the one 's which you reserved in memory , so there being 5 remaining in this proposition , there were 2 halfings : for which last ! account 4 , but because it could not exactly be halved without rejecting of 1. i adde the 1 therefore to this 4 , makes 5 , which halfe or summe alwayes multiplied by 4 , makes 20. from which subtract the first 3 and 2 , because the halfe could not be formerly added , leaves 15 , the number thought upon . other examples . the number thought upon . the number thought 12 the halfe of it 6 the summe 18 the halfe of it 9 the summe of it 27 the double of the number , 24 which taken away , rests 3 the halfe of it 1 for which account 2 and 1 put to it because the 3 could not be halfed , makes 3 this multiplied by 4 makes 12 the number thought 79 the greatest halfe 40 3 the summe 119 the greatest halfe of which is 60 2 the summe of it is 179 the double of 79 is 158 which taken from it , rests 21 the lesser half 10. which halve :   the halfe of this is 5 which makes   the half of this is 2 which is 10   the half of this is 1 , with 10 and 11 is 21.   this 21 which is the double of the last halfe with the remainder being multiplied by 4. makes 84 , from which take the aforesaid 3 and 2 , ●●st 79 , the number thought upon .   problem . xx. how to make an uniforme , & an inflexible body , to passe through two small holes of divers formes , as one being circular , and the other square , quadrangular , and triangular-wise , yet so that the holes shall be exactly filled . this probleme is extracted from geometricall observations , and seemes at the first somewhat obscure , yet that which may be extracted in this nature , will appeare more difficult and admirable . now in all geometricall practises , the lesser or easier problemes do alwayes make way to facilitate the greater : and the aforesaid probleme is thus resolved . take a cone or round pyramide , and make a circular hole in some board , or other hard material , which may be equall to the bases of the cone , and also a triangular hole , one of whose sides may be equall to the diameter of the circle , and the other two sides equall to the length of the cone : now it is most evident , that this conicall or pyramidall body , will fill up the circular hole , and being placed side-wise will fill up the triangular hole . moreover , if you cause a body to be turned , which may be like to two pyramides conjoyned , then if a circular hole be made , whose diameter is equal to the diameter of the cones conjoyned , and a quadrangular hole , whose sloping sides be equall to the length of each side of the pyramide , and the breadth of the hol equal to the diameter of the circle , this conjoyned pyramide shall exactly fill both the circular hole , and also the quadrangle hole . problem . xxi . how with one uniforme body or such like to fill three severall holes : of which the one is round , the other a just square , and the third an ovall forme ? this proposition seemes more subtill then the former , yet it may be practised two wayes : for the first , take a cylindricall body as great or little as you please : now it is evident that it will fill a circular hole , which is made equall to the basis of it , if it be placed downe right , and will also fill a long square ; whose sides are equall unto the diameter and length of the cylinder , and acording to pergeus , archimedes , &c. in their cylindricall demonstrations , a true ovall is made when a cylinder is cut slopewise , therefore if the oval have breadth equall unto the diameter of the basis of the cylinder , & any length whatsoever : the cylinder being put into his owne ovall hole shall also exactly fill it . the second way is thus , make a circular hole in some board , & also a square hole , the side of which square may be equall to the diameter of the circle : and lastly , make a hole oval-wise , whose breadth may be equal unto the diagonall of the square ; then let a cylindricall body be made , whose basis may be equall unto the circle , and the length equall also to the same : now being placed downe right shall fall in the circle , and flat-wise will fit the square hole , and being placed sloping-wise will fill the ovall . examination . you may note upon the last two problemes farther , that if a cone be cut ecliptick-wise , it may passe through an issoc●●● triangle through many scalen triangles , and through an ellipsis ; and if there be a cone cut scalen-wise , it will passe through all the former , only for the ellipsis place a circle : and further , if a solid colume be cut ecliptick-wise it may fill a circle , a square , divers parallelogrammes , and divers ellipses , which have different diameters . problem xxii . to finde a number thought upon ●fter another manner , then what is formerly delivered bid him that he multiply the number thought upon , by what number he pleaseth , then bid him divide that product by any other number , and then multiply that quotient by some other number ; and that product againe divide by some other , and so as often as he will : and here note , that he declare or tell you by what number he did multiply & divide now in the same time take a number at pleasure , and secretly multiply and divide as often as he did : then bid him divide the last number by that which he thought upon . in like manner do yours privately , then will the quotient of your divisor be the same with his , a thing which seemes admirable to those which are ignorant of the cause . now to have the number thought upon without seeming to know the last quotient , bid him adde the number thought upon to it , and aske him how much it makes : then subtract your quotient from it , there will remaine the number thought upon for example , suppose the number thought upon were 5 , multiply it by 4 makes 20. this divided by 2 , the quotient makes 10 , which multiplyed by 6 , makes 60 , and divided by 4 , makes 1● . in the same time admit you think upon 4 , which multiplied by 4 , makes 16 , this divided by 2 , makes 8 , which multiplied by 6 makes 48 , and divided by 4 makes 1● ; then divide 1● by the number thought , which was 5 , the quotient is ● ; divide also 12 by the number you took , viz. 4 , the quotient is also 3. as was declared ; therefore if the quo●ient ● be added unto the number thought , viz. ● , it makes 8 , which being known , the number thought upon is also knowne . problem xxiii . to finde out many numbers that sundry persons , or one man hath thought upon . if the multitude of numbers thought upon be odde , as three numbers , five numbers , seven , &c. as for example , let 5 numbers thought upon be these ● 2 , 3 , 4 , 5 , 6. bid him declare the sum of the first and second , which will be 5 , the second and third , which makes 7 , the third and fourth , which makes 9 , the fourth and fifth , vvhich makes 11 , and so alvvayes adding the tvvo next together , aske him hovv much the first and last makes together , vvhich is 8. then take these summes , and place them in order , and adde all these together , vvhich vvere in the odde places : that is the first , third , and fifth , viz. 5 , 9 , ● , makes 22. in like manner adde all these numbets together , vvhich are in the even places , that is in the second and fourth places , viz. 7 and 1● makes 18 , substract this from the former 22 , then there vvill remaine the double of the first number thought upon , viz. 4. which known , the rest is easily known : seeing you know the summe of the first and second ; but if the multitude of numbers be even as these six numbers , viz. 2 , ● , 4 , 5 , 6 , 7 , cause the partie to declare the summe of each two , by antecedent and consequent , and also the summe of the second and last , which will be 5 , 7 , 9 , 11 , 13 , 10 , then adde the odde places together , except the first , that is 9 , and 13 , makes 22 , adde also the even places together , that is 7 , 11 , 10 , which makes 28 , substract the one from the other , there shall remaine the double of the second number thought upon , which known all the rest are knowne . problem xxiv . how is it that a man in one and the same time , may have his head upward , and his feet upward , being in one and the same place ? the answer is very facill , for to be so he must be supposed to be in the centre of the earth : for as the heaven is above on every side , coelum undique sursum , all that which looks to the heavens being distant from the centre is upward ; and it is in this sense that ma●●olyeus in his cosmographie , & first dialogue , reported of one that thought he was led by one of the muses to hell , where he saw lucifer sitting in the middle of the world , and in the centre of the earth , as in a throne : having his head and feet upward . problem . xxv . of a ladder by which two men ascending at one time ; the more they ascend , the more they shall be asunder , notwithstanding one being as high as another this is most evident , that if there were a ladder halfe on this side of the centre of the earth , and the other halfe on the other side : and that two at the centre of the world at one instant being to ascend , the one towards us , and the other towards our antipodes , they should in ascending go farther and farther , one from another ; notwithstanding both of them being of like height . problem . xxvi . how it is that a man having but a rod or pole of land , doth bragge that he may in a right line passe from place to place above 3000 miles . the opening of this is easie , forasmuch as he that possesseth a rod of ground possesseth not only the exterior surface of the earth , but is master also of that which extends even to the centre of the earth , and in this wise all heritages & possessions are as so many pyramides , whose summets or points meet in the centre of the earth , and the basis of them are nothing else but each mans possession , field , or visible quantity ; and therefore if there were made or imagined so to be made , a descent to go to the bottome of the heritage , which would reach to the centre of the earth ; it would be above 3000 miles in a right line as before . problem . xxvii . how it is , that a man standing upright , and looking which way he will , he looketh either true north or true south . this happeneth that if the partie be under either of the poles , for if he be under the north-pole , then looking any way he looketh south , because all the meridians concurre in the poles of the world , and if he be under the south-pole , he looks directly north by the same reason . problem xxviii . to tell any one what number remaines after certaine operations being ended , without asking any question . bid him to think upon a number , and will him to multiply it by what number you think convenient : and to the pro●●ct bid him adde what number you please , or 〈◊〉 that secretly you consider , that it ma● be divided by that which multiplied , and 〈…〉 divide the sum by the number which he 〈…〉 by , and substract from this quotient the number thought upon : in the same time divide apart the number which was add●d by that which multiplied , so then your quotient shall be equall to his remainder , wherefore without asking him any thing , you shall tell him what did remaine , which will seem strange to him that knoweth not the cause : for example , suppose he thought 7 , which multiplied by 5 makes 35 , to which adde 10 , makes 45 , which divided by 5 , yields 9 , from which if you take away one the number thought , ( because the multiplier divided by the divisor gives the quotient 1 , ) the rest will be two , which will be also proved , if 10 the number which was added , were divided by 5 , viz. 2. problem xxix . of the play with two severall things . it is a pleasure to see and consider how the science of numbers doth furnish us , not only 〈…〉 recreate the spirits , but also 〈…〉 knowledge of admirable things , 〈…〉 measure be shewen in this 〈…〉 the meane time to produce alwayes some of them : suppose that a man hold divers things in his hand , as gold and ●ilver ▪ and in one hand he held the gold , and in the other hand he held the silver : to know subtilly , and by way of divination , or artificially in which hand the gold or silver is ; attribu●e t● the gold , or suppose it have a certaine price , and so likewise attribute to the silver another price , conditionally that the one be odd , and the other even : as for example , bid h●m that the gold be valued at 4 crownes , or shillings , and the silver at ● crownes , or 3 shillings , or any other number , so that one be odde ▪ and the other even , as before ; then bid him triple that which is in the right hand , & double that which is in the left hand , and bid him adde these two products together , and aske him if it be even or odde ; if it be even , then the gold is in the right hand ; if odde , the gold is in the left hand . problem . xxx . two numbers being proposed unto two severall parties , to tell which of these numbers is taken by each of them . as for example : admit you had proposed unto two men whose names were peter and john , two numbers , or pieces of money , the one even , and the other odde , as 10. and 9. and let the one of them take one of the numbers , and the other partie take the other number , which they place privately to themselves : how artificially , according to the congruity , and excellency of numbers , to finde which of them did take 10. and which 9. without asking any qustion : and this seems most subtill , yet delivered howsoever differing little from the former , and is thus performed : take privately to your selfe also two numbers , the one even , and the other odde , as 4. and 3. then bid peter that he double the number which he took , and do you privately double also your greatest number ; then bid john to triple the number which he hath , and do you the like upon your last number : adde your two products together , & mark if it be even or odde , then bid the two parties put their numbers together , and bid them take the halfe of it , which if they cannot do , then immediately tell peter he took 10. and john 9. because the aggregate of the double of 4. and the triple of 3. makes odde , and such would be the aggregate or summe of the double of peters number and johns number , if peter had taken 10. if otherwise , then they might have taken halfe , and so john should have taken 10. and peter 9. as suppose peter had taken 10. the double is 20. and the triple of 9. the other ●umber is 27. which put together makes 47. odde : in like manner the double of your number conceived in minde , viz. 4. makes 8. and the triple of the 3. the other number , makes 9. which set together makes 17. odde . now you cannot take the halfe of 17 , nor 47. which argueth that peter had the greater number , for otherwise the double of 9. is 18. & the triple of 10. is 30. which set together makes 48. the halfe of it may be taken : therefore in such case peter the took lesse number : and john the greater , and this being don cleanly carries much grace with it . problem . xxxi . how to describe a circle that shall touch 3 : points placed howsoever upon a plaine , if they be not in a right line . let the three points be a.b.c. put one foot of the compasse upon a. and describe an arch of a circle at pleasure : and placed at b. crosse that arch in the two points e. and f. and placed in c. crosse the arch in g. and h. then lay a ruler upon g.h. and draw a line , and place a ruler upon e. and f. cut the other line in k ▪ so k is the centre of the circumference of a circle , which will passe by the said three points a.b.c. or it may be inverted , having a circle drawne ; to finde the centre of that circle , make 3. points in the circumference , and then use the same way : so shall you have the centre , a thing most facill to every practitioner in the principles of geometrie . problem . xxxii . how to change a circle into a square forme ? m●ke a circle upon past-board or other materiall , as the circle a.c.d.e. of which a. is the centre ; then cut it into 4 quarters , and dispose them so , that a. at the centre of the circle may alwayes be at the angle of the square , and so the foure quarters of the circle being placed so , it will make a perfect square , whose side a.a. is equall to the diameter b.d. now here is to be noted that the square is greater then the circle by the vacuity in the middle , viz. m. problem . xxxiii . with one and the same comp●sses , and at one and the same extent , or opening , how to describe many circles concentricall , that is , greater or lesser one then another ? it is not without cause that many admire how this proposition is to be resolved ; yea in the judgement of some it is thought impossible : who consider not the industrie of an ingenious geometrician , who makes it possible , and that most facill , sundry wayes ; for in the first place if you make a circle upon a fine plaine , and upon the centre of that circle , a small pegge of wood be placed , to be raised up and put downe at pleasure by help of a small ho●e made in the centre , then with the same opening of the compasses , you may describe circles concentricall , that is , one greater or lesser than another ; for the higher the center is lifted up , the lesser the circle will be . secondly , the compasse being at that extent upon a gibus body , a circle may be described , which will be lesse than the former , upon a plaine , and more artificially upon a globe , or round bowle : and this againe is most obvious upon a round pyramide , placing the compasses upon the top of it , which will be farre lesse than any of the former ; and this is demonstrated by the 20. prop. of the first of euclids , for the diameter ● . d. is lesse than the line ad.a.e. taken together , and the lines ad.ae. being equall to the diameter bc. because of the same distance or extent of opening the compasses , it followes that the diameter e.d. and all his circles together is much lesse than the diameter , and the circle bc. which was to be performed . problem xxxiv . any numbers under 10. being thought upon , to finde what numbers they were . let the first number be doubled , and unto it adde 5. and multiply that summe by 5. and unto it adde 10. and unto this product add the next number thought upon ; multiply this same againe by 10. and adde unto it the next number , and so proceed : now if he declare the last summe ; marke if he thought but upon one figure , for then subtract only 35. from it , and the first figure in the place of tennes is the number thought upon : if he thought upon two figures , then subtract also the said ●5 . from his last summe , and the two figures which remaine are the number thought upon : if he thought upo● three figures , then subtract 350. and then the first three figures are the numbers thought upon , &c. so if one thought upon these numbers 5.7.9.6 . double the first , makes 1● . to which adde 5. makes 15. this multiplied by 5. makes 75. to which adde 1● . makes 85. to this adde the next number , viz. 7. makes 92. this multiplied by 10. makes 920. to which adde the next number , viz. 9. makes 929. which multiplied by 10. makes 9290. to which adde 6. makes 9296. from which subtract 3500. resteth 5796. the foure numbers thought upon . now because the two last figures are like the two numbers thought upon : to conceale this , bid him take the halfe of it , or put first 12. or any other number to it , and then it will not be so open . problem . xxxv . of the play with the ring . amongst a company of 9. or 10. persons , one of them having a ring , or such like : to finde out in which hand : upon which finger , & joynt it is ; this will cause great astonishment to ignorant spirits , which will make them beleeve that he that doth it works by magick , or witchcraft : but in effect it is nothing else but a nimble act of arithmetick , founded upon the precedent probleme : for first it is supposed that the persons stand or sit in order , that one is first , the next second , &c. likewise there must be imagined that of these two hands the one is first , and the other second : and also of the five fingers , the one is first , the next is second , and lastly of the joynts , the one is as 1. the other is as 2. the other as 3. &c. from whence it appeares that in performing this play there is nothing else to be done than to think 4. numbers : for example , if the fourth person had the ring in his left hand , and upon the fifth finger , and third joynt , and i would divine and finde it out : thus i would proceed , as in the 34 problem : in causing him to double the first number : that is , the number of persons , which was 4. and it makes 8. to which add 5. makes 13. this multiplied by 5. makes 65. put 10. to it , makes 75. unto this put ● . for the number belonging to the left hand , and so it makes 77. which multiplied by 10. makes 770. to this adde the number of the fingers upon which the ring is , viz. 5. makes 775. this multiplied by 10. makes 7750. to which adde the number for the joynt upon which the ring is , viz the third joynt , makes 7●53 . to which cause him to adde 14. or some other number , to conceale it the better : and it makes 7767. which being declared unto you , substract 3514 ▪ and there will remaine 4.2.5.3 . which figures in order declares the whol mystery of that which is to be known : 4. signifieth the fourth person , 2. the left hand , 5. the fifth finger , and 3. the third joynt of that finger . problem . xxxvi . the play of 34. or more dice . that which is said of the two precedent problemes may be applied to this of dice ( and many other particular things ) to finde what number appeareth upon each dice being cast by some one , for the points that are upon any side of a dice are alwayes lesse than 10 and the points of each side of a dice may be taken for a number thought upon : therefore the rule will be as the former : as for example , one having thrown three dice , and you would declare the numbers of each one , or how much they make together , bid him double the points of one of the dice , to which bid him adde 5 , then multiply that by 5. and to it adde 10 , and to the summe bid him adde the number of the second dice : and multiply that by 10 : lastly , to this bid him adde the number of the last dice , and then let him declare the whole number : then if from it you subtract ●50 . there will remaine the number of the three dice throwne . problem . xxxvii . how to make water in a glasse seeme to boyle and sparkle ? take a glasse neere full of water or other liquor ; and setting one hand upon the foot of it , to hold it fast : turne slightly one of the fingers of your other hand upon the brimme , or edge of the glasse ; having before privately wet your finger : and so passing softly on with your finger in pressing a little : for then first , the glasse will begin to make a noyse : secondly , the parts of the glasse will sensibly appeare to tremble , with notable rarefaction and condensation : thirdly , the water will shake , seeme to boyle : fourthly , it will cast it selfe out of the glasse , and leap out by small drops , with great astonishment to the standers by ; if they be ignorant of the cause of it , which is onely in the rarefaction of the parts of the glasse , occasioned by the motion and pressure of the finger . examination . the cause of this , is not in the rarefaction of the parts of the glasse , but it is rather in the quick locall motion of the finger , for reason sheweth us that by how much a body draweth nearer to a quality , the lesse is it subject or capable of another which is contrary unto it ? now condensation , and rarefaction are contrary qualities , and in this probleme there are three bodies considered , the glasse , the water , and the aire , now it is evident that the glasse being the most solid , and impenitrable body , is lesse subject and capable of rarefaction than the water , the water is lesse subject than the aire , and if there be any rarefaction , it is rather considerable in the aire then in the water , which is inscribed by the glasse , and above the water , and rather in the water then in the glasse : the agitation , or the trembling of the parts of the glasse to the sense appeares not : for it is a continued body ; if in part , why then not in the whole ? and that the water turnes in the glasse , this appeares not , but only the upper contiguous parts of the water : that at the bottome being lesse subiect to this agitation , and it is most certaine that by how much quicker the circular motion of the finger upon the edge of the glasse is , by so much the more shall the aire be agitated , and so the water shall receive some apparant affection more or lesse from it , according to that motion : as we see from the quicknesse of winde upon the sea , or c●lme thereof , that there is a greater or lesser agitation in the water ; and for further examination , we leave it to the search of those which are curious . problem . xxxviii . of a fine vessell which holds wine or water , being cast in●o it at a certaine height , but being filled higher , it will runne out of its owne accord . let there be a vessell a.b.c.d. in the middle of which place a pipe ; whose ends both above at e , and below at the bottom of the vessell as at ● ▪ are open ; let the end ● be somewhat lower than the brimme of the glasse : about this pipe , place another pipe as h. l , which mounts a little above e and let it most diligently be closed at h , that no aire enter in thereby , and this pipe at the bottome may have a small hole to give passage unto the water ; then poure in water or wine , and as long as it mounts not above e , it is safe ; but if you poure in the water so that it mount above it , farewell all : for it will not cease untill it be all gone out ; the same may be done in disposing any crooked pipe in a vessell in the manner of a faucet or funnell , as in the figure h , for fill it under h , at pleasure , and all will go well ; but if you fill it unto h. you will see fine sport , for then all the vessell will be empty incontinent , and the subtiltie of this will seeme more admirable , if you conceale the pipe by a bird , serpent , or such like , in the middle of the glasse . now the reason of this is not difficult to those which know the nature of a cock or faucet ; for it is a bowed pipe , one end of which is put into the water or liquor , and sucking at the other end untill the pipe be full , then will it run of it selfe , and it is a fine secret in nature to see , that if the end of the pipe which is out of the water , be lower then the water , it will run out without ceasing : but if the mouth of the pipe be higher then the water or levell with it , it will not runne , although the pipe which is without be many times bigger than that which is in the water : for it is the property of water to keep alwayes exactly levell ▪ examination . here is to be noted , that if the face of the water without be in one and the same plaine , with that which is within , though the outtermost pipe be ten times greater than that which is within ; the water naturally will not runne , but if the plaine of the water without be any part lower then that which is within , it will freely runne : and here may be noted further , that if the mouth of the pipe which is full of water , doth but only touch the superficies of the water within , although the other end of the pipe without be much lower than that within , the water it will not run at all : which contradicts the first ground ; hence we gather that the pressure or ponderosity of the water within , is the cause of running in some respect . problem . xxxix . of a glasse very pleasant . sometimes there are glasses which are made of a double fashion , as if one glasse were within another , so that they seem but one , but there is a little space between them . no● poure wine or other liquor between the two edges by help of a tunnell , into a little hole left to this end , so vvill there appeare tvvo fine delusions or fallacies ; for though there be not a drop of wine vvithin the hollovv of the glasse , it vvill seem to those vvhich behold it that it is an ordinary glasse full of wine , and that especially to those vvhich are side-vvise of it , and if any one move it , it vvill much confirme it , because of the motion of the wine ; but that vvhich vvill give most delight , is that , if any one shall take the glasse , and putting it to his mouth shall think to drink the wine , instead of vvhich he shall sup the aire , and so vvill cause laughter to those that stand by , vvho being deceived , vvill hold the glass to the light , & thereby considering that the raies or beames of the light are not reflected to the eye , as they vvould be if there vvere a liquid substance in the glasse , hence they have an assured proofe to conclude , that the hollovv of the glasse is totally empty . problem . xl. if any one should hold in each hand , as many pieces of money as in the other , how to finde how much there is ? bid him that holds the money that he put out of one hand into the other vvhat number you think convenient : ( provided that it may be done , ) this done , bid him that out of the hand that he put the other number into , that he take out of it as many as remaine in the other hand , and put it into that hand : for then be assured that in the hand which was put the first taking away : there will be found just the double of the number taken away at the first . example , admit there were in each hand 12 shi●lings or counters , and that out of the right hand you bid him take 7. and put it into the left : and then put into the right hand from the left as many as doth remaine in the right , which is 5. so there will be in the left hand ●4 , which is the double of the number taken out of the right hand , to wit 7. then by some of the rules before delivered , it is easie to finde how much is in the right hand , viz. 10. problem . xli . many dice being cast , how artificially to discover the number of the points that may arise . svppose any one had cast three dice secretly , bid him that he adde the points that were upmost together : then putting one of the dice apart , unto the former summe adde the points which are under the other two , then bid him throw these two dice , and mark how many points a paire are upwards , which adde unto the former summe : then put one of these dice away not changing the side , mark the points which are under the other dice , and adde it to the former summe : lastly , throw that one dice , and whatsoever appeares upward adde it unto the former summe ; and let the dice remaine thus : this done , comming to the table , note what points do appeare upward upon the three dice , which adde privately together , and unto it adde ●1 or 3 times 7 : so this addition or summe shall be equall to the summe which the party privately made of all the operations which he formerly made . as if he should throw three dice , and there should appeare upward 5 , 3 , 2. the sum of them is 10. and setting one of them apart , ( as 5. ) unto 10 , adde the points which are under 3 and 2 , which is 4 and 5 , and it makes 19. then casting these two dice suppose there should appeare 4 and 1 , this added unto 19 makes 24. and setting one of these two dice apart as the 4. unto the former 24 , i adde the number of points which is under the other dice , viz. under 1 , that is 6 , which makes 30. last of all i throw that one dice , and suppose there did appeare 2 , which i adde to the former 30 , and it makes 32 , then leaving the 3 dice thus , the points which are upward will be these , 5 , 4 , 2 unto which adde secretly 21 , ( as before was said ) so have you 32 , the same number whi●h he had ; and in the same manner you may practise with 4 , 5 , 6 , or many dice or other bodies , observing only that you must adde the points opposite of the dice ; for upon which depends the whole demonstration or secret of the play ; for alwayes that which is above and underneath makes 7. but if it make another number , then must you adde as often that number . problem . xlii . two mettals , as gold and silver , or of other kin●● weighing alike , being privately placed into two like boxes , to finde which of them the gold or silver is in . but because that this experiment in water hath divers accidents , and therefore subject to a caution ; and namely , because the matter of the chest , mettall or other things may hinder . behold here a more subtill and certaine invention to finde and discover it out without weighing it in the water ▪ now experience and reason sheweth us that two like bodies or magnitudes of equall weight , and of divers mettalls , are not of equal quantity : and seeing that gold is the heaviest of all mettalls , it will occupie less roome or place ; from which will follow that the like weight of lead in the same forme , will occupie or take up more roome or place . now let there be therefore presented two globes or chests of wood or other matter alike , & equall one to the other , in one of which in the middle there is another globe or body of lead weighing 12. l. ( as c , ) and in the other a globe or like body of gold weighing 12 pound ( as b. ) now it is supposed that the wooden globes or chests are of equall weight , forme , and magnitude : and to discover in which the gold or lead is in , take a broad paire of compasses , and clip one of the coffers or globes somewhat from the middle , as at d. then fix in the chest or globe a small piece of iron between the feet of the compasses , as ek , at the end of which hang a vveight g , so that the other end may be counterpoysed , and hang in aequilibrio : and do the like to the other chest or globe . novv if that the other chest or globe being clipped in like distance from the end , and hanging at the other end the same weight g. there be found no difference ; then clip them nearer tovvards the middle , that so the points of the compasse may be against some of the mettall vvhich is inclosed ; or just against the extremitie of the gold as in d , and suppose it hang thus in aequilibrio ; it is certaine that in the other coffer is the lead ; for the points of the compasses being advanced as much as before , as at f , vvhich takes up a part of the lead , ( because it occupies a greater place than the gold ) therefore that shall help the vveight g. to vveigh , and so vvill not hang in aequilibrio , except g be placed neare to f. hence vve may conclude , that there is the lead ; and in the other chest or globe , there is the gold. examination . if the two boxes being of equall magnitude weighed in the aire be found to be of equall weight , they shall necessarily take up like place in the water , and therefore weigh also one as much as another : hence there is no possibilitie to finde the inequalitie of the mettalls which are inclosed in these boxes in the water : the intention of archimedes was not upon contrary mettalls inclosed in 〈…〉 boxes , but consisted of comparing metta●●● , simple in the water one with another : therefore the inference is false and absurd . problem . xliii . two globes of diverse mettalls , ( as one gold , and the other copper ) yet of equall weight being put into a box , as bg , to finde in which end the gold or copper is . this is discovered by the changing of the places of the tvvo bovvles or globes , having the same counterpoyse h to be hung at the other side , as in n. and if the gold vvhich is the lesser globe , vvere before the nearest to the handle d● , having novv changed his place vvill be farthest from the handle de , as in k. therefore the centre of gravity of the two globes taken together , shall be farther separate from the middle of the handle ( under which is the centre of gravity of the box ) than it was before , and seeing that the handle is alwayes in the middle of the box , the vveight n. must be augmented ▪ to keep it in equil●●●● and by this way one may knovv , that if at the second time , the counterpoise be too light , it is a signe that the gold is farthest off the handle , as at the first triall it vvas nearest . problem . xliiii . how to represent diverse sorts of rainebowes here below ? the rainbovve is a thing admirable in the vvorld , vvhich ravisheth often the eyes and spirits of men in consideration of his rich intermingled colours vvhich are seen under the clouds , seeming as the glistering of the starres , precious stones , and ornaments of the most beauteous flovvers : some part of it as the resplendent stars , or as a rose , or burning cole of fire ▪ in it one may see dyes of sundry sorts , the violet , the blew , the orange , the saphir , the jacinct , and the emerald colours , as a lively plant placed in a green soile : and as a most rich treasure of nature , it is a high work of the sun who casteth his raies or beames as a curious painter drawes strokes with his pensill , and placeth his colours in an exquisite situation ; and solomon saith , eccles. 43. it is a chiefe and principall work of god. notwithstanding there is left to industrie how to represent it from above , here below , though not in perfection , yet in part , with the same intermixture of colours that is above . have you not seen how by oares of a boate it doth exceeding quickly glide upon the water with a pleasant grace ? aristotle sayes , that it coloureth the water , and makes a thousand atomes , upon which the beames of the sunne reflecting , make a kinde of coloured rainbowe : or may we not see in houses or gardens of pleasure artificiall fountaines , which poure forth their droppie streames of water , that being between the sunne and the fountaine , there will be presented as a continuall rainbowe ? but not to go farther , i will shew you how you may do it at your doore , by a fine and facill experiment . take water in your mouth , and turne your back to the sunne , and your face against some obscure place , then blow out the water which is in your mouth , that it may be sprinkled in small drops and vapours : you shall see those atomes vapours in the beames of the sunne to turne into a faire rainebowe , but all the griefe is , that it lasteth not , but soone is vanished . but to have one more stable and permanent in his colours : take a glasse full of water , and expose it to the sunne , so that the raies that passe through strike upon a shadowed place , you will have pleasure to see the fine forme of a rainebovve by this reflection . or take a trigonall glasse or crystall glasse of diverse angles , and look through it , or let the beames of the sunne passe through it ; or vvith a candle let the appearances be received upon a shadovved place : you vvill have the same contentment . problem xlv . how that if all the powder in the world were in closed within a bowle of paper or glasse , and being fired on all parts , it could not break that bowle ? if the bowle and the powder be uniforme in all his parts , then by that means the powder would presse and move equally on each side , in which there is no possibility whereby it ought to begin by one side more than another . now it is impossible that the bowle should be broken in all his parts : for they are infinite . of like fineness or subtiltie may it be that a bowle of iron falling from a high place upon a plaine pavement of thin glasse , it were impossible any wise to break it ; if the bowle were perfectly round , and the glasse flat and uniforme in all his parts ▪ for the bowle would touch the glasse but in one point , which is in the middle of infinite parts which are about it : neither is there any cause why it ought more on one side than on another , seeing that it may not be done with all his sides together ; it may be concluded as speaking naturally , that such a bovvle falling upon such a glasse vvill not break it . but this matter is meere metaphysicall , and all the vvorkmen in the vvorld cannot ever vvith all their industrie make a bovvle perfectly round , or a glasse uniforme . problem . xlvi . to finde a number which being divided by 2 , there will remaine 1 , being divided by 3 , there will remaine 1 ; and so likewise being divided by 4 , 5 , or 6 , there would still remaine 1 ; but being didivided by 7 , there will remaine nothing . in many authors of arithmetick this probleme is thus proposed : a vvoman carrying egges to market in a basket , met an unruly fellovv who broke them : who vvas by order made to pay for them : and she being demanded what number she had , she could not tell : but she remembred that counting them by 2 & 2 , there remained 1 ▪ likewise by 3 and 3 by 4 and 4 , by 5 and 5 , by 6 and 6 ; there still remained 1. but when she counted them by 7 and 7 , there remained nothing : now how may the number of egges be discovered ? finde a number which may exactly be measured by 7 , and being measured by 2 , 3 , 4 , 5 , and 6 ; there vvill still remaine a unite ▪ multiply these numbers together , makes 720 , to which adde 1 ; so have you the number , viz. 721. in like manner 301 vvill be measured by 2 , 3 , 4 , 5 , 6 ; so that 1 remaines : but being measured by 7 , nothing vvill remaine ; to vvhich continually adde 220 , and you have other numbers vvhich vvill do the same : hence it is doubtfull vvhat number she had , therefore not to faile , it must be knovvn vvhether they did exceed 400 , 800 , &c. in vvhich it may be conjectured that it could not exceed 4 or 5 hundred , seeing a man or vvoman could not carry 7 or 8 hundred egges , therefore the number vvas the former ●01 . vvhich she had in her basket : vvhich being counted by 2 and 2 , there vvill remaine 1 , by 3 and 3 , &c. but counted by 7 and 7 , there vvill remaine nothing . problem . xlvii . one had a certaine number of crownes , and counting them by 2 and 2 , there rested 1. counting them by 3 and 3 , there rested 2. counting them by 4 and 4 , there rested 3. counting them by 5 and 5 , there rested 4. counting them by 6 & 6 , there rested 5. but counting them by 7 and 7 , there remained nothing : how many crownes might he have ? this question hath some affinitie to the precedent , and the resolution is almost in the same manner : for here there must be found a number , vvhich multiplied by 7 , and then divided by 2 , 3 , 4 , 5 , 6 ; there may alvvayes remaine a number lesse by 1 than the divisor : novv the first number vvhich arrives in this nature is 119 , unto vvhich if 420 be added , makes 539 , vvhich also vvill do the same : and so by adding 420 , you may have other numbers to resolve this proposition . problem . xlviii . how many sorts of weights in the least manner must there be to weigh all sorts of things between 1 pound and 40 pound , and so unto 121 , & 364 pound . to vveigh things betvveen 1 and 40 , take numbers in triple proportion , so that their summe be equall , or somewhat greater than 40 , as are the numbers 1 3.9.27 . i say that with ● such weights , the first being of 1 pound , the second being 3 pound , the third being 9 pound , and the fourth being 27 : any weight between 1 and 40 pound may be weighed . as admit to weigh 21 pound , put unto the thing that is to be weighed the 9 pound weight , then in the other ballance put 27 pound and 3 pound , which doth counterpoise 21 pound and 9 pound , and if 20 pound were to be weighed , put to it in the ballance 9 and 1 , and in the other ballance put 27 and 3 , and so of others in the same manner take those 5 weights , 1 , 3 , 9 , 27 ▪ 81 , you may weigh with them between 1 pound , and 121 pound : and taking those 6 weights ▪ as 1 , 3 , 9 , 2● , 81 , 243 , you may weigh even from 1 pound unto 364 pound : this depends upon the property of continued proportionals , the latter of which containing twice all the former . problem . xlix . of a deceitfull ballance which being c●●●ty seemes i● be just , because it hangs in aequilibrio : not●ithstanding putting 12 pound in one ballance , and 11 in the other , it will remaine in aequilibrio . aristotle maketh mention of this ballance in his mechanick questions , and saith , that the merchants of purpose in his time used them to deceive the world : the subtiltie or craft of which is thus , that one arme of the ballance is longer then another , by the same proportion , that one weight is heavier then another : as if the beame were 23 inches long , and the handle placed so that 12 inches should be on one side of it , and 11 inches on the other side : conditionally that the shorter end should be as heavy as the longer , a thing easie to be done : then afterwards put into the ballance two unequal weights in such proportion as the parts of the beame have one unto another , which is 12 to 11 , but so that the greater be placed in the ballance which hangs upon the shorter part of the beame , and the lesser weight in the other ballance : it is most certaine that the ballances will hang in aequilibrio , which will seem most sincere and just ; though it be most deceitfull , abominable , and false . the reason of this is drawne from the experiments of archimedes , who shewes that two unequall weights will counterpoyse one another , when there is like proportion betweene the parts of the beame ( that the handle separates ) and the vveights themselves : for in one and the same counterpoise , by hovv much it is farther from the centre of the handle , by so much it seems heavier , therefore if there be a diversitie of distance that the ballances hang from the handle , there must necessarily be an ineqality of weight in these ballances to make them hang in aequilibrio , and to discover if there be deceit , change the weight into the other ballance , for as soone as the greater vveight is placed in the ballance that hangs on the longer parts of the beame : it vvill vveigh dovvne the other instantly . problem . l. to heave or lift up a bottle with a straw . take a stravv that is not bruised , bovv it that it make an angle , and put it into the bottle so that the greatest end be in the neck , then the reed being put in the bovved part vvil cast side-vvise , and make an angle as in the figure may be seen : then may you take the end vvhich is out of the bottle in your hand , and heave up the bottle , and it is so much surer , by how much the angle is acuter or sharper ; and the end which is bowed approacheth to the other perpendicular parts which come out of the bottle . problem . li. how in the middle of a wood or desert , without the sight of the sunne , starres , shadow or compasse , to finde out the north or south , or the foure cardinall points of the world , east , west , & c ? it is the opinion of some , that the windes are to be observed in this : if it be hot , the south is found by the windes that blow that way , but this observation is uncertaine and subject to much error : nature will help you in some measure to make it more manifest than any of the former , from a tree thus : cut a small tree off , even to the ground , and mark the many circles that are about the sap or pith of the tree , which seem nearer together in some part than in other , which is by reason of the suns motion about the tree : for that the humiditie of the parts of the tree towards the south by the heat of the sun is rarified , and caused to extend : and the s●n not giving such heat towards the north-part of the tree , the sap is lesser rarefied , but condensed ; by which the circles are nearer together on the north-part , than on the south-part : therefore if a line be drawne from the widest to the narrowest part of the circles , it shall shew the north & south of the world . another experiment may be thus : take a small needle , such as women work with : place it gently downe flatwise upon still water , and it will not sink , ( which is against the generall tenet that iron will not swimme ) which needle will by little and little turne to the north and south-points . but if the needle be great and will not swim , thrust it through a small piece of cork , or some such like thing , and then it will do the same : for such is the property of iron when it is placed in aequilibrio , it strives to finde out the poles of the world or points of north and south in a manner as the magnes doth . examination . here is observable , that the moisture which aideth to the growth of the tree , is dilated and rarefied by the meridionall heat , and contracted by the septentrionall cold : this rarefaction works upon the part of the humour or moisture that is more thinne , which doth easily dissipate and evaporate : which evaporation carries a part of the salt with it ; and because that solidation or condensation , so that there is left but a part of the nourishment which the heat bakes up and consumes : so contrarily on the other side the condensation and restrictive quality of the moisture causeth lesse evaporation and perdition : and so consequently there remaines more nourishment , which makes a greater increase on that side than on the other side : for as trees have their growth in winter , because of their pores and these of the earth are shut up : so in the spring when their pores are open , and when the sappe and moisture is drawne by it , there is not such cold on the north-side that it may be condensed at once : but contrarily to the side which is south , the heat may be such , that in little time by continuance , this moisture is dissipated greatly : and cold is nothing but that which hardneth and contracteth the moisture of the tree , and so converteth it into wood . problem . lii . three persons having taken counters , cards , or other things , to finde how much each one hath taken . cause the third party to take a number which ma● be divided by 4 , and as often as he takes 4 , let the second party take 7 , and the first take 13 , then cause them to put them all together , and declare the summe of it ; which secretly divide by 3 , and the quotient is the double of the number which the third person did take . or cause the third to give unto the second and first , as many as each of them hath ; then let the second give unto the first and third , as many as each of them hath ; lastly , let the third give unto the second and first , as many as each of them hath ; and then aske how much one of them hath ; ( for they will have then all alike , ) so halfe of that number is the number that the third person had at the first : which knowne all is knowne . problem . liii . how to make a consort of musick of many parts with one voyce , or one instrument only ? this probleme is resolved , so that a finger or player upon an instrument , be neare an echo which answereth his voice or instrument ; and if the echo answereth but once at a time , he may make a double ; if twice , then a triple , if three times , then an harmonie of foure parts , for it must be such a one that is able to exercise both tune and note as occasion requires . as when he begins ut , before the echo answer , he may begin sol , and pronounce it in the same tune that ●he echo answereth , by which meanes you ●ave a fifth , agreeable consort of musick : then in the same time that the echo followeth , to sound the second note sol , he may sound forth another sol higher or lower to make an eight , the most perfect consort of musick , and so of others , if he will continue his voice with the echo , and sing alone with two parts . now experience sheweth this to be true , which often comes to passe in many churches , making one to beleeve that there are many more parts in the musick of a quire , then in effect truly there are because of the resounding and multiplying of the voic● , and redoubling of the quire. problem . liiii . t● make or describe an ovall form , or that which neare resembles unto it , at one turning with a paire of common compasses . there are many fine wayes in geometricall practices , to make an ovall figure or one neare unto it , by severall centres : any of which i will not touch upon , but shew how it may be done promptly upon one centre only . in which i will say nothing of the ovall forme , which appeares , when one describeth circles with the points of a common compasses , somewhat deep upon a skinne stretched forth hard : which contracting it selfe in some parts of the skinne maketh an ovall forme . but it will more evidently appeare upon a columne or cylinder : if paper be placed upon it , then with a paire of compasses describe as it were a circle upon it , which paper afterwards being extended , will not be circular but ovall-wise : and a paire of compasses may be so accommodated , that it may be done also upon a plaine thus . as let the length of the ovall be h. k , fasten 2 pinnes or nailes neare the end of that line as f. g , and take a threed which is double to the length of g. h , or f. k , then if you take a compasse which may have one foot lower than another , with a spring between his legges : and placing one foot of this compasse in the centre of the ovall , and guiding the threed by the other foot of the compasses , and so carrying it about : the spring will help to describe and draw the ovall forme . but in stead of the compasses it may be done with ones hand only , as in the figure may appeare . problem . lv. of a pu●se difficult to be opened . it is made to shut and open with rings : first at each side there is a strap or string , as ab . and cd , at the end of which are 2 rings , b & d , and the string cd passeth through the ring b , so that it may not come out againe ; or be parted one from another : and so that the ring b , may slide up and downe upon the string cd , then over the purse , there is a piece of leather efgh , which covers the opening of the purse , and there is another piece of leather ae , which passeth through many rings : which hath a slit towards the end i , so great that the string bc may slide into it : now all the cunning or craft is how to make fast or to open the purse , which consists in making the string bc slide through the side at i , therefore bring down b to i , then make the end i passe through the ring b , and also d with his string to passe through the slit i , so shall the purse be fast , and then may the strings be put as before , and it will seem difficult to discover how it was done . now to open the purse , put through the end i through the ring b , and then through the slit i , by which you put through the string dc , by this way the purse will be opened . problem . lvi . whether it is more hard and admirable without compasses to make a perfect circle , or being made to finde out the centre of it ? it is said that upon a time past , two mathematicians met , and they would make tryall of their industry : the one made instantly a perfect circle without compasses , and the other immediately pointed out the centre thereof with the point of a needle ; now which is the chiefest action ? it seems the first , for to draw the most noblest figure upon a plaine table without other help than the hand , and the minde , is full of admiration ; to finde the centre is but to finde out only one point , but to draw a round , there must be almost infinite points , equidistant from the centre or middle ; that in conclusion it is both the circle and the centre together . but contrarily it may seem that to finde the centre is more difficult , for what attention , vivacitie , and subtiltie must there be in the spirit , in the eye , in the hand , which will chuse the true point amongst a thousand other points ? he that makes a circle keeps alwayes the same distance , and is guided by a halfe distance to finish the rest ; but he that must finde the centre , must in the same time take heed to the parts about it , and choose one only point which is equall distant from an infinite of other points which are in the circumference ; which is very difficult . aristotle confirmes this amongst his morals , and seems to explaine the difficultie which is to be found in the middle of vertue ; for it may want a thousand wayes , and be farre separated from the true centre of the end of a right mediocritie of a vertuous action ; for to do well it must touch the middle point which is but one , and there must be a true point which respects the end , and that 's but one only . now to judge which is the most difficult , as before is said , either to draw the round or to finde the centre , the round seems to be harder than to finde the centre , because that in finding of it , it is done at once , and hath an equall distance from the whole ; but , as before , to draw a round there is a visible point imagined , about which the circle is to be drawne . i esteeme that it is as difficult therefore , if not more , to make the circle without a centre , as to finde the middle or centre of that circle . problem . lvii . any one having taken 3 cards , to finde how many points they containe this is to be exercised upon a full pack of cards of 52 , then let one choose any three at pleasure secretly from your sight , and bid him secretly account the points in each card , and will him to take as many cards as will make up 15 to each of the points of his cards , then will him to give you the rest of the cards , for 4 of them being rejected , the rest shew the number of points that his three cards which he took at the first did conteine . as if the 3 cards were 7 , 10 , and 4 ; now 7 wants of 15 , 8. take 8 cards therefore for your first card : the 10 wants of 15 ▪ 5 , take 5 cards for your second card : lastly 4 wants of 15 , 11 , take 11 cards for your third card , & giving him the rest of the cards , there will be 25 ; from which take 4 , there remaines 21 , the number of the three cards taken , viz. 7 , 10 , and 4. whosoever would practise this play with 4 , 5 , 6 , or more cards , and that the whole number of cards be more or lesse than 52 ; and that the terme be 15 , 14 , 12 , &c , this generall rule ensuing may serve : multiply the terme by the number of cards taken at first : to the product adde the number of cards taken , then subtract this summe from the whole number of cards ; the remainder is the number which must be subtracted from the cards , which remaines to make up the game : if there remaine nothing after the subtraction , then the number of cards remaining doth justly shew the number of points which were in the cards chosen . if the subtraction cannot be made , then subtract the number of cards from that number , and the remainder added unto the cards that did remaine , the summe will be the number of points in the cards taken , as if the cards were 7 , 10 , 5 , 8 , and the terme given were 12 ; so the first wants 5 , the second wants 2 , the third wants 7 , and the fourth wants 4 cards , which taken , the party gives you the rest of the cards : then secretly multiply 12 by 4 , makes 48 ; to which adde 4 , the number of cards taken makes 52 , from which 52 should be taken , rest nothing : therefore according to the direction of the remainder of the cards which are 30 , is equall to the points of the foure cards taken , viz. 7 , 10 , 5 , 8. againe , let these five cards be supposed to be taken , 8 , 6 , 10 , 3 , 7 ; their differences to 15 , the termes are 7 , 9 , 5 , 12 , 8 , which number of cards taken , there will remaine but 6 cards : then privately multiply 15 by 5 , makes 75 , to which adde 5 makes 80 , from this take 52 the number of cards , rest 28 , to vvhich add the remainder of cards , make 34. the summe with 8 , 6 , 10 , 3 , 7. problem . lvii . many cards placed in diverse ranks , to finde which of these cards any one hath thought . take 15 cards , and place them in 3 heaps in rank-wise , 5 in a heap : now suppose any one had thought one of these cards in any one of the heaps , it is easie to finde vvhich of the cards it is , and it is done thus ; ask him in vvhich of the heaps it is , vvhich place in the middle of the other tvvo ; then throvv dovvne the cards by 1 and 1 into three severall heaps in rank-vvise , untill all be cast dovvne , then aske him in which of the rankes his card is , which heap place in the middle of the other two heaps alwayes , and this do foure times at least , so in putting the cards altogether , look upon the cards , or let their back be towards you , and throw out the eight card , for that was the card thought upon without faile . problem . lviii . many cards being offered to sundry persons , to finde which of these cards any one thinketh upon . admit there were 4 persons , then take 4 cards , and shew them to the first , bid him think one of them , and put these 4 away , then take 4 other cards , and shew them in like manner to the second person , and bid him think any one of these cards , and so do to the third person , and so the fourth , &c. then take the 4 cards of the first person , and dispose them in 4 rankes , and upon them the 4 cards of the second person , upon them also these of the third person , and lastly , upon them these of the fourth person , then shew unto eaeh of these parties each of these ranks , and aske him if his card be in it which he thought , for infallibly that vvhich the first partie thought upon vvill be in the first rank , and at the bottome , the card of the second person vvill be in the second ranke , the card of the third thought upon will be in the third rank , and the fourth mans card will be in the fourth rank , and so of others , if there be more persons use the same method . this may be practised by other things , ranking them by certaine numbers : allotted to pieces of money , or such like things . problem . lix . how to make an instrument to help hearing , as galileus made to help the sight ? think not that the mathematickes ( which hath furnished us with such admirable helps for seeing ) is wanting for that of hearing , it s well knowne that long trunks or pipes make one heare well farre off , and experience shewes us that in certaine places of the orcades in a hollow vault , that a man speaking but softly at one corner thereof , may be audibly understood at the other end : notwithstanding those which are between the parties cannot heare him speak at all : and it is a generall principle , that pipes do greatly help to strengthen the activitie of naturall causes : we see that 〈◊〉 contracted in a pipe , burnes 4 or 5 foot high , which would scarce heat , being in the open aire : the rupture or violence of water issuing out of a fountaine , shewes us that vvater being contracted into a pipe , causeth a violence in its passage . the glasses of galeileus makes us see how usefull pipes or trunkes are to make the light and species more visible , and proportionable to our eye . it is said that a prince of italy hath a faire hall , in which he can with facility heare distinctly the discourses of those which walk in the adjacent gardens , which is by certaine vessels and pipes that answer from the garden to the hall. vitruvius makes mention also of such vessels and pipes , to strengthen the voice and action of comedians : and in these times amongst many noble personages ▪ the new kinde of trunkes are used to help the hearing , being made of silver , copper , or other resounding materiall ; in funnell-wise putting the widest end to him which speaketh , to the end to contract the voice , that so by the pipe applied to the eare it may be more uniform and lesse in danger to dissipate the voice , and so consequently more fortified . problem . lx. of a fine lamp which goes not out , though one carry it in ones pocket : or being rolled upon the ground will still burne . it must be observed that the vessell in which the oile is put into , have two pinnes on the sides of it , one against another , being included within a circle : this circle ought to have two other pinnes , to enter into another circle of brasse , or other solid matter : lastly , this second circle hath two pinnes , which may hang within some box to containe the whole lamp , in such manner , that there be 6 pinnes in different position : now by the aid of these pegges or pinnes , the lamp that is in the middle will be alwayes well situated according to his centre of gravity , though it be turned any way : though if you endeavour to turne it upside downe , it will lie levell ▪ which is pleasant and admirable to behold to those which know not the cause : and it is facil from his to make a place to rest quiet in , though there be great agitation in the outvvard parts . problem . lxi . any one having thought a card amongst many cards , how artificially to discover it out ? take any number of cards as 10 , 12 , &c. and open some 4 or 5 to the parties sight , and bid him think one of them , but let him note vvhether it be the first , second , third , &c. then vvith promptness learn vvhat number of cards you had in your hands , and take the other part of the cards , and place them on the top of these you hold in your hand ; and having done so , aske him whether his card were the first , second , &c. then before knowing the number of cards that were at the bottome , account backwards untill you come to it : so shall you easily take out the card that he thought upon . problem . lxii . three women ab.c. carried apples to a marke to sell , a had 20 , b 30 ▪ and c 40 , they sold as many for a penny , the one as the other : and brought home one as much money as another , how could this be ? the answer to the probleme is easie ▪ as suppose at the beginning of the market : a ▪ sold her apples at a penny an apple : and sold but 2. which was 2 pence , and so she had 18 left : but b. sold 17. which was 17 pence , and so had 13 left : c. sold 32. which was 32 pence , and so had 8 apples left ▪ then a said she would not sell her apples so cheap , but would sell them for 3 pence the peece , which she did : and so her apples came to 54 pence , and b having left but 13 apples sold them at the same rate , which came to 39 pence : and lastly ▪ c. had but 8 apples , which at the same rate came to 24 pence : these summes of money which each others before received come to 56 pence , and so much each one received ; and so consequently brought home one as much as another . problem . lxiii . of the properties of some numbers . first , any two numbers is just the summe of a number , that have equall distance from the halfe of that number ▪ the one augmenting , and the other diminishing , as 7 and 7 , of 8 and ● , of 9 and 5 , of 10 and 4 , of 11 and 3 , of 12 and 2 , of 13 and ● . as the one is more than the halfe , the other is lesse . secondly , it is difficult to finde two numbers whose summe and product is alike , ( that is ) if the numbers be multiplied one by another , and added together , will be equall , which two numbers are 2 and 2 , for to multiply 2 by 2 makes 4 , and adding 2 unto 2 makes the same : this property is in no other two whole numbers , but in broken numbers there are infinite , whose summe and product will be equall one to another . as clavius shewes upon the 36 pro. of the 9t h book of euclide . thirdly , the numbers 5 and 6 are called circular numbers , because the circle turnes to the point from whence it begins : so these numbers multiplied by themselves , do end alwayes in 5 and 6 , as 5 times 5 makes 25 , that againe by 5 makes 125 , so 6 times 6 makes 36 , and that by 6 makes 216 , &c. fourthly , the number 6 , is the first which arithmeticians call a perfect number , that is , whose parts are equall unto it , so the 6 part of it is 1 , the third part is 2 , the halfe is 3 , which are all his parts : now 1 , 2 , and 3 , is equall to 6. it is wonderfull to conceive that there is so few of them , and how rare these numbers are ▪ 50 of perfect men : for betwixt 1 & 1000000000000 numbers there is but ten , that is ; 6 , 28 , 486. 8128. 120816. 2096128. 33550336. 536854528. 8589869056 , & 137438691328 ▪ with this admirable property , that alternately they end all in 6 and 8 , & the twentieth perfect number is 151115727451553768931328. fiftly , the number 9 amongst other priviledges carries with it an excellent property : for take what number you will , either in grosse or in part , the nines of the whole or in its parts rejected , and taken simply will be the same , as ●7 it makes 3 times 9 , so vvhether the nines be rejected of 27 , or of the summe of 2 and 7 , it is all one , so if the nines vvere taken avvay of 240. it is all one , if the nines vvere taken avvay of 2 , 4 , and 0 ; for there vvould remaine 6 in either ; and so of others . sixtly , 11 being multiplied by 2 , 4 , 5 , 6 , 7 , 8 , or 9 , will end and begin with like numbers ; so 11 multiplied by 5 makes 55 , if multiplied by 8 , it makes 88 , &c. seventhly , the numbers 220 and 284 being unequall , notwithstanding the parts of the one number do alwayes equalize the other number : so the aliquot parts of 220 are 110 , 54 , 44 , 22 , 20 , 11 , 10 , 5 , 4 , 2 , 1 , which together makes 284. the aliquot parts of 284 , are 142 , 71 , 4 , 2 , 1. which together makes 220 , a thing rare and admirable , and difficult to finde in other numbers . i● one be taken from any square number which is odde , the square o● halfe of it being added to the first square , will make a square number . the square of halfe any even number + . 1 being added to that even number makes a square number , and the even number taken from it leaves a square number . if odde numbers be continually added from the unitie successively , there will be made all square numbers , and if cubick numbers be added successively from the unitie , there will be likewise made square numbers . problem . lxiv . of an excellent lamp , which serves or furnisheth it selfe with oile , and burnes a long time . i speak not here of a common lamp which ca●danus writes upon in his book de subtilita●● , for that 's a little vessell in columne-wise , which is full of oile , and because there is but one little hole at the bottome neare the weeke or match ; the oile runnes not , for feare that there be emptinesse above : when the match is kindled it begins to heat the lamp , and rarefying the oile it issueth by this occasion : and so sends his more airie parts above to avoid vacuitie . it is certaine that such a lampe the atheniaus used , which lasted a whole yeare without being touched : which was placed before the statue of minerva , for they might put a certaine quantitie of oile in the lamp cd , and a match to burne without being consumed : such as the naturalists write of , by which the lamp will furnish it selfe , and so continue in burning : and here may be noted that the oile may be poured in , at the top of th● vessell at a little hole , and then made fast againe that the aire get not in . problem . lxv . of the play at keyles or nine pinnes . you will scarce beleeve that with one bowle and at one blow playing freely , one may strike downe all the keyles at once : yet from mathematicall principles it is easie to be demonstrated , that if the hand of him that playes were so well assured by experience , as reason induceth one thereto ; one might at one blow strike downe all the keyles , of at least 7 or 8 , or such a number as one pleaseth . for they are but 9 in all disposed or placed in a perfect square , having three every way . let us suppose then that a good player beginning to play at 1 somewhat low , should so strike it , that it should strike down the keyles 2 and 5 , and these might in their violence strike downe the keyles 3 , 6 , and 9 , and the bowle being in motion may strike down the keyle 4 , and 7 ; which 4 keyle may strike the keyle 8 , & so all the 9 keyles may be striken down at once . problem . lxiv . of spectacles of pleasure . simple spectacles of blew , yellow , red or green colour , are proper to recreate the sight , and will present the objects died in like colour that the glasses are , only those of the greene do somewhat degenerate ; instead of shewing a lively colour it will represent a pale dead colour , and it is because they are not dyed greene enough , or receive not light enough for greene : and colour these images that passe through these glasses unto the bottome of the eye . examination . it is certaine , that not onely glasses dyed green , but all other glasses coloured , yield the app●arances of objects strong or weak in colour according to the quantity of the dye , more or lesse , as one being very yellow , another a pale yellow ; now all colours are not proper to glasses to give colour , hence the defect is not that they want facultie to receive light , or resist the penetration of the beams ; for in the same glasses those which are most dyed , give alwayes the objects more high coloured and obscure , and those which are lesse dyed give them more pale and cleare : and this is daily made manifest by the painting of glasse , which hinders more the penetration of the light than dying doth , where all the matter by fire is forced into the glasse , leaving it in all parts transparent . spectacles of crystall cut with divers angles diamond-wise do make a marvellous multiplication of the appearances , for looking towards a house it becomes as a towne , a towne becomes like a citie , an armed man seems as a whole company caused solely by the diversity of refractions , for as many plaines as there are on the outside of the spectacle , so many times will the object be multiplied in the appearance , because of diverse images cast into the eye . these are pleasurable spectacles for avaricious persons that love gold and silver , for one piece will seeme many , or one heap of money will seeme as a treasury : but all the mischiefe is , he will not have his end in the enjoying of it , for indeavouring to take it , it will appeare but a deceitfull image , or delusion of nothing . here may you note that if the finger be directed by one and the same ray or beam , which pointeth to one and the same object , then at the first you may touch that visible object without being deceived : otherwise you may faile often in touching that which you see . againe , there are spectacles made which do diminish the thing seen very much , and bring it to a faire perspective forme , especially if one look upon a faire garden plat , a greater walk , a stately building , or great court , the industry of an exquisite painter cannot come neare to expresse the lively forme of it as this glasse will represent it ; you will have pleasure to see it really experimented , and the cause of this is , that the glasses of th●se spectacles are hollow and thinner in the middle , than at the edges by which the visuall angle is made lesser : you may observe a further secret in these spectacles , for in placing them upon a window one may see those that passe to and fro in the streets , without being seen of any , for their property is to raise up the objects that it lookes upon . now i would not passe this probleme without saying something of galileus admirable glasse , for the common simple perspective glasses give to aged men but the eyes or sight of young men , but this of galileus gives a man an eagles eye , or an eye that pierceth the heavens : first it discovereth the spottie and shadowed opacous bodies that are found about the sunne , which darknet and diminisheth the splendor of that beautifull and shining luminary : secondly , it shewes the new planets that accompany saturne and jupiter : thirdly , in venus is seen the new , full , and quartill increase ; as in the moon by her separation from the sunne : fourthly , the artificiall structure of this instrument helpeth us to see an innumerable number of stars , which otherwise are obscured , by reason of the naturall weaknesse of our sight , yea the starres in via lactea are seen most apparantly ; where there seem no starres to be , this instrument makes apparantly to be seen , and further delivers them to the eye in their true and lively colour , as they are in the heavens : in which the splendor of some is as the sunne in his most glorious beauty . this glasse hath also a most excellent use in observing the body of the moone in time of eclipses , for it augments it manifold , and most manifestly shewes the true forme of the cloudy substance in the sunne ; and by it is seene when the shadow of the earth begins to eclipse the moon , & when totally she is over shadowed : besides the celestiall uses which are made of this glasse , it hath another noble property ; it farre exceedeth the ordinary perspective glasses , which are used to see things remote upon the earth , for as this glasse reacheth up to the heavens and excelleth them there in his performance , so on the earth it claimeth preheminency , for the objects which are farthest remote , and most obscure , are seen plainer than those which are neere at hand , scorning as it were all small and triviall services , as leaving them to an inferiour help : great use may be made of this glass in discovering of ships , armies , &c. now the apparell or parts of this instrument or glasse , is very meane or simple , which makes it the more admirable ( seeing it performes such great service ) having but a convex glasse thickest in the middle , to unite and amasse the rayes , and mak the object the greater : to the augmenting the visuall angle , as also a pipe or trunk to amasse the species , and hinder the greatness of the light which is about it : ( to see well , the object must be well inlightened , and the eye in obscurity ; ) then there is adjoyned unto it a glasse of a short sight to distinguish the rayes , which the other would make more confused if alone . as for the proportion of those glasses to the trunk , though there be certaine rules to make them , yet it is often by hazard that there is made an excellent one there being so many difficulties in the action , therefore many ought to be tryed , seeing that exact proportion , in geometricall calculation cannot serve for diversity of sights in the observation . problem . lxvii . of the adamant or magnes , and the needles touched therewith . who would beleeve if he saw not with his eyes , that a needle of steel being once touched with the magnes , turnes not once , not a yeare ▪ but as long as the world lasteth ; his end towards the north and south , yea though one remove it , and turne it from his position , it will come againe to his points of north and south . who would have ever thought that a brute stone black and ill formed , touching a ring of iron , should hang it in the aire , and that ring support a second , that to support a third , and so unto 10 , 12 , or more , according to the strength of the magnes ; making as it vvere a chaine without a line , without souldering together , or without any other thing to support them onely ; but a most occult and hidden vertue , yet most evident in this effect , which penetrateth insensibly from the first to the second , from the second to the third , &c. what is there in the world that is more capable to cast a deeper astonishment in our minds than a great massie substance of iron to hang in the aire in the middest of a building without any thing in the world touching it , only but the aire ? as some histories assure us , that by the aid of a magnes or adamant , placed at the roof of one of the turkish synagogues in meca : the sepulchre of that infamous mah●met rests suspended in the aire ; and plinie in his naturall historie writes that the architect or democrates did begin to vault the temple of a●sin●e in alexandria , with store of magnes to produce the like deceit , to hang the sepulchre of that goddesse likewise in the aire . i should passe the bounds of my counterpoise , if i should divulge all the secrets of this stone , and should expose my selfe to the laughter of the world : if i should brag to shew others the cause how this appeareth , than in its owne naturall sympathy , for why is it that a magnes with one end will cast the iron away , & attract it with the other ? from whence commeth it that all the magnes is not proper to give a true touch to the needle , but only in the two poles of the stone : which is known by hanging the stone by a threed in the aire untill it be quiet , or placed upon a peece of cork in a dish of water , or upon some thinne board , for the pole of the stone will then turne towards the poles of the world , and point out the north and south , and so shew by which of these ends the needle is to be touched ? from whence comes it that there is a variation in the needle , and pointeth not out truly the north and south of the world , but only in some place of the earth ? how is it that the needle made with pegges and inclosed within two glasses , sheweth the height of the pole , being elevated as many degrees as the pole is above the horizon ? what 's the cause that fire and garlick takes away the propertie of the magnes ? there are many great hidden mysteries in this stone , which have troubled the heads of the most learned in all ages ; and to this time the world remaines ignorant of declaring the rrue cause thereof . some say , that by help of the magnes persons which are absent may know each others minde , as if one being here at london , and another at prague in germany : if each of them had a needle touched with one magnes , then the vertue is such that in the same time that the needle which is at prague shall move , this that is at london shall also ; provided that the parties have like secret notes or alphabets , and the observation be at a set houre of the day or night ; and when the one party will declare unto the other , then let that party move the needle to these letters which will declare the matter to the other , and the moving of the other parties needle shall open his intention . the invention is subtile , but i doubt whether in the world there can be found so great a stone ▪ or such a magnes which carries with it such vertue : neither is it expedient , for treasons would be then too frequent and open . examination . the experimentall difference of rejection , and attraction proceeds not from the different nature of stones , but from the quality of the iron ; and the vertue of the stone consisteth only , and especially in his poles , which being hanged in the aire , turnes one of his ends alwayes naturally towards the south , and the other towards the north : but if a rod of iron be touched with one of the ends thereof , it hath the like property in turning north and south , as the magnes hath : notwithstanding the end of the iron rod touched , hath a contrary position , to that end of the stone that touched it ; yet the same end will attract it , and the other end reject it : and so contrarily this may easily be experimented upon two needles touched with one or different stones , though they have one and the same position ; for as you come unto them apply one end of the magnes neare unto them , the north of the one will abhorre the north of the other , but the north of the one will alwayes approach to the south of the other : and the same affection is in the stones themselves . for the finding of the poles of the magnes , it may be done by holding a small needle between your fingers softly , and so moving it from part to part over the stone untill it be held perpendicular , for that shall be one of the poles of the stone which you may marke out ; in like manner finde out the other pole : now to finde out which of those poles is north or south , place a needle being touched with one of the poles upon a smooth convex body , ( as the naile of ones finger or such like , ) and marke which way the end of the needle that was touched turneth : if to the south , then the point that touched it was the south-pole , &c. and it is most certain and according to reason and experience : that if it be suspended in aequilibrio in the aire , or supported upon the water , it will turne contrary to the needle that toucheth it ; for then the pole that was marked for the south shall turne to the north , &c. problem . lxviii . of the properties of aeolipiles or bowels to blow the fire . these are concave vessels of brass or copper or other material , which may indure the , fire : having a small hole very narrow , by which it is filled with water , then placing it to the fire , before it be hot there is no effect seen ; but assoone as the heat doth penetrate it , the water begins to rarefie , & issueth forth with a hidious and marvelous force ; it is pleasure to see how it blowes the fire with great noise . novv touching the forme of these vessels , they are not made of one like fashion : some makes them like a bovvle , some like a head painted representing the vvinde , some make them like a peare : as though one vvould put it to rost at the fire , vvhen one vvould have it to blovv , for the taile of it is hollovv , in forme of a funnell , having at the top a very little hole no greater than the head of a pinne . some do accustome to put vvithin the aeolipile a crooked funnell of many foldings , to the end that the vvinde that impetuously rolles ▪ to and fro vvithin , may imitate the noise of thunder . others content themselves vvith a simple funnell placed right upvvard , somevvhat vvider at the top than elsevvhere like a cone , vvhose basis is the mouth of the funnell : and there may be placed a bovvle of iron or brasse , vvhich by the vapours that are cast out vvill cause it to leap up , and dance over the mouth of the aeolipile . lastly , some apply near to the hole smal windmils , or such like , vvhich easily turne by reason of the vapours ; or by help of tvvo or more bovved funnels , a bowle may be made to turne● these aeolipiles are of excellent use for the melting of mettalls and such like . now it is cunning and subtiltie to fill one of these aeolipiles with water at so little a hole , and therefore requires the knowledge of a philosopher to finde it out : and the way is thus . heat the aeolipiles being empty , and the aire which is within it will become extreamely rarefied ; then being thus hot throw it into water , and the aire will begin to be condensed : by which meanes it will occupie lesse roome , therefore the water will immediately enter in at the hole to avoide vacuitie : thus you have some practicall speculation upon the aeolipile . problem . lxix . of the thermometer : or an instrument to measure the degrees of heat and cold in the aire . this instrument is like a cylindricall pipe of glasse , which hath a little ball or bowle at the toppe ▪ the small end of which is placed into a vessell of water below , as by the figure may be seene . then put some coloured liquor into the cylindricall glasse , as blew , red , yellow , green , or such like : such as is not thick . this being done the use may be thus . those that will determine this change by numbers and degrees , may draw a line upon the cylinder of the thermometer ; and divide it into 4 degrees , according to the ancient philosophers , or into 4 degrees according to the physicians , dividing each of these 8 into 8 others : to have in all 64 divisions , & by this vvay they may not only distinguish upon vvhat degree the vvater ascendeth in the morning , at midday , & at any other houre : but also one may knovv hovv much one day is hotter or colder than another : by marking hovv many degrees the vvater ascendeth or descendeth , one may compare the hottest and coldest dayes in a vvhole year together vvith these of another year : againe one may knovv hovv much hotter one roome is than another , by vvhich also one might keep a chamber , a furnace , a stove , &c. alvvayes in an equalitie of heat , by making the vvater of the thermometer rest alvvayes upon one & the same degree : in brief , one may judge in some measure the burning of fevers , and neare unto what extension the aire can be rarefied by the greatest heat . many make use of these glasses to judge of the vveather : for it is observed that if the vvater fall in 3 or 4 hours a degree or thereabout , that raine insueth ; and the vvater vvill stand at that stay , untill the vveather change : marke the water at your going to bed , for if in the morning it hath descended raine followeth , but if it be mounted higher , it argueth faire weather : so in very cold weather , if it fall suddenly , it is snow or some sleekey weather that wiil insue , problem . lxx . of the proportion of humane bodies of statues , of colossus or huge images , and of monstrous giants . pythagoras had reason to say that man is the measure of all things . first , because he is the most perfect amongst all bodily creatures , & according to the maxime of philosophers , that which is most perfect and the first in rank , measureth all the rest . secondly , because in effect the ordinary measure of a foot , the inch , the cubit , the pace , have taken their names and greatnesse from humane bodies . thirdly , because the symmetrie and concordancie of the parts is so admirable , that all workes which are well proportionable , as namely the building of temples , of shippes , of pillars , and such like pieces of architecture , are in some measure fashioned and composed after his proportion . and we know that the arke of noah built by the commandement of god , was in length 300 cubits , in breadth 50 cubits , in height or depth 30 cubits , so that the length containes the breadth 6 times , and 10 times the depth : now a man being measured you will finde him to have the same proportion in length , breadth , and depth . vilalpandus treating of the temple of solomon ( that chieftaine of works ) was modulated all of good architecture , and curiously to be observed in many pieces to keep the same proportion as the body to his parts : so that by the greatnesse of the work and proportionable symmetrie , some dare assure themselves that by knowledge of one onely part of that building , one might know all the measures of that goodly structure . some architects say that the foundation of houses , and basis of columnes , are as the foot ; the top , and roofe as the head ; the rest as the body : those which have beene somewhat more curious , have noted that as in humane bodies , the parts are uniforme , as the nose , the mouth , &c. these which are double are put on one side or other , with a perfect equality in the same architecture . in like manner , some have been yet more curious than solid ; comparing all the ornaments of a corinth to the parts of the face , as the brow , the eyes , the nose , the mouth ; the rounding of pillars , to the vvrithing of haire , the channells of columnes , to the fouldings of vvomens robes , &c. novv building being a vvork of the best artist , there is much reason vvhy man ought to make his imitation from the chiefe vvork of nature ; vvhich is man. hence it is that vitru●ius in his third book , and all the best architectes , treate of the proportion of man ; amongst others albert durens hath made a whole book of the measures of mans body , from the foot to the head , let them read it who wil , they may have a prefect knowledge thereof : but i will content my selfe and it may satisfie some with that which followeth . first , the length of a man well made , which commonly is called height , is equall to the distance from one end of his finger to the other : when the armes are extended as wide as they may be . secondly , if a man have his feet and hands extended or stretched in forme of s. andrews crosse , placing one foot of a paire of compasses upon his navill , one may describe a circle which will passe by the ends of his hands and feet , and drawing lines by the termes of the hands and feet , you have a square within a circle . thirdly , the breadth of man , or the space which is from one side to another ; the breast , the head , and the neck , make the 6 part of all the body taken in length or height . fourthly , the length of the face is equall to the length of the hand , taken from the small of the arme , unto the extremity of the longest finger . fiftly , the thicknesse of the body taken from the belly to the back ; the one or the other is the tenth part of the whole body , or as some will have it , the ninth part , little lesse . sixtly , the height of the brow , the length of the nose , the space between the nose and the chinne , the length of the eares , the greatnesse of the thumbe , are perfectly equall one to the other . what would you say to make an admirable report of the other parts , if i should reckon them in their least ? but in that i desire to be excused , and will rather extract some conclusion upon ▪ that which is delivered . in the first place , knowing the proportion of a man , it is easie to painters , image-makers , &c. perfectly to proportionate their work ; and by the same is made most evident , that which is related of the images and statues of greece , that upon a day diverse workmen having enterprised to make the face of a man , being severed one from another in sundry places , all the parts being made and put together , the face was found in a most lively and true proportion . secondly , it is a thing most cleare , that by the help of proportion , the body of hercules was measured by the knowledge of his foot onely , a lion by his claw , the giant by his thumb , and a man by any part of his body . for so it was that pythagoras having measured the length of hercules foot , by the steps which were left upon the ground , found out all his height : and so it was that phidias having onely the claw of a lion , did figure and draw out all the beast according to his true type or forme , so the exquisite painter timantes , having painted a pygmey or dwarfe , which he measured with a fadome made with the inch of a giant , it was sufficient to know the greatnesse of that giant to be short , we may by like methode come easily to the knowledge of many fine antiquities touching statues , colossus , and monstrous giants , onely supposing one had found but one only part of them , as the head , the hand , the foot or some bone mentioned in ancient histories . of statues , of colossus , or huge images . vitruvius relates in his second book , that the architect dinocrates was desirous to put out to the world some notable thing , went to alexander the great , and proposed unto him a high and speciall piece of work which he had projected : as to figure out the mount athos in forme of a great statue , which should hold in his right hand a towne capable to receive ten thousand men : and in his left hand a vessell to receive all the water that floweth from the mountaine , which with an ingine should cast into the sea. this is a pretty project , said alexander , but because there was not field-roome thereabout to nourish and reteine the citizens of that place , alexander was wise not to entertaine the designe . now let it be required of what greatnesse this statue might have been , the towne in his right hand , and the receiver of water in his left hand if it had been made . for the statue , it could not be higher than the mountaine it selfe , and the mountaine was about a mile in height plumb or perpendicular ; therefore the hand of this statue ought to be the 10th part of his height , which would be 500 foot , and so the breadth of his hand would be 250 foot , the length now multiplyed by the breadth , makes an hundred twenty five thousand square feet , for the quantitie of his hand to make the towne in , to lodge the said 10 thousand men , allowing to each man neere about 12 foot of square ground : now judge the capacitie of the other parts of this collossus by that which is already delivered . secondly , plinie in his 34 book of his natural history , speakes of the famous colossus that was at rhodes , between whose legges a shippe might passe with his sailes open or displayed , the statue being of 70 cubits high : and other histories report that the sarasens having broken it , did load 900 camels with the mettal of it , now what might be the greatnesse and weight of this statue ? for answer , it is usually allowed for a camels burthen 1200 pound weight , therefore all the collosus did weigh 1080000 pound weight , which is ten hundred and fourescore thousand pound vveight . novv according to the former rules , the head being the tenth part of the body , this statues head should be of 7 cubits , that is to say , 10 foot and a halfe , and seeing that the nose , the brovv , and the thumbe , are the third part of the face , his nose vvas 3 foot and a halfe long , and so much also vvas his thumbe in length : novv the thicknesse being alvvayes the third part of the length , it should seem that his thumb was a foot thick at the least . thirdly , the said plinie in the same place reports that nero did cause to come out of france into italy , a brave and bold statue-maker called zenodocus , to erect him a colossus of brasse , which was made of 120 foot in height , which nero caused to be painted in the same height . now would you know the greatnesse of the members of this colossus , the breadth would be 20 foot , his face 12 foote , his thumb and his nose 4 foot , according to the proportion before delivered . thus i have a faire field or subject to extend my selfe upon , but it is upon another occasion that it was undertaken , let us speak therefore a word touching the giants , and then passe away to the matter . of monstrous giants . you will hardly beleeve all that which i say touching this , neither will i beleeve all that which authors say upon this subject : notwithstanding you nor i cannot deny but that long ago there have been men of a most prodigious greatnesse ; for the holy vvritings vvitnesse this themselves in deut , chap. 3. that there vvas a certaine giant called og , of the town of rabath , vvho had a bed of iron , the length thereof vvas 9 cubits , and in breadth 4 cubits . so in the first of kings chap. 17. there is mention made of goliah , vvhose height vvas a palme and 6 cubits , that is more then 9 foot , he was armed from the head to the foot , and his curiat onely with the iron of his lance , weighed five thousand and six hundred shekels , which in our common weight , is more than 233 pound , of 12 ounces to the pound : now it is certaine , that the rest of his armes taking his target , helmet , bracelets , and other armour together , did weigh at the least 5 hundred pound , a thing prodigious ; seeing that the strongest man that now is , can hardly beare 200 pound , yet this giant carries this as a vesture without paine . solinus reporteth in his 5 chap. of his historie , that during the grecians warre after a great overflowing of the rivers , there was found upon the sands the carcase of a man , whose length was 33 cubits , ( that is 49 foot and a halfe ) therefore according to the proportion delivered , his face should be 5 foot in length , a thing prodigious and monstrous . plinie in his 7. book and 16 chap. saith , that in the isle of crete or candie , a mountaine being cloven by an earth-quake , there was a body standing upright , which had 46 cubits of height : some beleeve that it was the body of orion or othus , ( but i think rather it was some ghost or some delusion ) whose hand should have beene 7 foot , and his nose two foot and a half long . but that which plutarch in the l●fe of sertorius reports of , is more strange , who saith , that in timgy a morative towne , where it is thought that the giant antheus was buried , sertorius could not beleeve that which was reported of his prodigious greatnesse , caused his sepulchre to be opened , and found that his body did containe 60 cubits in length , then by proportion he should be 10 cubits or 15 foot in breadth ; 9 foot for the length of his face , 3 foot for his thumb , which is neare the capacitie of the colossus at rhodes . but behold here a fine fable of symphoris campesius , in his book intituled hortus gallicus , who sayes that in the kingdome of sicilie , at the foot of a mountaine neare trepane , in opening the foundation of a house , they found a cave in which was ●aid a giant , which held in stead of a staffe a great post like the mast of a ship : and going to handle it , it mouldered all into ashes , except the bones which remained of an exceeding great measure , that in his head there might be easily placed 5 quarters of corn , and by proportion it should seeme that his length was 200 cubits , or 300 foot : if he had said that he had been 300 cubits in length , then he might have made us beleeve that noahs ark was but great enough for his sepulchre . who can believe that any man ever had 20 cubits , or 30 foot in length for his face , and a nose of 10 foot long ? but it is very certaine that there have been men of very great stature , as the holy scriptures before witnesse , and many authours worthy of beliefe relate : josephus acosta in his first book of the indian history , chap. 19. a late writer , reporteth , that at peru was found the bones of a giant , which was 3 times greater than these of ours are , that is 18 foot , for it is usually attributed to the tallest ordinary man in these our times but 6 foot of length ; and histories are full of the description of other giants of 9 , 10 , and 12 foot of height , and it hath been seen in our times some which have had such heights as these . problem . lxxi . of the game at the palme , at trap , at bowles , paile-maile , and others . the mathematickes often findeth place in sundry games to aid and assist the gamesters , though not unknowne unto them , hence by mathematicall principles , the games at tennis may be assisted , for all the moving in it is by right lines and reflections . from whence comes it , that from the appearances of flat or convex glasses , the production and reflection of the species are explained ; is it not by right lines ? in the same proportion one might sufficiently deliver the motion of a ball or bowle by geometrical lines and angles . and the first maxime is thus : when a bowle toucheth another bowle ▪ or when a trapstick striketh the ball , the moving of the ball is made in a right line , which is drawne from the centre of the bowle by the point of contingencie . secondly , in all kinde of such motion ; when a ball or bowle rebounds , be it either against wood , a wall , upon a drumme , a pavement , or upon a racket ; the incident angle is alwayes equall to the angle of reflection . now following these maximes , it is easie to canclude , first , in what part of the wood or wall , one may make the bowle or ball go to reflect or rebound , to such a place as one would . secondly , how one may cast a bowle upon another , in such sort that the first or the second shall go and meet with the third , keeping the reflection or angle of incidence equal . thirly , how one may touch a bowle to send it to what part one pleaseth : such and many other practices may be done . at the exercises at keyls there must be taken heed that the motion slack or diminish by little and little , and may be noted that the maximes of reflections cannot be exactly observed by locall motion , as in the beames of light and of other quallities , whereof it is necessary to supply it by industry or by strength , otherwise one may be frustrated in that respect . problem . lxxii . of the game of square formes . nvmbers have an admirable secrecie , diversly applied , as before in part is shewed , and here i will say something by way of transmutation of numbers . it 's answered thus , in the first forme the men were as the figure a , then each of these 4 souldiers placed themselves at each gate , and removing one man from each angle to each gate , then would they be also 9 in each side according to the figure b. lastly , these 4 souldiers at the gates take away each one his cumrade , and placing two of these men which are at each gate to each angle , there will be still 9 for each side of the square , according to the figure c. in like manner if there were 12 men , how might they be placed about a square that the first side shall have 3 every way , then disordered , so that they might be 4 every way ; and lastly , being transported might make 5 every way ? & this is according to the figures , f. g.h problem . lxxiii . how to make the string of a viole sensibly shake , without any one touching it ? this is a miracle in musick , yet easie to be experimented . take a viole or other instrument , and choose two strings , so that there be one between them ; make these two strings , agree in one and the same tune : then move the viole-bowe upon the greater string , and you shall see a wonder : for in the same time that that shakes which you play upon , the other will likewise sensibly shake without any one touching it ; and it is more admirable that the string which is between them will not shake at all : and if you put the first string to another tune or note , and loosing the pin of the string , or stopping it with your finger in any fret , the other string will not shake : and the same will happen if you take two violes , and strike upon a string of the one , the string of the other will sensibly shake . now it may be demanded , how comes this shaking , is it in the occult sympathie , or is it in the strings being wound up to like notes or tunes , that so easily the other may receive the impression of the aire , which is agitated or moved by the shaking or the trembling of the other ? & whence is it that the viole-bowe moved upon the first string , doth instantly in the same time move the third string , and not the second ? if the cause be not either in the first or second ? i leave to others to descant on . examination . in this examination we have something else to imagine , than the bare sympathie of the cords one to another : for first there ought to be considered the different effect that it produceth by extention upon one and the same cord in capacitie : then what might be produced upon different cords of length and bigness to make them accord in a unisone or octavo , or some consort intermediate : this being naturally examined , it will be facill to lay open a way to the knowledge of the true and immediate cause of this noble and admirable phaenomeny . now this will sensibly appeare when the cords are of equall length and greatnesse , and set to an unisone ; but when the cords differ from their equalitie , it will be lesse sensible : hence in one and the same instrument , cords at a unisone shall excite or shake more than that which is at an octavo , and more than those which are of an intermediate proportionall consort : as for the other consorts they are not exempted , though the effect be not so sensible , yet more in one than in another : and the experiment will seem more admirable in taking 2 lutes , viols , &c. & in setting them to one tune : for then in touching the cord of the one , it will give a sensible motion to the cord of the other : and not onely so but also a harmony . problem . lxxiiii . of a vessell which containes three severall kindes of liquor , all put in at one bung-hole , and drawn out at one tap severally without mixture . the vessell is thus made , it must be divided into three cells for to conteine the three liquors , which admit to be sack , claret , and white-wine : now in the bung-hole there is an engine with three pipes , each extending to his proper cell , into which there is put a broach or funnell pierced in three places , in such sort , that placing one of the holes right against the pipe which answereth unto him , the other tvvo pipes are stopped ; then vvhen it is full , turne the funnel , and then the former hole vvill be stopped , and another open , to cast in other vvine vvithout mixing it vvith the other . novv to dravv out also vvithout mixture , at the bottome of the vessell there must be placed a pipe or broach , vvhich may have three pipes ; and a cock piersed vvith three holes so artificially done , that turning the cock , the whole vvhich ansvvereth to such of the pipes that is placed at the bottom , may issue forth such vvine as belongeth to that pipe , & turning the cock to another pipe , the former hole vvil be stopped ; and so there will issue forth another kinde of wine without any mixtures ; but the cocke may be so ordered that there may come out by it two wines together , or all three kindes at once : but it seems best when that in one vessell and at one cocke , a man may draw severall kindes of wine , and which he pleaseth to drink . problem . lxxv . of burning-glasses . in this insuing discourse i will shew the invention of prom●theus , how to steale fire from heaven , and bring it down to the earth ; this is done by a little round glasse , or made of steele , by which one may light a candle , and make it flame , kindle fire-brands to wake them burne , melt lead , ●inne , gold , and silver , in a little time ▪ with as great ease as though it had been put into a cruzet over a great fire . but this is nothing to the burning of those glasses which are hollow , namely those which are of steele well polished , according to a par●bolicall or ovall section . a sphericall glasse , or that which is according to the segment of a sphere , burnes very effectually about the fourth part of the diameter ; notwithstanding the parabolie and ecliptick sections have a great effect : by which glasses there are also diverse figures represented forth to the eye . the cause of this burning is the uniting of the beames of the sunne , which heat mightily in the point of concourse or inflammation , which is either by transmissi●n or reflection ▪ now it is pleasant to behold when one breatheth in the point of concourse , or throweth small dust there , or sprinkles vapours of hot water in that place ; by which the pyramidall point , or point of inflammation is knowne . now some authors promise to make glasses which shall burne a great distance off , but yet not seen vulgarly produced , of which if they were made , the parabolie makes the greatest eff●ct , and is g●nerally held to be the invention of archimedes or pro●●us . maginus in the 5 chap. of his treatise of sphericall glasses , shewes how one may serve himselfe with a concave glasse , to light fire in the shadow , or neare such a place where the sunne shines not , which is by help of a flat glasse , by which may be made a percussion of the beames of the sun into the concave glasse , adding unto it that it serves to good use to put fi●e to a mine , provided that the combustible matter be well applyed before the concave glasse ; in which he saies true : but because all the effect of the practice depends upon the placing of the glasse and the powder which he speaks not of : i will deliver here a rule more generall . how one may place a burning-glasse with his combust●ble matter in such sort , that at a convenient houre of the day , the sun shining , it shall take fire and burne : now it is certaine that the point of inflammation or burning , is changed as the sun changeth place , and no more nor lesse , than the shadow turnes about the style of a dyall ; therefore have regard to the suns motion , and ●is height and place : a bowle of crystall in the same place that the top of the style is , and the powder or other combustible matter under the meridian , or houre of 12 , 1 , 2 , 3 , &c. or any other houre , and under the suns arch for that day : now the sunne comming to the houre of 12 , to ● , 2 , ● , &c. the sunne casting his beames through the crystall bowle , will fire the materiall or combustible thing , which meets in the point of burning : the like may be observed of other burning-glasses . examination . it is certaine in the first part of this probleme that conicall , ●oncave and sphericall glasses , of what matter soever , being placed to receive the beames of the sun will excite heat , and that heat is so much the greater , by how much it is neere the point of conc●rse or inflamatio● . but that archimedes or proclus d●d fire or burne shipps with such glasses , the ancient histories are silent , yea the selves say nothing : besides the great difficultie that doth oppose it in remotenesse , and the matter that the effect is to work upon : now by a common glasse we fire things neare at hand , from which it seems very facil to such which are lesse read , to do it at a farre greater distance , and so by re●ation some deliver to the world by supposition that which never was done in action : this we say the rather , not to take away the most excellent and admirable effects which are in burning-glasses , but to shew the variety of antiquity , and truth of history : and as touching to burne at a great distance , as is said of some , it is absolutely impossible ; and that the parabolicall and ovall glasses were of archimedes and ●roclus invention is much uncertaine : for besides the construction of such glasses , they are more difficult than the obtuse concave ones are ; and further , they cast not a great heat but neere at hand ; for if it be cast farre off , the effect is little , and the heat weake , or otherwise such glasses must be greatly extended to contract many beames to amasse a sufficient quantity of beames in parabolicall and conicall glasses , the point of inflammation ought to concur in a point , which is very difficult to be done in a due proportion . moreover if the place be farre remote , as is supposed before , such a glasse cannot be used but at a great inclination of the sunne ▪ by which the eff●ct of ●urning is d●min●shed , by reason of the weaknesse of the sunne-beames . and here may be noted in the last part of this probleme , that by r●ason of obstacles if one plaine glasse be not sufficient , a second glasse may be applyed to help it : that so if by one simple reflection it cannot be done , yet by a double reflection the sun-beames may be ●ast into the said caverne or mine , and though the reflected beams in this case be weak ▪ yet upon a 〈◊〉 c●mbustible matter it will not faile to do the effect . problem . lxxvi . containing m●ny ple●sant questions by way of arithmetick● . j will not in●ert i● this probleme that which is drawne from the ●reek epigrams , but proposing the question immediately will give the an●wer also , without ●●aying to shew the manner how they are answered ; in this j will 〈◊〉 be tied to the ●reek tearms , w●●ch j account no● proper to this place , nei●●er to my purpose : ●et t●o●e ●ead that will di●phanta s●●●●biliu● upon eu●li●● and others , and they may be satisf●ed of the 〈…〉 the mule. jt 〈◊〉 ●hat ●he mule and the asse upon a day 〈◊〉 a voyage each of them carried a barrell full of wine : now the las●e asse f●lt her selfe over-loaden , complained and bowed under her burthen ; which th● mule seeing said unto her being angry , ( for it was in the time when beasts spake ) thou great asse , wherefore complainest thou ? if i had but onely one measure of that which thou carriest , i should be loaden twice as much as thou art , and if j should give a measure of my loading to thee , yet my burthen would be as much as thine . now how many measures did each of them carry ? answer , the mule did carry 7 measures , and the asse 5 measures : for if the mule had one of the measures of the asses loading , then the mule would have 8 measures , which is double to 4 , and giving one to the asse , each of them would have equall burthens : to wit , 6 measures apiece . of the number of souldiers that fought before old troy. homer being asked by he●iodus how many grecian souldiers came against troy ? who answered him thus ; the grecians , said homer , made 7 fires , or had 7 kitchins , and before every fire , or in every kitchin there were 50 broaches turning to rost a great quantitie of flesh , and each broach had meat enough to satisfie 900 men : now judge how many men there might be . answer , 315000. that is , three hundred and fifteen thousand men , which is cleare by multiplying 7 by 50 , and the product by 900 makes the said 315000. of the number of crownes that two men had . john and peter had certaine number of crowns : john said to peter , if you give me 10 of your crownes , i shall have three times as much as you have : but peter said to j●hn , if you give me 10 of your crownes i shall have 5 times as much as you have : how much had each of them ? answere , john had 15 crownes and 5 sevenths of a crowne , and peter had 18 crownes , and 4 sevenths of a crowne . for if you adde 10 of peters crownes to those of johns , then should john have 25 crownes and 5 sevenths of a crowne , which is triple to that of peters , viz. 8 ▪ and 4 sevenths : and john giving 10 to peter , peter should have then 28 crownes , and 4 sevenths of a crowne , which is quintupla , or 5 times as much as john had left , viz. 5 crownes and 5 sevenths . in like manner two gamesters playing together , a and b ▪ after play a said to b , give me 2 crownes of thy money , and i shall have twice as much as thou hast : and b said to a , give me 2 crownes of thy money , and i shall have 4 times as much as thou hast : now how much had each ? answer , a had 3 and 5 seventhes , and b had 4 and 6 seventhes . about the houre of the day . some one asked a mathemacian what a clocke it was ; who answered that the rest of the day is foure thirds of that which is past : now judge what a clock it is . answer , if the day were according to the jewes and ancient romanes , which ma●e it alwayes to be 12 houres , it was then the ● houre , and one seventh of an hou●e , so there remained of the whole day 6 , that is , 6 houres , and 6 sevenths of an hour . now if you take the 1 / ● of 5 ● / 7 it is ●2 / 7 or ● and ● 7 , which multipled by 4 makes 6 and 6 / 7 , which is the remainder of the day , as before : but if the day had been 24 houres , then the houre had been 10 of the clock ▪ and two seventhes of an houre , which is found ▪ out by dividing 12 , or 24 by ● . there might have been added many curious propositions in this kinde , but they vvould be too difficult for the most part of people ▪ therefore i have omi●ted them ▪ of pythagoras his schollers . pythagoras being asked what number of schollers he had , ansvvered , that halfe of them studied mathematickes , the fourth part physick , the seventh part rethorick , and besides he had 3 vvomen : novv judge you saith he , hovv many schollers i have . ansvver , he had in all 28 , the halfe of vvhich is 14 , the quarter of which is 7 , and the seventh part of which is which 14 , 7 , and 4 , makes 25 , and the other 3 to make up the 28 , were the 3 women . of the number of apples given amongst the graces and the muses . the three graces carrying apples upon a day , the one as many as the other , met with the 9 muses , who asked of them some of their apples ; so each of the graces gave to each of the muses alike , and the distribution being made , they found that the graces & the muses had one as many as the other : the question is how many apples each grace had , and how many they gave to each muse ? ●o ansvver the qeustion , joyne the number of graces and muses together vvhich makes 12 , and so many apples had each grace : novv may you take the double , triple , &c. of 12 that is 24 , 36 , &c. conditionally , that if each grace had but 12 , then may there be allotted to each muse but one onely ; if 24 , then to each 2 apples , if ●6 , then to each muse 3 apples , and so the distribution being made , they have a like number , that is one as many as the other . of the testament or last will of a dying father . a dying father left a thousand crovvnes amongst his tvvo children ; the one being legitimate , and the other a bastard , conditionally that the fifth part which his legittimate sonne should have , should exceed by 10 , the fourth part of that which the bastard should have : what was each 〈◊〉 part ? answer , the legitimate sonne had 577 crownes and 7 / ● , and the bastard 42● crownes and 2 / 9 now the fifth part of 577 and 7 ninthes is 1●5 , and 5 / 9 , and the fourth part of 422 and ● is 105 and ● which is lesse then ●15 ● by 10 , according to the will of the testator . of the cups of croesus . croesus gave to the temple of the ●ods six cups of gold ▪ which weighed together ●00 drammes , but each cup was heavier one than another by one dram : how much did each of them therefore weigh ? answer , the first weighed 102 drammes and a halfe ; the second 101 drammes and a halfe , the third 100 drammes and ● , the fourth 99 a & halfe , the fifth 98 & a halfe ; and the sixt cup weighed 97 drammes and a halfe ▪ which together makes 600 drams as before . of cupids apples . cvpid complained to his mother that the muses had taken away his apples , clio , said he , took from me the fifth part , euterp the twelfth part , thalia the eighth part , m●lp●meno the twentieth part , erates the seventh part ▪ terpomene the fourth part , polyhymnia took away 30 , vrania 220 , and calliope 300. so there vvere left me but 5 appls , hovv many had he in all at the first ? i ansvver 3●60 . there are an infinite of such like questions amongst the greek epigrams : but it would be unpleasant to expresse them all : i will onely adde one more , and shew a generall rule for all the rest . of a mans age. a man vvas said to passe the sixth part of his life in childe-hood , the fourth part in his youth , the ●hird part in manhood , and 18 yeares besides in old age : what might his age be ? the ansvver is , 72 yeares : vvhich and all others is thus resolved : multiply 1 / ● ▪ ¼ and ⅓ ▪ together , that is , 6 by 4 makes 24 , and that againe by 3 makes 72 , then take the third part of 72 , vvhich is 24 , the fourth part of it , vvhich is 18 , and the sixth part of it vvhich is 12 , these added together make 54 , vvhich taken from 72 , rests 18 this divided by 18 ( spoken in the question ) gives 1 , which multiplied by the summe of the parts , viz. 72 , makes 72 , the ansvver as before . of the lion of bronze placed upon a fountaine with this epigramme . ovt of my right eye if i let vvater passe , i can fill the cisterne in 2 dayes : if i let it passe out of the left eye , it vvill be filled in 3 dayes : if it passe out of my feet , the cistern vvill be 4 dayes a●filling ; but if i let the vvater passe out of my mouth , i can fill the cistern then in 6 houres : in vvhat time should i fill it , if i poure forth the vvater at all the passages at once ? the greeks ( the greatest talkers in the vvorld ) variously apply this question to divers statues , and pipes of fountaines : and the solution is by the rule of ● , by a generall rule , or by ●lgebra . they have also in their anthologie many other questions , but because they are more proper to exercise , than to recreate the spirit , i passe them over ( as before ) with silence . problem . lxxvii . divers excellent and admirable experiments upon glasses . there is nothing in the world so beautifull as light : and nothing more recreative to the sight , than glasses vvhich reflect : therefore i vvill novv produce some experiments upon them , not that vvill dive into their depth ( that vvere to lay open a mysterious thing ) but that vvhich may delight and recreate the spirits : let us suppose therefore these principles , upon which is built the demonstration of the appar●nces which are made ●n all sort of glasses . first , that the rayes or beames , vvhich reflect upon a glasse , make the angle of incident equall to the angle of reflection , by the first theo. of the catoptick of euc. secondly , that in all plain glasses , the images are seen in the perpendicular line to the glasse , as far within the glass as the object is without it . thirdly , in concave , or convex glasses , the images are seen in the right line which passeth from the object and through the centre in the glasse . theo. 17. and 18. and here you are to understand , that there is not meant only those which are simple glasses or glasses of steele , but all other bodies , which may represent the visible image of things by reason of their reflection , as water , marble , mettal , or such like . now take a glasse in your hand and make experiment upon that which followeth . experiment upon flat and plaine glasses . first , a man cannot see any thing in these glasses , if he be not directly and in a perpendicular line before it , neither can he see an object in these glasses , if it be not in such a place , that makes the angle of incidence equall to the angle of reflexion : therefore when a glasse stands upright , that is , perpendicular to the horizon , you cannot see that which is above , except the glasse be placed down flat : and to see that on the right hand , you must be on the left hand , &c. secondly , an image cannot be seen in a glass if it be not raised above the surface of it ; or place a glasse upon a wall , you shall see nothing which is upon the plaine of the wall , and place it upon a table or horizontal plaine , you shall see nothing of that which is upon the table . thirdly , in a plaine glasse all that is seene appeares or seemes to sink behinde the glasse , as much as the image is before the glasse , as before is said . fourthly , ( as in water ) a glasse lying downe flat , or horizontall , towers , trees , men , or any height doth appeare , inversed or upside downe ; and a glasse placed upright , the right hand of the jmage seems to be the left , and the left seems to be the right . fifthly , will you see in a chamber that which is done in the street , without being seen ▪ then a glasse must be disposed , that the line upon which the jmages come on the glasse , make the angle of incidence equall to that angle of reflexion . seventhly , present a candle upon a plaine glasse , and look flaunting upon it , so that the candle and the glasse be neere in a right line , you shall see 3 , 4 , 5 , &c. images , from one and the same candle . eightly , take tvvo plaine glasses , and hold them one against the other , you shall alternately see them oftentimes one vvithin the other , yea vvithin themselves , againe and againe . ninthly , if you hold a plaine glasse behinde your head , and another before your face , you may see the h●nder part of your head , in that glasse vvhich you hold before your face . tenthly , you may have a fine experiment if you place tvvo glasses together , that they make an acute angle , and so the lesser the angle is , the more apparances you shall see , the one direct , the other inversed , the one approaching , and the other retiring . eleventhly , it is a vvonder & astonishment to some , to see within a glasse an image vvithout knovving from vvhence it came , and it may be done many vvayes : as place a glass higher than the eye of the beholder , and right against it is some image ; so it resteth not upon the beholder , but doth cast the image upvvards . then place another object , so that it reflect , or cast the image downeward to the eye of the spectator ▪ without perceiving it being hid behinde something , for then the glasse will represent a quite contrary thing , either that which is before the glasse , or that which is about it , to wit , the other hidden object . twelfthly , if there be ingraved behinde the backside of a glasse , or drawne any image upon it , it will appeare before as an image , without any appearance : o● portraicture to be perceived . examination . this 12 article of ingraving an image behinde the glasse , will be of no great consequence ▪ because the lineaments will seem so obscure , but if there were painted some image , and then that covered according to the usuall covering of glasses behinde , and so made up like an ordinary looking-glasse having an image in the middle , in this respect it would be sufficiently pleasant : and that which would admire the ignorant , and able to exercise the most subtillest , and that principally if the glasse be in an obscure place , and the light which is given to it be somewhat farre off . place a glasse neare the floor of a chamber , & make a hole through the place under the glasse , so that those which are below may not perceive it , and dispose a bright image under the hole so that it may cast his species upon the glasse , and it will cause admiration to those which are below that know not the cause ; the same may be done by placing the image in a chamber adjoyning , and so make it to be seen upon the side of the wall. 14 in these channel-images which shew one side a deaths head , & another side a faire face : and right before some other thing : it is a thing evident , that setting a plaine glasse sidewise to this image you shall see it in a contrary thing , then that which was presented before sidewise . 15 lastly , it is a fine secret to present unto a plaine glasse writing with such industry , that one may read it in the glasse , and yet out of the glasse there is nothing to be known , which will thus happen , if the writing be writ backward : but that which is more strange , to shew a kinde of writing to a plaine glasse , it shall appear another kinde of writing both against sense and forme , as if there were presented to the glasse wel it would shew it met ; if it were written thus miv , and presented to the glasse , it would appeare thus vim ; for in the first , if the glasse ly flat , then the things are inversed that are perpendicular to the glass , if the glass and the object be upright , then that on the right hand , is turned to the left , as in the latter . and here i cease to speak further of these plaine glasses , either of the admirable multiplications , or appearances , which is made in a great number of them ; for to content the sight in this particular , one must have recourse to the cabinets of great personages who inrich themselves with most beautifull ones . experiments upon gibbous , or convex sphericall glasses . if they be in the forme of a bowle , or part of a great globe of glasse , there is singular contentment to contemplate on them . first , because they present the objects lesse and more gracious , and by hovv much more the images are separated from the glasse , by so much the more they diminish in magnitude . secondly , they that shew the images plaiting , or foulding , which is very pleasant , especially when the glasse is placed downe , and behold in it some blanching , feeling , &c. the upper part of a gallerie , the porch of a hall , &c. for they will be represented as a great vessel having more belly in the middle then at the two ends , and posts , and joists of timber will seeme as circles . thirdly , that which ravisheth the spirits , by the eye , and which shames the best perspective painting that a painter can make , is the beautifull contraction of the images , that appeare within the sphericity of these small glasses : for present the glasse to the lower end of a gallarie , or at the corner of a great court full of people , or towards a great street , church , fortification , an army of men , to a whole cittie ; all the faire architecture , and appearances will be seene contracted within the circuit of the glasse with such varietie of colours , and distinctions in the lesser parts , that i know not in the world what is more agreeable to the sight , and pleasant to behold , in which you will not have an exact proportion , but it will be variable , according to the distance of the object from the glasse . exptriments upon hollow , or concave sphericall glasses . i have heretofore spoken how they may burne , being made of glasse , or metall , it remaines now that i deliver some pleasant uses of them , which they represent unto our sight , and so much the more notable it will be , by how much the greater the glasse is , and the globe from whence it is extracted for it must in proportion as a segment of some be made circle or orbe . examination . in this we may observe that a section of 2.3 . or 4. inches in diameter , may be segments of spheres of 2.3 . or 4. foot ● nay of so many fadome , for it is certaine that amongst those which comprehend a great portion of a lesser sphere , and those which comprehend a little segment of a great spheere , whether they be equall or not in section , there will happen an evident difference in one and the same experiment , in the number , situation , quantitie , and figure of the images of one or many different objects , and in burning there is a great difference . maginus , in a little tractate that he had upon these glasses , witnesseth of himselfe that he hath caused many to be polished for sundry great lords of italy , and germanie , which were segments of globes of 2.3 . and 4. foot diameter ; and i wish you had some such like to see the experiments of that which followeth ; it is not difficult to have such made , or bought here in town , the contentment herein would beare with the cost . examination . touching maginus he hath nothing ayded us to the knowledge of the truth by his extract out of vitellius , but left it : expecting it from others , rather than to be plunged in the search of it himselfe , affecting rather the forging of the matter , and composition of the glasses , than geometrically to establish their effects . first therefore in concave glasses , the images are seene sometimes upon the surface of the glasses , sometimes as though they were within it and behinde it , deeply sunk into it , sometimes they are seene before , and without the glasse , sometimes between the object and the glasse ; sometimes in the place of the eye , sometimes farther from the glasse then the object is : which comes to passe by reason of the divers concourse of the beames , and change of the place of the images in the line of reflection . examination . the relation of these appearances passe current amongst most men , but because the curious may not receive prejudice in their experiments , something ought to be said thereof to give it a more lively touch : in the true causes of these appearances , in the first place it is impossible that the image can be upon the surface of the glasse , and it is a principall point to declare truly in which place the image is seen in the glasse those that are more learned in opticall knowledge affirme the contrary , and nature it selfe gives it a certaine place according to its position being alwayes seen in the line of reflection which alhazen , vitellius , and others full of grea● knowledge , have confirmed by their writings : but in their particular they were too much occupied by the authority of the ancients who were not s●fficiently ci●cumspect in experience upon which the principles of this sub●ect ought to be built , an● searched not fully into the true cause of these appearances , seeing they leave unto posterities many 〈◊〉 in their writings , ●nd those that followed them for the most part fell into the like errors . as for the jmages to bid● in the eye ▪ it cannot be but is imp●rtinent and absurd ; but it followeth that , by how much neerer the ob●ect appro●cheth to the glasse , by so much the more the appearances seem to come to the eye : and if the eye be without the point of concourse , and the object also ; as long as the object approacheth thereto , the representation of the image cometh neere the eye , but passing the point of concourse it goes back againe : these appearances thus approaching do not a little astonish those which are ignorant of the cause : they are inversed , if the eye be without the point of concourse untill the object be within , but contrarily if the eye be between the point of concourse and the glasse , then the jmages are direct : and if the eye or the object be in the point of concourse , the glasse will be enlightened and the jmages confused , and if there were but a spark of fire in the said point of concourse , all the glasse would seeme a burning fire-brand , and we dare say it would occurre without chance , and in the night be the most certaine and subtilest light that can be , if a candle were placed there . and whosoever shall enter into the search of the truth of new experiments in this subject without doubt he will confirme what we here speak of : & will finde new lights with a conveniable position to the glasse , he will have reflection of quantities , of truth , and fine secrets in nature , yet not known , which he may easily comprehend if he have but an indifferent sight , and may assure himselfe that the images cannot exceed the fight , nor trouble it , a thing too much absurd to nature . and it is an absolute verity in this science , that the eye being once placed in the line of reflection of any object , and moved in the same line : the obect is seene in one and the same place immutable ; or if the image and the eye move in their owne lines , the representation in the glasse seemes to invest it selfe continually with a different figure . now the image comming thus to the eye , those which know not the secret , draw their sword when they see an image thus to issue out of the glasse , or a pistoll which some one holds behinde : and some glasses will shew a sword wholly drawne out , sepa●ated from the glasse , as though it were in the aire : and it is daily exercised , that a man may touch the image of his hand or his face out of the glasse , which comes out the farther , by how much the glasse is great and the centre remote . examination . now that a pistoll being presented to a glasse behinde a man , should come out of the glasse , and make him afraid that stands before , seeming to shoot at him , this cannot be : for no object whatsoever presented to a concave glasse , if it be not neerer to the g●asse then the eye is it comes not out to the sight of the party ; therefore he needs not feare that which is said to be behinde his back , and comes out of the glasse ; for if it doth come out , it must then necessarily be before his face , so in a concave glasse whose centre is farre remote of a sword , stick , or such like be presented to the glasse , it shall totally be seen to come forth of the glasse and all the hand that holds it . and here generally note that if an image be seen to issue out of the glasse to come towards the face of any one that stands by , the object shall be likewise seen to thrust towards that face in the glass and may easily be knowne to all the standers by : so many persons standing before a glasse , if one of the company take a sword , and would make it issue forth towards any o●her that stands there : let him chuse his image in the gl●sse and carry the sword right towards it and the effect will follow . in like manner ones hand being presented to the glosse as it is thrust towards the centre , s● the representation of it comes towards it , and so the hands will seeme to be united , or to touch one another . from which may be concluded , if such a glasse be placed at the seeling or planching of a hall , so that the face be horizontall and look downward ; one may see under it as it were a man hanging by the feet , and if there were many placed so , one could not enter into that place without great feare or scaring : for one should see many men in the aire as if they were hanging by the feet . examination . touching a glasse tyed at a seeling or planching , that one may see a man hang by the feet in the aire , and so many glasses , many men may be seen : without caution this is very absurd for if the glasse or glasses be not so great that the centre of the sphere upon which it was made , extend not neere to the head of him that is under it , it will not pleasantly appeare , and though the glasse should be of that capacity that the centre did extend so farre , yet will not the images be seene to them which are from the glasse but on●y to those which are under it , or neere unto it : and to them it will not ably appeare , and it would be most admirable to have a gallerie vaulted over with such glasses which would wonderfully astonish any one that enters into it : for a●l the things in the gallery would be seen to hang in the aire , and you could not walk without incountering airie apparitions . secondly , in flat or plaine glasses the image is seen equall to his object , and to represent a whole man , there ought to be a glasse as great as the image is : in convex glasses the images are seen alwayes lesse , in concave glasses they may be seen greater or lesser , but not truly proportionable , by reason the diverse reflexions which contracts or inlargeth the species : when the eye is between the centre and the surface of the glasse ; the image appeares sometimes very great and deformed , and those which have but the appearance of the beginning of a beard on their chinne , may cheare up themselves to see they have a great beard ; those that seeme to be faire will thrust away the glasse with despight , because it will transforme their beauty : those that put their hand to the glasse vvill seeme to have the hand of a giant , and if one puts his finger to the glasse it vvill be seen as a great pyramide of flesh , inversed against his finger . thirdly , it is a thing admirable that the eye being approached to the point of concourse of the glasse , there vvill be seen nothing but an intermixture or confusion : but retiring back a little from that point , ( because the rayes do there meet ▪ ) he shall see his image inversed , having his head belovv and his feet above . fourthly , the divers appearances caused by the motion of objects , either retiring or approaching : whether they turne to the right hand or to the left hand , whether the glasse be hung against a wall , or whether it be placed upon a pavement , as also what may be represented by the mutuall aspect of concave glasses with plaine and convex glasses but i will with silence passe them over , only say something of two rare experiments more as followeth . the first is to represent by help of the sun , such letters as one would upon the front of a house : so that one may read them : maginus doth deliver the way thus . write the letters , saith he , sufficiently bigge , but inversed upon the surface of the glasse , with some kinde of colour , or these letters may be written with wax , ( the easier to be taken out againe : ) for then placing the glasse to the sunne , the letters which are written there will be reverberated or reflected upon the wall : hence it was perhaps that pythagoras did promise with this invention to write upon the moone . in the second place , how a man may sundry wayes help himselfe with such a glasse , with a lighted torch or candle , placed in the point of concourse or inflammation , which is neare the fourth part of the diameter : for by this meanes the light of the candle will be reverberated into the glasse , and vvill be cast back againe very farre by parrallel lines , making so great a light that one may clearly see that vvhich is done farre off , yea in the camp of an enemie : and those which shall see the glasse a farre off , will think they see a silver basin inlightened , or a fire more resplendent then the torch . it is this way that there are made certaine lanthorns which dazell the eyes of those which come against them ; yet it serves singular well to enlighten those which carry them , accommodating a candle with a little hollow glasse , so that it may successively be applyed to the point of inflammation . in like manner by this reflected light , one may reade farre off , provided that the letters be indifferent great , as an epitaph placed high , or in a place obscure ; or the letter of a friend which dares not approach without perill or suspicion . examination . this will be scarce sensible upon a wall remote from the glasse , and but indifferently seen upon a wall which is neare the glasse , and withall it must be in obscuritie or shadowed , or else it will not be seen . to cast light in the night to a place remote , with a candle placed in the point of concourse or inflammation , is one of the most notablest properties which can be shewne in a concave glasse : for if in the point of inflammation of a parabolicall section , a candle be placed , the light will be reflected by parallel lines , as a columne or cylinder ; but in the sphericall section it is defective in part , the beames being not united in one point , but somewhat scattering : notwithstanding it casteth a very great beautifull light . lastly , those which feare to hurt their sight by the approach of lampes or candles , may by this artifice place at some corher of a chamber , a lamp with a hollow glasse behinde it , which will commodiously reflect the light upon a table , or to a place assigned : so that the glasse be somewhat raised to make the light to streeke upon the table with sharp angles , as the sunne doth when it is but a little elevated above the horizon , for this light shall exceed the light of many candles placed in the roome , and be more pleasant to the sight of him that useth it . of other glasses of pleasure . first , the columnary and pyramidall glasses that are contained under right lines , do represent the images as plaine glasses do ; and if they be bowing , then they represent the image , as the concave and convex glasses do . secondly , those glasses which are plaine , but have ascents of angels in the middle , will shew one to have foure eyes , two mouthes , two noses , &c. examination . th●se experiments will be found different according to the diverse meeting of the glasses , which commonly are made scuing-wise at the end , 〈◊〉 which there will be two divers superficies in the glasse , making the exteriour angle somewhat raised , at the interiour onely one superficies , which may be covered according to ordinary glasses to c●use a reflexion , and so it will be but one glasse , which by refraction according to the different thicknesse of the glasse , and different angles of the scuing forme , do differently present the images to the eye , as foure eyes , two mouthes , two noses ; sometimes three eyes one mouth , and one nose , the one large and the other long , sometimes two eyes onely : with the mouth and the nose deformed , which the glasse ( impenetrable ) will not shew . and if there be an interiour solid angle , according to the difference of it ( as if it be more sharp ) there will be represented two distinct double images , that is , two entire visages and as the angle is open , by so much the more the double images will reunite and enter one within another , which will present sometimes a whole visage extended at large , to have foure eyes , two noses , and two mouthes : and by moving the glasse the angle will vanish , and so the two superficies will be turned into one , and the duplicity of images will also vanish and appeare but one onely : and this is easily experimented with two little glasses of steel , or such like so united , that they make divers angles and inclinations . thirdly , there are glasses which make men seeme pale , red , and coloured in diverse manners , which is caused by the dye of the glasse , or the diverse refraction of the species : and those which are made of silver , latine , steele , &c. do give the images a diverse colour also . in which one may see that the appearances by some are made fairer , younger or older than they are ; and contrarily others will make them foule and deformed : and give them a contrary visage : for if a glasse be cut as it may be , or if many pieces of glasse be placed together to make a conveniable reflexion : there might be made of a mole ( as it were ) a mountaine , of one haire a tree , a fly to be as an elephant , but i should be too long if i should say all that which might be said upon the property of glasses . i will therefore conclude this discourse of the properties of these glasses with these foure recreative problemes following . problem . lxxviii . 1 how to shew to one that is suspitious , what is done in another chamber or roome : notwithstanding the interposition of the wall . for the performance of this , there must be placed three glasses in the two chambers , of which one of them shall be tyed to the planching or seeling , that it may be common to communicate the species to each glasse by reflexion , there being left some hole at the top of the wall against the glasse to this end : the two other glasses must be placed against the two walls at right angles , as the figure here sheweth at b. and c. then the sight at e by the line of incidence fe , shall fall upon the glasse ba , and reflect upon the superficies of the glasse bc , in the point g ; so that if the eye be at g , it should see e , and e would reflect upon the third glass in the point h , and the eye that is at l , will see the image that is at e. in the point of the cath●r● : which image shall come to the eye of the suspicious , viz. at l. by help of the third glasse , upon which is made the second reflexion , and so brings unto the eye the object , though a wall be between it . corolarie . 1. by this invention of reflections the besiegers of a towne may be seene upon the rampart : notwithstanding the parapet , which the besieged may do by placing a glasse in the hollow of the ditch , and placing another upon the toppe of the wall , so that the line of incidence comming to the bottom of the ditch , make an angle equall to the angle of reflexion , then by this situation and reflexion , the image of the besiege● 〈◊〉 will be seen to him is upon the rampart corolarie 2. by which also may be inferred , that the same reflexions may be seen in a regular polygon , and placing as many glasses as there are sides , counting two for one ; for then the object being set to one of the glasses , and the eye in the other , the jmage will be seen easily . corolarie 3. farther , notwithstanding the interposition of many walls , chambers , or cabinets , one may see that which passeth through the most remotest of them , by placing of many glasses as there are openings in the walls , making them to receive the incident angles equall : that is , placing them in such sort by some geometricall assistant , that the incident points may meet in the middle of the glasses : but here all the defect will be , that the jmages passing by so many reflexions , will be very weak and scarce observable . problem . lxxix . how with a musket to strike a mark , not looking towards it , as exact as one aiming at it . as let the eye be at o ▪ and the mark c , place a plaine glasse perpendicular as ab . so the marke c shall be seen in catheti ca , viz. in d , and the line of reflexion is d , now let the musket fe , upon a rest ▪ be moved to and fro untill it be seen in the line od , which admit to be hg , so giving fire to the musket , it shall undoubtedly strike the mark . corolaries . from which may be gathered , that one may exactly shoot out of a musket to a place which is not seen , being hindered by some obstacle , or other interposition . as let the eye be at m , the mark c , and the wall which keeps it from being seene , admit to be qr , then set up a plaine glass as ab , and let the musket by gh , placed upon his rest po. now because the marke c is seen at d , move the musket to and fro , untill it doth agree with the line of reflection mb , which suppose at li , so shall it be truly placed , and giving fire to the musket , it shall not faile to strike the said mark at c. problem . lxxx . how to make an image to be seen hanging in the aire , having his head downeward . take two glasses , and place them at right angles one unto the other , as admit ab , and cb , of which admit cb , ho●izontall , and let the eye be at h , and the object or image to be de ; so d will be reflected at f , so to n , so to he : then at g , so to ● and then to h , and by a double reflection ed will seeme in qr , the highest point d in r , and the point l in q inversed as was said , taking d for the head , and e for the feet ; so it will be a man inversed , which will seem to be flying in the aire , if the jmage had wings unto it , and had secretly 〈◊〉 motion : and if the glasse were bigge enough to receive many reflexions , it would deceive the sight the more by admiring the changing of colours that would be seen by that motion . problem . lxxxi . how to make a company of representative souldiers seeme to be a regiment , or how few in number may be multiplyed to seem to be many in number . to make the experiment upon men , there must be prepared two great glasses ; but in stead of it we will suppose two lesser , as gh . and fi , one placed right against another perpendicular to the horizon , upon a plaine levell table : betvveene vvhich glasses let there be ranged in battalia-vvise upon the same table a number of small men according to the square g , h , i , f , or in any other forme or posture : hen may you evidently see hovv the said battel vvill be multiplyed and seem farre bigger in the appearance than it is in effect . corolarie . by this invention you may make a little cabinet of foure foot long , and tvvo foot large , ( more or lesse ) vvhich being filled vvith rockes or such like things , or there being put into it silver , gold , stones of luster , jewels , &c. and the walls of the said cabinet being all covered , or hung with plaine glasse ; these visibles will appeare manifoldly increased , by reason of the multiplicitie of reflexions , and at the opening of the said cabinet , having set something which might hide them from being seen , those that look into it will be astonished to see so few in number which before seemed to be so many . problem . lxxxii . of fine and pleasant dyal● . could you choose a more ridiculous one than the natural dyall written amongst the greek epigrams , upon which some sound poet made verses ; shewing that a man carrieth about him alwayes a dyall in his face by meanes of the nose and teeth ? and is not this a jolly dyall ? for he need not but open the mouth , the lines shall be all the teeth , and the nose shall serve for the style . of a dyall of hearbes . can you have a finer thing in a garden , or in the middle of a compartemeet , than to see the lines and the number of houres represented with little bushie hearbes , as of hysope or such which is proper to be cut in the borders ; and at the top of the style to have a fanne to shew which way the winde b●oweth ? this is very pleasant and useful . of the dyall upon the fingers and the hand . is it nor a commoditie very agreeable , when one is in the fie●d or in some vil●age vvithout any other dyall , to see onely by the hand what of the clock it is ? vvhich gives it very neare ; and may be practised by the left hand , in this manner . take a stravv or like thing of the length of the index or the second finger , hold this straw very right betvveen the thumb and the fore-finger , then stretch forth the hand ▪ and turne your back , and the palm of your hand tovvards the sunne ; so that the shadovv of the muscle vvhich is under the thumb , touch the line of life , vvhich is betvveen the middle of the tvvo other great lines , vvhich is seen in the palme of the hand , this done , the end of the shadovv vvill shevv vvhat of the clock it is : for at the end of the first finger it is 7 in the morning , or 5 in the evening , at the end of the ring-finger it is 8 in the morning , or 4 in the evening , at the end of the little finger or first joynt , it is 9 in the morning , or 3 in the after-noone , 10 & 2 at the second joynt , 11 and 1 at the third joynt , and midday in the line follovving , vvhich comes from the end of the index . of a dyall which was about an obeliske at rome . was not this a pretty fetch upon a pavement , to choose an obeliske for a dyall , having 106 foot in height , without removing the basis of it ? plinie assures us in his 26 book and 8 chap. that the emperour augustus having accom●odated in the field of mars an obeliske of this height , he made about it a pavement , and by the industry of man●lius the mathematician , there were enchaced markes of copper upon the pavement , and placed also an apple of gold upon the toppe of the said obeliske , to know the houre and the course of the sunne , with the increase and decrease of dayes by the same shadow : and in the same manner do some by the shadow of their head or other style , make the like experiments in astronomie . of dyals with glasses . pt●lomie w●ites , as cardanus reports , that long ago there were glasses which served for dyals , and presented the face of the beholder as many times as the houre ought to be , twice if it were 2 of the clock , 9 if it were 9 , &c. but this was thought to be done by the help of water , and not by glasses , which did leake by little and little out of the vessell , discovering anon one glasse , then anon two glasses , then 3 , 4 , 5 glasses , &c. to shew so many faces as there were houres , which was onely by leaking of water . of a dyall which hath a glasse in the place of the style . what will you say of the invention of mathematicians , which finde out daily so many fine and curious novelties ? they have now a way to make dyals upon the wainscot or seeling of a chamber , and there where the sunne can never shine , or the beames of the sunne cannot directly strike : and this is done in placing of a little glasse in the place of the style which reflecteth the light , with the same condition that the shadow of the style sheweth the houre : and it is easie to make experiment upon a common dyall , changing only the disposition of the dyall , and tying to the end of the style a piece of plaine glasse . the almaines use it much , who by this way have no greater trouble , but to put their noses out of their beds and see what a clock it is , which is reflected by a little hole in the window upon the wall or seeling of the chamber . examination . in this there are two experiments considerable , the first is with a very little glasse placed so that it may be open to the beames of the sunne , the other hath respect to a spacious or great glasse placed to a very little hole so that the sun may shine on it , for then the shadow which is cast upon the dyall is converted into beames of the sunne , and will reflect and becast upon a plain opposite : and in the other it is a hole in the window or such like , by which may passe the beames of the sun , which represent the extreamity of the style , & the glasse representeth the plaine of the dyall , upon which the beames being in manner of shadowes reflect cast upon a plaine opposite : and it is needfull that in this second way the glasse may be spacious , as before , to receive the delineaments of the dyall . otherwise you may draw the lineaments of a dyall upon any plaine looking-glasse which reflecteth the sunne-beames , for the applying a style or a pearle at the extreamitie of it : and placed to the sunne , the reflexion will be answerable to the delineaments on the glasse : but here note , that the glasse ought to be great , and so the delineaments thereon . but that which is most noble , is to draw houre-lines upon the outside of the glasse of a window , and placing a style thereto upon the outside , the shadow of the style will be seen within , and so you have the hour , more certaine without any difficulty . of dyals with water . svch kinde of dyals were made in ancient times , and also these of sand : before they had skill to make sun-dyals or dyals with wheeles ; for they used to fill a vessell with water , and having experience by tryall thar it would runne out all in a day , they did marke within the vessell the houres noted by the running of the water ; and some did set a piece of light board in the vessell to swimme upon the top of the water , carrying a little statue , which with a small stick did point out the houre upon a columne or wall , figured with houre-notes , as the vessell was figured within . novv it seemes a safer vvay that the vvater passe out by drop and drop , and drop into a cylindricall glasse by help of a pipe : for having marked the exterior part of the cylinder in the houre notes , the vvater it selfe vvhich falls vvithin it , vvill shevv vvhat of the clock it is , farre better than the running of sand , for by this may you have the parts of the houres most accurate , vvhich commonly by sand is not had : and to vvhich may be added the houres of other countreys vvith greater ease . and here note , that as soone as the vvater is out ▪ of one of the glasses , you may turne it over into the same againe out of the other , and so let it runne anevv . problem . lxxxiii . of cannons or great artillery . souldiers , and others would willingly see 〈◊〉 problems , which containe : three or foure subtile questions : the first is , how to charge a cannon without powder ? this may be done vvith aire and vvater , only having throvvn cold vvater into the cannon , vvhich might be squirted forceably in by the closure of the mouth of the piece , that so by this pressure the aire might more condense ; then having a round piece of vvood very just , and oiled vvell for the better to slide , and thrust the bullet vvhen it shall be time : this piece of vvood may be held fast vvith some pole , for feare it be not thrust out before his time : then let fire be made about the trunion or hinder part of the piece to heat the aire and vvater , and then vvhen one vvould shoot it , let the pole be quickly loosened , for then the aire searching a greater place , and having vvay novv offered , vvill thrust out the vvood and the bullet very quick : the experiment vvhich vve have in long trunkes shooting out pellats vvith aire only , shevveth the verity of this probleme . 2 in the second question it may be demanded , how much time doth the bull●● of a cannon spend in the aire before i● falls to the ground ? the resolution of this question depends upon the goodnesse of the piece & charge thereof , seeing in each there is great difference . it is reported , that tich● bra●e , and the landsgrave did make an experiment upon a cannon in germany , which being charged and shot off ; the bullet spent two minutes of time in the aire before it fell : and the distance was a germane mile , which distance proportionated to an hours time , makes 120 italian miles . 3. in the third question it may be asked , how it comes to passe , that a cannon shooting upwards , the bullet flies with more violence than being shot point-blanke , or shooting downeward ? if we regard the effect of a cannon when it is to batter a wall , the question is false , seeing it is most evident that the blowes which fall perpendicular upon a wall , are more violent than those which strike byas-wise or glaunsingly . but considering the strength of the blow only , the question is most true , and often experimented to be found true : a piece mounted at the best of the randon , which is neare halfe of the right , conveyes her bullet with a farre greater violence then that which is shot at point blanke , or mounted parallel to the horizon . the common reason is , that shooting high , the fire carries the bowle a longer time in the aire , and the aire moves more ●acill upwards , than dovvnevvards , because that the airy circles that the motion of the bullet makes , are soonest broken . hovvsoever this be the generall tenet , it is curious to finde out the inequality of moving of the aire ; vvhether the bullet fly upvvard , dovvnevvard , or right forvvard , to produce a sensible dfference of motion ; & some think that the cannon being mounted , the bullet pressing the povvder maketh a greater resistance , and so causeth all the povvder to be inflamed before the bullet is throvvne out , vvhich makes it to be more violent than othervvise it vvould be . when the cannon is othervvise disposed , the contrary arives , the fire leaves the bullet , and the bullet rolling from the povvder resists lesse : and it is usually seene , that shooting out of a musket charged onely vvith povvder , to shoot to a marke of paper placed point blanke , that there are seene many small holes in the paper , vvhich cannot be other than the graines of powder which did not take fire : but this latter accident may happen from the over-charging of the piece , or the length of it , or windy , or dampenesse of the powder . from which some may think , that a cannon pointed right to the zenith , should shoot with greater violence , then in any other mount or forme whatsoever : and by some it hath beene imagined , that a bullet shot in this fashion hath been consumed , melted , and lost in the aire , by reason of the violence of the blow , and the activity of the sire , and that sundry experiments have been made in this nature , and the bullet never found . but it is hard to believe this assertion : it may rather be supposed that the bullet falling farre from the piece cannot be discerned where it falls : and so comes to be lost . 4. in the fourth place it may be asked , whether the discharge of a cannon b● so much the greater , by how much it is longer ? it seemeth at the first to be most true , that the longer the piece is , the more violent it shoots : and to speak generally , that which is direction by a trunke , pipe , or other concavitie , is conveyed so much the more violent , or better , by how much it is longer , either in respect of the sight , hearing , water , fire , &c. & the reason seems to hold in cannons , because in those that are long , the fire is retained a longer time in the concavitie of the piece , and so throwes out the bullet with more violence ; and experience lets us see that taking cannons of the same boare , but of diversitie of length from 8 foot to 12 , that the cannon of 9 foot long hath more force than that of 8 foot long , and 10 more than that of 9 , and so unto 12 foote of length . now the usuall cannon carries 600 paces , some more , some lesse , yea some but 200 paces from the piece , and may shoot into soft earth 15 or 17 foot , into sand or earth which is loose , 22 or 24 foot , and in firme ground , about 10 or 12 foot , &c. it hath been seen lately in germany , where there were made pieces from 8 foot long to 17 foot of like boare , that shooting out of any piece which was longer than 12 foot ; the force was diminished , and the more in length the piece increaseth , the lesse his force was : therefore the length ought to be in a meane measure , and it is often seene , the greater the cannon is , by so much the service is greater : but to have it too long or too short , is not convenient , but a meane proportion of length to be taken , otherwise the flame of the fire will be over-pressed with aire : whic hinders the motion in respect of substance , and distance of getting out . problem . lxxxiiii . of predigious progression and multiplication , of creatures , plants , fruits , numbers , gold , silver , &c. when they are alwayes augmented by certaine proportion . here we shall shew things no lesse admirable , as recreative , and yet so certaine and easie to be demonstrated , that there needs not but multiplication only , to try each particular : and first , of graines of mustard-seed . first , therefore it is certaine that the increase of one graine of mustard-seed for 20 yeares space , cannot be contained within the visible world , nay if it were a hundred times greater than it is : and holding nothing besides from the centre of the earth even unto the firmament , but only small grains of mustard-seed : now because this seems but words , it must be proved by art , as may be done in this wise , as suppose one mustard-seed sowne to bring forth a tree or branch , in each extendure of which might be a thousand graines : but we will suppose onely a thousand in the whole tree , and let us proceed to ●0 yeares , every seed to bring forth yearely a thousand graines , now multiplying alwayes by a thousand , in lesse then 17 years you shall have to many graines which will surpasse the sands , which are able to fill the whole firmament : for following the supposition of archimedes , and the most probable opinion of the greatness of the firmament which ●i●ho brahe hath left us ; the number of graines of sand will be sufficiently expressed with 49 ciphers , but the number of graines of mustard-seed at the end of 17 yeares will have 52 ciphers : and moreover , graines of mustard-seed , are farre greater than these of the sands : it is therefore evident that at the seventeenth yeare , all the graines of mustard-seed which shall successively spring from one graine onely , cannot be contained within the limits of the whole firmament ; what should it be then , if it should be multiplied againe by a thousand for the ●8 yeare : and that againe by a thousand for every yeares increase untill you come to the 20 yeare , it 's a thing as cleare as the day , that such a heap of mustard-seed would be a hundred thousand times greater than the earth : and bring onely but the increase of one graine in 20 yeares . of pigges . secondly , is it not a strange proposition , to say that the great turke with all his revenues , is not able to maintaine for one yeares time , all the pigges that a sow may pigge with all her race , that is , the increase with the increase unto 12 years : this seemes impossible , yet it is most true , for let us suppose and put , the case , that a sow bring forth but 6 , two males , and 4 females , and that each female shall bring forth as many every yeare , during the space of 12 yeares , at the end of the time there will be found above 3● millions of pigges : now allowing a crowne for the maintenance of each pigge for a yeare , ( which is as little as may be , being but neare a halfe of a farthing allowance for each day ; ) there must be at the least so many crownes to maintaine them , one a year , viz. 33 millions , which exceeds the turkes revenue by much . of graines of corne. thirdly , it will make one astonished to think that a graine of corne , with his increase successively for the space of 12 yeares will produce in grains 24414062●000000000000 , which is able to load almost al the creatures in the world. to open which , let it be supposed that the first yeare one graine being sowed brings forth 50 , ( but sometimes there is seen 70 , sometimes 100 fold ) which graines sowen the next yeare , every one to produce 50 , and so consequently the whole and increase to be sowen every yeare , until 12 yeares be expired , there will be of increase the aforesaid prodigious summe of graines , viz. 244140625000000000000 , which will make a cubical heap of 6258522 graines every way , which is more than a cubicall body of 31 miles every way : for allowing 40 graines in length to each foot , the cube would be 156463 foot every way : from which it is evident that if there were two hundred thousand cities as great as london , allowing to each 3 miles square every way , and 100 foot in height , there would not be sufficient roome to containe the aforesaid quantitie of corne : and suppose a bushel of corne were equal unto two cubicke feet , which might containe twenty hundred thousand graines then would there be 122070462500000. bushells , and allowing 30 bushels to a tunne , it would be able to load 81380●0833 vessels , which is more than eight thousand one hundred and thirty eight millions , ship loadings of ●00 tunne to each ship a : quantity so great that the sea is scarce able to beare , or the universal world able to finde vessels to carry it . and if this corne should be valued at halfe a crown the bushel , it would amount unto 15258807812500 pounds sterling , which i think exceeds all the treasures of all the princes , and of other particular men in the whole world : and is not this good husbandry to sowe one grain of corne ; and to continue it in sowing , the increase only for 12 yeares to have so great a profit ? of the increase of sheep . fourthly , those that have great flocks of sheep may be quickly rich , if they would preserve their sheep without killing or selling of them : so that every sheep produce one each yeare , for at the end of 16 yeares , 100 sheepe will multiply and increase unto 6553600 , which is above 6 millions , 5 hundred 53 thousand sheep : now supposing them worth but a crown a piece , it would amount unto 1638400 pounds sterling , vvhich is above 1 million 6 hundred 38 thousand pounds , a faire increase of one sheep : and a large portion for a childe if it should be allotted . of the increase of cod-fish , carpes , &c. fifthly , if there be any creatures in the vvorld that do abound vvith increase or fertilitie , it may be rightly attributed to fish ; for they in their kindes produce such a great multitude of eggs , and brings forth so many little ones , that if a great part vvere not destroyed continually , vvithin a ●ittle vvhile they vvould fill all the sea , ponds , and rivers in the vvorld ; and it is easie to shevv hovv it vvould come so to passe , onely by supposing them to increase without taking or destroying them for the space of 10 or 12 yeares : having regard to the soliditie of the waters which are allotted for to lodge and containe these creatures , as their bounds and place of rest to live in . of the increase and multiplication of men . sixthly , there are some that cannot conceive how it can be that from eight persons ( which were saved after the deluge or noahs flood ) should spring such a world of people to begin a monarchie under nimrod , being but 200 yeares after the flood , and that amongst them should be raised an army of two hundred thousand fighting men : but it is easi●y proved if vve take but one of the children of noah , and suppose that a nevv generation of people begun at every 30 yeares , and that it be continued to the seventh generation vvhich is 200 yeares ; for then of one only family there vvould be produced one hundred and eleven thousand soules , three hundred and five to begin the vvorld : though in that time men lived longer , and vvere more capable of multiplication and increase : vvhich number springing onely from a simp●e production of one yearly , vvould be farre greater , if one man should have many vvives , vvhich in ancient times they had : from vvhich it is also that the children of israel , vvho came into egypt but onely 70 soules , yet after 210 yeares captivity , they came forth vvith their hostes , that there vvere told six hundred thousand fighting men , besides old people , women and children ; and he that shall separate but one of the families of joseph , it would be sufficient to make up that number : how much more should it be then if we should adjoyne many families together ? of the increase of numbers . seventhly , what summe of money shall the city of london be worth , if it should be sold , and the money be paid in a yeare after this manner : the first week to pay a pinne , the second week 2 pinnes , the third week 4 pinnes , the fourth week 8 pinnes , the fifth week 16 pinnes . and so doubling untill the 52 weeks , or the yeare be expired . here one would think that the value of the pinnes would amount but to a small matter , in comparison of the treasures , or riches of the whole city : yet it is most probable that the number of pinnes would amount unto the sum of 4519599628681215 , and if we should allow unto a quarter a hundred thousand pinnes , the whole would contain ninetie eight millions , foure hundred thousand tunne : which is able to load 45930 shippes of a thousand tunne apiece : and if we should allow a thousand pins for a penny , the summe of money would amount unto above eighteen thousand , eight hundred and thirty millions of pounds sterling , an high price to sell a citie at , yet certain , according to that first proposed . so if 40 townes were sold upon condition to give for the first a penny , for the second 2 pence , for the third 4 pence , &c. by doubling all the rest unto the last , it would amount unto this number of pence , 109951●62●●76 , which in pounds is 4581298444 , that is foure thousand five hundred and fourescore millions of pounds and more . of a man that gathered up apples , stones , or such like upon a condition . eightly , admit there were an hundred apples , stones , or such like things that were plac'd in a straight line or right forme , a pace one from another , and a basket being placed a pace from the first : how many paces would there be made to put all these stones into the basket , by fetching one by one ? this would require near halfe a day to do it , for there would be made ten thousand and ninety two paces before he should gather them all up . of changes in bells , in musicall instruments , transmutation of places , in numbers , letters , men or such like . ninethly , is it not an admirable thing to consider how the skill of numbers doth easily furnish us with the knowledge of mysterious and hidden things ? which simply looked into by others that are not versed in arithmetick , do present unto them a world of confusion and difficultie . as in the first place , it is often debated amongst our common ringers , what number of changes there might be made in 5 , 6 , 7 , 8 , or more bells : who spend much time to answer their owne doubts , entring often into a labyrinth in the search thereof : or if there were 10 voyces , how many severall notes might there be ? these are propositions of such facility , that a childe which can but multiply one number by another , may easily resolve it , which is but only to multiply every number from the unite successively in each others product , unto the terme assigned : so the 6 number that is against 6 in the table , is 720 , and so many ( hanges may be made upon 6 bells , upon 5 there are 120 , &c. in like manner against 10 in the table is 3618800 , that is , three millions , six hundred twenty eight thousand , eight hundred : which shews that 10 voices may have so many consorts , each man keeping his owne note , but only altering his place ; and so of stringed instruments , and the gamat may be varied according to which , answerable to the number against x , viz. 1124001075070399680000 notes , from which may be drawne this , or the like proposition . suppose that 7 schollers were taken out of a free schoole to be sent to an vniversitie , there to be entertained in some colledge at commons for a certaine summe of money , so that each of them have two meales daily , and no longer to continue there , then that sitting all together upon one bench or forme at every meale , there might be a divers transmutation of place , of account in some one of them , in comparison of another , and never the whole company to be twice alike in situation : how long may the steward entertaine them ? ( who being not skilled in this fetch may answere unadvisedly . ) it is most certaine that there will be five thousand and forty several 1 a 1 2 b 2 6 c 3 24 d 4 120 e 5 7●0 f 6 5040 g 7 403●0 h 8 362880 i 9 3628800 k 10 39916800 l 11 479001600 m 12 6227020800 n 13 87178291200 o 14 1307674368000 p 15 20922789888000 q 16 355687537996000 r 17 6402375683928000 s 18 121645137994632000 t 19 2432902759892640000 u 20 51090957957745440000 w 21 1124001075070399680000 x 22 25852024726619192640000 y 23 6●0448593438860623360000 z 24 positions or changings in the seatings , which maks 14 years time wanting 10 weeks and 3 dayes . hence from this mutability of transmutation , it is no marvell tha● by 24 letters there ariseth and is made such variety of languages in the world , & such infinite number of words in each language ; seeing the diversity of syllables produceth that effect , and also by the interchanging & placing of letters amongst the vowels , & amongst themselves maketh these syllables : vvhich alphabet of 24 letters may be varied so many times , viz. 620448593438860623360000 vvhich is six hundred tvventy thousand , foure hundred forty eight millions of millions of millions five hundred ninety three thousand , foure hundred thirty eight milions of milions , & more . novv allovving that a man may reade or speak one hundred thousand vvords in an houre vvhich is tvvice more vvords than there are conteined in the psalmes of david , ( a taske too great for any man to do in so short a time ) and if there were foure thousand six hundred and fifty thousand millions of men , they could not speak these words ( according to the hourely proportion aforesaid ) in threescore and ten thousand yeares ; which variation & transmutation of letters , if they should be written in bookes , allowing to each leaf 28000 words , ( which is as many as possibly could be inserted , ) and to each book a reame or 20 quire of the largest and thinnest printing paper , so that each book being about 15 inches long , 12 broad , and 6 thick : the books that would be made of the transmutation of the 24 letters aforesaid , would be at least 38778037089928788 : and if a library of a mile square every way , of 50 foot high , were made to containe 250 galleries of 20 foot broad apiece , it would containe foure hundred mill●ons of the said books : so there must be to containe the rest no lesse than 9●945092 such libraries ; and if the books were extended over the surface of the globe of the earth , it would be a decuple covering unto it : a thing seeming most incredible that 24 letters in their transmutation should produce such a prodigious number , yet most certaine and infallible in computation . of a servant hired upon certaine conditions . a servant said unto his master , that he would dvvell vvith him all his life-time , if he would but onely lend him land to sowe one graine of corne with all his increase for 8 years time ; how think you of this bargaine ? for if he had but a quarter of an inch of ground for each graine , and each graine to bring forth yearely of increase 40 graines , the whole sum would amount unto , at the terme aforesaid , 6553600000000 graines : and seeing that three thousand and six hundred millions of inches do but make one mile square in the superficies , it shall be able to receive foureteene thousand and foure hundred millions of graines , which is 14400000000. thus dividing the aforesaid 6553600000000 , the quotient will be 455 , and so many square miles of land must there be to sowe the increase of one graine of corne for 8 yeares , which makes at the least foure hundred and twenty thousand acres of land , which rated but at five shillings the acre per annum , amounts unto one hundred thousand pound ; which is twelve thousand and five hundred pound a yeare , to be continued for 8 yeares ; a pretty pay for a masters servant 8 yeares service . problem . lxxxv . of fountaines , hydriatiques , machinecke , and other experiments upon water , or other liquor . 1. first how to make water at the foot of a mountaine to ascend to the top of it , and so to descend on the other side ? to do this there must be a pipe of lead , which may come from the fountaine a , to the top of the mountaine b ; and so to descend on the other side a little lower then the fountaine , as at c. then make a hole in the pipe at the top of the mountaine , as at b , and stop the end of the pipe at a and c ; and fill this pipe at b with water : & close it very carefully againe at b , that no aire get in : then unstop the end at a , & at c ; then will the water perpetually runne up the hill , and descend on the other side , which is an invention of great consequence to furnish villages that want water . 2. secondly , how to know what wine or other liquor there is in a vessell without opening the bung-hole , and without making any other hole , than that by which it runnes out at the top ? in this problem there is nothing but to take a bowed pipe of glasse , and put it into the faucets hole , and stopping it close about : for then you shall see the wine or liquor to ascend in this pipe , untill it be just even with the liquor in the vessel ; by which a man may fill the vessel , or put more into it : and so if need were , one may empty one vessel into another without opening the bung-hole . 3. thirdly , how is it that it is said that a vessell holds more water being placed at the foot of a mountaine , than standing upon the top of it ? this is a thing most certaine , because that water and all other liquor disposeth it selfe sphericaliy about the centre of the earth ; and by how much the vessel is nearer the centre , by so much the more the surface of the water makes a lesser sphere , and therefore every part more gibbous or swelling , than the like part in a greater sphere : and therefore when the same vessell is farther from the centre of the earth , the surface of the water makes a greater sphere , and therefore lesse gibbous , or swelling over the vessell : from whence it is evident that a vessell near the centre of the earth holds more water than that which is farther remote from it ; and so consequently a vessel placed at the bottome of the mountaine holds more water , than being placed on the top of the mountaine . first , therefore one may conclude , that one and the same vessel will alwayes hold more : by how much it is nearer the centre of the earth . secondly , if a vessell be very neare the centre of the earth , there will be more water above the brims of it , than there is within the vessel . thirdly , a vessel full of water comming to the centre wil spherically increase , and by little and little leave the vessel ; and passing the centre , the vessel will be all emptied . fourthly , one cannot carry a paile of water from a low place to a higher , but it will more and more run out and over , because that in ascending it lies more levell , but descending it swels and becomes more gibbous . 4. fourthly , to conduct water from the top of one mountaine , to the top of another . as admit on the top of a mountaine there is a spring , and at the toppe of the other mountaine there are inhabitants which want water : now to make a bridge from one mountaine to another , were difficult and too great a charge ; by way of pipes it is easie and of no great price : for if at the spring on the top of the mountaine be placed a pipe , to descend into the valley , and ascend to the other mountlaine , the water will runne naturally , and continually , provided that the spring be somewhat higher than the passage of the water at the inhabitants . 5. fifthly , of a fine fountaine which spouts water very high , and with great violence by turning of a cock. let there be a vessell as ab , made close in all his parts , in the middle of which let cd be a pipe open at d neare the bottome , and then with a squirt squirt in the water at c , stopped above by the cock or faucet c , vvith as great violence as possible you can ; and turne the cock immediatly . novv there being an indifferent quantity of vvater and aire in the vessel , the vvater keeps it selfe in the bottome , and the aire vvhich vvas greatly pressed , seeks for more place , that turning the cock the water issueth forth at the pipe , and flyes very high , and that especially if the vessell be a little heated : some make use of this for an ewer to wash hands withall , and therefore putting a moveable pipe above c , such as the figure sheweth : which the water will cause to turne very quick , pleasurable to behold . 6. sixtly , of archimedes screw , which makes water ascend by descending . this is nothing else but a cylinder , about the which is a pipe in form of a screw , and when one turnes it , the water descends alwayes in respect of the pipe : for it passeth from one part which is higher to that which is lower , and at the end of the engine the water is found higher than it was at the spring . this great enginer admirable in all mathematicall arts invented this instrument to wash king hieroies great vessells , as some authors saye , also to water the fields of egypt , as diodorus witnesseth : and cardanus reporteth that a citizen of milan having made the like engine , thinking himselfe to be the first inventer , conceived such exceeding joy , that he be came mad , foll . 2. againe a thing may ascend by descending , if a spiral line be made having many circulations or revolutions ; the last being alwayes lesser than the first , yet higher than the plaine supposed it is most certaine that then putting a ball into it , and turning the spirall line so , that the first circulation may be perpendicular , or touch alwayes the supposed plain : the ball shall in descending continually ascend , untill at last it come to the highest part of the spirall line , & so fall out . and here especially may be noted , that a moving body as water , or a bullet , or such like , will never ascend if the helicall revolution of the screw be not inclining to the horizon : so that according to this inclination the ball or liquor , may descend alwayes by a continuall motion and revolution . and this experiment may be more usefull , naturally made with a thred of ●ron , or latine turned or bowed helically about a cylinder , with some distinction of distances between the heli●es , for then having drawn out the cylinder , or having hung or tyed some weight at it in such sort , that the water may easily drop if one lift up the said thred : these helices or revolutions , notwithstanding will remaine inclining to the horizon , and then turning it about forward , the said weight will ascend , but backward it will descend . now if the revolutions be alike , and of equallity amongst themselves , and the whirling or turning motion be quicke , the sight vvill be so deceived , that producing the action it vvill seeme to the ignorant no lesse than a miracle . 7. seventhly , of another fine fountaine of pleasure . this is an engine that hath two wheeles with cogges , or teeth as ab , which are placed within an ovall cd , in such sort , that the teeth of the one , may enter into the notches of the other ; but so just that neither aire nor water may enter into the ovall coffer , either by the middle or by the sides , for the wheele must joyne so neare to the sides of the coffer , that there be no vacuitie : to this there is an axeltree with a handle to each wheele , so that they may be turned , and a being turned , that turneth the other wheele that is opposite : by which motion the aire that is in e , & the water that is carried by the hollow of the wheeles of each side , by continuall motion , is constrained to mount and flie out by the funnell f : now to make the water runne what way one would have it , there may be applied upon the top of the pipe f , two other moveable pipes inserted one within another ; as the figure sheweth . but here note , that there may acrue some inconveniency in this machine seeing that by quick turning the cogges or teeth of the wheeles running one against another , may neare break them , and so give way to the aire to enter in , which being violently inclosed vvill escape to occupie the place of the vvater , vvhose vveight makes it so quick : hovvsoever , if this machine be curiously made as an able vvorkeman may easily do , it is a most sovereigne engine , to cast vvater high and farre off for to quench fires . and to have it to raine to a place assigned , accommodate a socket having a pipe at the middle , vvhich may point tovvards the place being set at the top thereof , and so having great discretion in turning the axis of the vvheele , it may vvork exceeding vvell , and continue long . 8. eightly , of a fine watering pot for gardens . this may be made in forme of a bottle according to the last figure or such like , having at the bottome many small holes , and at the neck of it another hole somevvhat greater than those at the bottome , vvhich hole at the top you must unstop vvhen you vvould fill this vvatering pot , for then it is nothing but putting the lovver end into a paile of vvater , for so it vvill fill it selfe by degrees : and being full , put your thumb on the hole at the neck to stop it , for then may you carry it from place to place , and it vvill not sensibly runne out , som●thing it vvill , and all in time ( if it vvere never so close stopped ) contrary to the ancient tenet in philosophy , that aire will not penetrate . 9. ninthly , how easily to take wine out of a vessell at the bu●g-hole , without piercing of a hole in the vessell ? in this there is no need but to have a cane or pipe of glasse or such like , one of the ends of which may be closed up almost , leaving some small hole at the end ; for then if that end be set into the vessell at the bung-hole , the whole cane or pipe will be filled by little and little ; and once being full , stop the other end which is without and then pull out the cane or pipe , so will it be ful of wine , then opening a little the top above , you may fill a glasse or other pot with it , for as the wine issueth out , the aire commeth into the cane or pipe to supply vacuity . 10. tenthly , how to measure irregular bodies by help of water ? some throw in the body or magnitude into a vessell , and keep that which floweth out over , saying it is alwayes equal to the thing cast into the water : let i● is more nea●er this way to poure into a vessell such a quantity of water , which may be thought sufficient to cover the body or magnitude , and make a marke how high the water is in the vessell , then poure out all this water into another vessell , and let the body or magnitude be placed into the first vessel ; then poure in water from the second vessell , until it ascend unto the former marke made in the first vessell , so the vvater vvhich remaines in the second vessel is equall to the body or magnitude put into the water : but here note that this is not exact or free from error , yet nearer the truth than any geometrician can otherwise possibly measure , and these bodies that are not so full of pores are more truly measured this way , than others are . 11. to finde the weight of water . seeing that 574 / 1000 part of an ounce weight , makes a cubicall inch of water : and every pound weight haverdepoize makes 27 cubicall inches , and 1 9 / ● ; fere , and that ● gallons and a halfe wine measure makes a foot cubicall , it is easie by inversion , that knowing the quantity of a vessel in gallons , to finde his content in cubicall feet or weight : and that late famous geometrician master brigs found a cubical foot of vvater to vveigh neare 62 pound vveight haverdepoize but the late learned simon stevin found a cubicall foot of vvater to vveigh 65 pound , vvhich difference may arise from the inequalitie of vvater ; for some vvaters are more ponderous than others , and some difference may be from the weight of a pound , and the measure of a foot : thus the weight and quantitie of a solid foot settled , it is easie for arithmeticians to give the contents of vessells or bodies which containe liquids . 12. to finde the charge that a vessell may carry as shippes , boates , or such like . this is generally conceived , that a vessell may carry as much weight as that water weigheth , which is equall unto the vessell in bignesse , in abating onely the weight of the vessell : we see that a barrel of wine or water cast into the water , will not sink to the bottome , but swim easily , and if a ship had not iron and other ponderosities in it , it might swim full of water without sinking : in the same manner if the vessell were loaden with lead , so much should the watter weigh : hence it is that marriners call shippes of 50 thousand tunnes , because they may containe one or two thousand tunne , and so consequently carry as much . 13. how comes it that a shippe having safely sayled in the vast ocean , and being come into the port or harbour , without any tempest will sink down right ? the cause of this is that a vessel may carry more upon some kinde of water than upon other ; now the water of the sea is thicker and heavier than that of rivers , wels , or fountains ; therefore the loading of a vessell which is accounted sufficient in the sea , becomes too great in the hurbour or sweet water . now some think that it is the depth of the water that makes vessells more easie to swimme , but it is an abuse ; for if the loading of a ship be no heavier than the water that would occupie that place , the ship should as easily swim upon that water , as if it did swim upon a thousand fathom deep of water , and if the vvater be no thicker than a leafe of paper , and weigheth but an ounce under a heavy body , it vvill support it , as vvell as if the vvater under it vveighed ten thousand pound vveight : hence it is if there be a vessell capable of a little more than a thousand pound vveight of vvater , you may put into this vessell a piece of vvood , vvhich shall vveigh a thousand pound vveight ; ( but lighter in his kinde than the like of magnitude of vvater : ) for then pouring in but a quart of vvater or a very little quantitie of vvater , the vvood vvill svvim on the top of it , ( provided that the vvood touch not the sides of the vessell : ) vvhich is a fine experiment , and seems admirable in the performance . 14. how a grosse body of mettle may swimme upon the water ? this is done by extending the mettle into a thin plate , to make it hollovv in forme of a vessel ; so that the greatnesse of the vessell which the aire vvith it containeth , be equal to the magnitude of the vvater , vvhich vveighes as much as it , for all bodies may svvim vvithout sinking , if they occupie the place of vvater equal in vveight unto them , as if it vveighed 12 pound it must have the place of 12 pound of vvater : hence it is that vve see floating upon the vvater great vessells of copper or brasse , vvhen they are hollovv in forme of a caldron . and how can it be otherwise conceived of islands in the sea that swim and float ? is it not that they are hollow and some part like unto a boat , or that their earth is very light and spongeous , or having many concavities in the body of it , or much wood within it ? and it would be a pretty proposition to shew how much every kinde of metall should be inlarged , to make it swim upon the water : which doth depend upon the proportion that is between the vveight of the vvater and each metall . novv the proportion that is betvveene metalls and water of equall magnitude , according to some authors , is as followeth . a magnitude of 10 pound weight of water will require for the like magnitude of gold. 187 ½ lead . 116 ½ silver . 104 copper . 91 iron . 81 tinne . 75 from which is inferred , that to make a piece of copper of ●0 pound weight to swimme , it must be so made hollow , that it may hold 9 times that weight of water and somewhat more , that is to say , 91 pound : seeing that copper and water of like magnitudes in their ponderosities , are as before , as ●0 to 91. 15. how to weigh the lightnesse of the aire ? place a ballance of wood turned upside downe into the water , that so it may swim , then let water be inclosed within some body , as within a bladder or such like , and suppose that such a quantitie of aire should weigh one pound , place it under one of the ballances , and place under the other as much weight of lightnesse as may counter-ballance and keep the other ballance that it rise not out of the water : by which you shall see how much the lightnesse is . but without any ballance do this ; take a cubicall hollow vessell , or that which is cylindricall , which may swimme on the water , and as it sinketh by placing of weights upon it , marke hovv much , for then if you vvould examine the vveight of any body , you have nothing to do but to put it into this vessell , and marke hovv deep it sinkes , for so many pound it vveighes as the vveights put in do make it so to sinke . 16. being given a body , to marke it about , and shew how much of it will sink in the water , or swim above the water . this is done by knovving the vveight of the body vvhich is given , and the quantity of vvater , vvhich vveighes as much as that body ; for then certainly it vvill sink so deep , untill it occupieth the place of that quantitie of vvater . 17. to finde how much severall mettle or other bodies doe weigh lesse in the water than in the aire : take a ballance , & vveigh ( as for example ) 9 pound of gold , silver , lead , or stone in the aire , so it hang in aequilibrio ; then comming to the vvater , take the same quantity of gold silver , lead , or stone , and let it softly dovvne into it , and you shall see that you shall need a lesse counterpoise in the other ballance to counter-ballance it : vvherefore all solids or bodies vveigh lesse in the vvater than in the aire , and so much the lesse it vvill be , by hovv much the vvater is grosse and thick , because the vveight findes a greater resistance , and therefore the vvater supports more than aire ; and further , because the vvater by the ponderositie is displeased , and so strives to be there againe , pressing to it , by reason of the other vvaters that are about it , according to the proportion of his weight . archimedes demonstrateth , that all bodies weigh lesse in the water ( or in like liquor ) by how much they occupie place : and if the water weigh a pound weight , the magnitude in the water shall weigh a pound lesse than in the aire . now by knowing the proportion of water and mettles , it is found that gold loseth in the water the 19 part of his weight , copper the 9 part , quicksilver the 15 part , lead the 12 part , silver the 10 part , iron the 8 part , tinne the 7 part and a little more : wherefore in materiall and absolute weight , gold in respect of the water that it occupieth weigheth 18 , and ¾ times heavier than the like quantitie of water , that is , as 18 ¾ to the quicksilver 15 times , lead 11 and ⅗ , silver 10 and ⅔ , copper 9 and 1 / 10 , iron 8 and ½ , and tinne 8 and 1 / ● . contrarily in respect of greatnesse , if the water be as heavy as the gold , then is the water almost 19 times greater than the magnitude of the gold , and so may you judge of the rest . 18. how is it that a ballance having like weight in each scale , and hanging in aequilibrio in the aire , being placed in another place , ( without removing any weight ) it shall cease to hang in aequilibrio sensibly : yea by a great difference of weight ? this is easie to be resolved by considering different mettles , which though they vveigh equall in the aire , yet in the vvater there vvill be an apparant difference ; as suppose so that in the scale of each ballance be placed 18 pound vveight of severall metalls , the one gold , and the other copper , vvhich being in aequilibrio in the aire , placed in the vvater , vvill not hang so , because that the gold los●eth neare the 18 part of his vveight , vvhich is about 1 pound , and the copper loseth but his 9 part , vvhich is 2 pound : vvherefore the gold in the vvater vveigheth but 17 pound , and the copper 16 pound , vvhich is a difference most sensible to confirme that point . 19. to shew what waters are heavier one than another , and how much . physicians have an especiall respect unto this , judging that vvater vvhich is lightest is most healthfull and medicinall for the body , & sea-men knovv that the heaviest vvaters do beare most , and it is knovvne vvhich water is heaviest thus . take a piece of wax , and fasten lead unto it , or some such like thing that it may but precisely swimme , for then it is equal to the like magnitude of water , then put it into another vessell which hath contrary water , and if it sinke , then is that water lighter than the other : but if it sinke not so deep , then it argueth the water to be heavier or more grosser than the first water , or one may take a piece of vvood , and marke the quantitie of sinking of it into severall waters , by vvhich you may judge which is lightest or heaviest , for in that which it sinkes most , that is infallibly the lightest , and so contrarily . 20. how to make a pound of water weigh as much as 10 , 2● , ●0 , or a hundred pound of lead ; nay as much as a thousand , or ten thousand and pound weight ? this proposition seems very impossible , yet water inclosed in a vessell , being constrained to dilate it selfe , doth weigh so much as though there were in the concavitie of it a solid body of water . there are many wayes to experiment this proposition , but to verifie it , it may be sufficient to produce two excellent ones onely : which had they not been really acted , little credit might have been given unto it . the first way is thus . take a magnitude which takes up as much place as a hundred or a thousand pound of water , and suppose that it were tied to some thing that it may hang in the aire ; then make a ballance that one of the scales may inviron it , yet so that it touch not the sides of it : but leave space enough for one pound of water : then having placed 100 pound weight in the other scale , throw in the water about the magnitude , so that one pound of water shall weigh downe the hundred pound in the other ballance . problem . lxxxvi . of sundry questions of arithmetick , and first of the number of sands . it may be said incontinent , that to undertake this were impossible , either to number the sands of lybia , or the sands of the sea ; and it vvas this that the poets sung , and that vvhich the vulgar beleeves ; nay , that vvhich long ago certaine philosophers to gelon king of sicily reported , that the graines of sand vvere innumerable : but i ansvvere vvith archimedes , that not only one may number those vvhich are at the border and about the sea ; but those vvhich are able to fill the vvhole vvorld , if there vvere nothing else but sand ; and the graines of sands admitted to be so small , that 10 may make but one graine of poppy : for at the end of the account there need not to expresse them , but this number 30840979456 , and 35 ciphers at the end of it . clavius and archimedes make it somevvhat more ; because they make a greater firmament than ticho brahe doth ; and if they augment the vniverse , it is easie for us to augment the number , and declare assuredly how many graines of sand there are requisite to fill another vvorld , in comparison that our visible vvorld vvere but as one graine of sand , an atome or a point ; for there is nothing to do but to multiply the number by it selfe , vvhich vvill amount to ninety places , vvhereof tvventie are these , 95143798134910955936 , and 70 ciphers at the end of it : vvhich amounts to a most prodigious number , and is easily supputated : for supposing that a graine of poppy doth containe 10 graines of sand , there is nothing but to compare that little bovvle of a graine of poppy , vvith a bovvle of an inch or of a foot , & that to be compared vvith that of the earth , and then that of the earth vvith that o the firmament ; and so of the rest . 2. divers metalls being melted together in one body , to finde the mixture of them . this wat a notable invention of archimedes , related by vitrivius in his architecture , where he reporteth that the gold-smith which king hiero imployed for the making of the golden crowne , which was to be dedicated to the gods , had stolen part of it , and mixed silver in the place of it : the king suspicious of the work proposed it to archimedes , if by art he could discover without breaking of the crowne , if there had been made mixture of any other metall with the gold. the way which he found out was by bathing himselfe ; for as he entred into the vessell of water , ( in which he bathed himselfe ) so the water ascended or flew out over it , and as he pulled out his body the water descended : from which he gathered that if a bowle of pure gold , silver , or other metall were cast into a vessell of water , the water proportionally according to the thing cast in would ascend ; and so by way of arithmetick the question lay open to be resolved : who being so intensively taken with the invention , leapes out of the bath all naked , crying as a man transported , i have found , i have found , and so discovered it . now some say that he took two masses , the one of pure gold , and the other of pure silver ; each equall to the weight of the crowne , and therefore unequall in magnitude or greatnesse ; and then knowing the severall quantities of water which was answerable to the crown , and the severall masses , he subtilly collected , that if the crowne occupied more place within the water than the masse of gold did : it appeared that there was silver or other metall melted with it . now by the rule of position , suppose that each of the three masses weighed 18 pound a piece , and that the masse of gold did occupie the place of one pound of water , that of silver a pound and a halfe ▪ and the crown one pound and a quarter only : then thus he might operate the masse of silver which weighed 18 pounds , cast into the water , did cast out halfe a pound of water more then the masse of gold , which weighed 18 pound , and the crowne which weighed also 18 pound , being put into a vessell full of water , threw out more water than the masse of gold by a quarter of a pound , ( because of mixt metall which was in it : ) therefore by the rule of proportion , if halfe a pound of water ( the excesse ) be answerable to 18 pound of silver , one quarter of a pound of excesse shall be answerable to 9 pound of silver , and so much was mixed in the crowne . some judge the way to be more facill by weighing the crowne first in the aire , then in the water ; in the aire it weighed 18 pound , and if it were pure gold , in the water it would weigh but 17 pound ; if it were copper it would weigh but 16 pound ; but because vve vvill suppose that gold and copper is mixed together , it vvill vveigh lesse then 17 pound , yet more than 16 pound , and that according to the proportion mixed : let it then be supposed that it vveighed in the vvater 16 pound and 3 quarters , then might one say by proportion , if the difference of one pound of losse , vvhich is betvveen 16 and 17 ) be ansvverable to 18 pound , to vvhat shall one quarter of difference be ansvverable to , vvhich is betvveen 17 and 16 ¾ , and it vvill be 4 pound and a halfe ; and so much copper vvas mixed vvith the gold. many men have delivered sundry vvayes to resolve this proposition since archimedes invention , and it vvere tedious to relate the diversities . baptista benedictus amongst his arithmeticall theoremes , delivers his vvay thus : if a masse of gold of equall bignesse to the crovvne did vveigh 20 pound , and another of silver at a capacity or bignesse at pleasure , as suppose did vveigh 12 pound , the crovvne or the mixt body would vveigh more than the silver , and lesser than the gold , suppose it vveighed 16 pound vvhich is 4 pound lesse than the gold by 8 pound , then may one say , if 8 pound of difference come from 12 pound of silver , from vvhence comes 4 pound vvhich vvill be 6 pound and so much silver vvas mixed in it , &c. 3. three men bought a quantitie of wine , each paid alike , and each was to have alike ; it happened at the last partition that there were 21 barrells , of which 7 were full , 7 halfe full , and 7 empty , how must they share the wine and vessells , that each have as many vessells one as another , & as much wine one as another ? this may be answered two wayes as followeth , and these numbers 2 , 2 , 3 , or 3 , 3 , 1 , may serve for direction , and signifies that the first person ought to have 3 barrells full , and as many empty ones , and one which is halfe full ; so he shall have 7 vessells and 3 barrels , and a halfe of liquor ; and one of the other shall in like manner have as much , so there will remaine for the third man 1 barrell full , 5 which are halfe full , and 1 empty , and so every one shall have alike both in vessells and wine . and generally to answer such questions , divide the number of vessells by the number of persons , and if the quotient be not an intire number , the question is impossible ; but when it is an intire number , there must be made as many parts as there are 3 persons , seeing that each part is lesse than the halfe of the said quotient : as dividing 21 by 3 there comes 7 for the quotient , which may be parted in these three parts , 2 , 2 , 3 , or 3 , 3 , 1 , each of which being lesse than ha●fe of 7. 4. there is a ladder which stands upright against a wall of 10 foot high , the foot of it is pulled out 6 foot from the wall upon the pavement : how much hath the top of the ladder descended ? the ansvver is , 2 foot : for by pythagoras rule the square of db , the hypothenusal is equall to the square of da 6 , & ab 10. novv if da be 6 foot , and ab 10 foot , the squares are 36 and 100 , vvhich 36 taken from 100 rests 64 , vvhose roote-quadrate is 8 so the foot of the ladder being novv at d , the toppe vvill be at c , 2 foot lovver than it vvas vvhen it vvas at b. problem . lxxxvii . witty suits or debates between caius and sempronius , upon the forme of f●gures , which geometricians call isoperimeter , or equall in circuit or compasse . marvell ●ot at it if i make the mathematicks take place at the ba●●e , and if i set forth here b●rtoleus , who witnesseth of himselfe , that being then an ancient doctor in the law , he himselfe took upon him to learne the elements and principles of geometry , by which he might set forth certaine lawes touching the divisions of fields , waters , islands , and other incident places : now this shall be to shew in passing by , that these sciences are profitable and behovefull for judges , counsellors , or such , to explaine many things which fall out in lawes , to avoid ambiguities , contentions , and suits often . 1. incident . caius had a field which was directly square , having 24 measures in circuit , that was 6 on each side : sempronius desiring to fit himselfe , prayed caius to change with him for a field which should be equivalent unto his , and the bargaine being concluded , he gave him for counterchange a piece of ground which had just as much in circuit as his had , but it was not square , yet quadrangular and rectangled , having 9 measures in length for each of the two longest sides , and 3 in breadth for each shorter side : now caius which was not the most subtillest nor wisest in the world accepted his bargaine at the first , but afterward● having conferred with a land-measures and mathematician , found that he was over-reached in his bargaine , and that his field contained 36 square measures , and the other field had but 27 measures , ( a thing easie to be knowne by multiplying the length by the breadth : ) sempronius contested with him in suite of law , and argued that figures which have equall perimeter or circuit , are equall amongst themselves : my field , saith he , hath equall circuit with yours , therefore it is equall unto it in quantitie . now this was sufficient to delude a judge which was ignorant in geometricall proportions , but a mathematician will easily declare the deceit , being assured that figures which are isoperemiter , or equall in circuit , have not alwayes equall capacitie or quantitie : seeing that with the same circuit , there may be infinite figures made which shall be more and more capable , by how much they have more angles , equall sides , and approach nearer unto a circle , ( which is the most capablest figure of all , ) because that all his parts are extended one from anothes , and from the middle or centre as much as may be : so we see by an infa●lible rule of experience , that a square is more capable of quantitie than a triangle of the same circuit , and a pentagone more than a square , and so of others , so that they be regular figures that have their sides equall , otherwise there might be that a regular triangle , having 24 measures in circuit might have more capacitie than a rectangled parallelogram , which had also 24 measures of circuit , as if it were 11 in length , and 1 inbreadth , the circuit is still 24 , yet the quantitie is but 11. and if it had 6 every way , it gives the same perimeter , viz. 24. but a quantitie of 36 as before . 2. incident . sempronius having borrowed of caius a sack of corne , which was 6 foot high and 2 foot broad , and when there was question made to repay it , sempronius gave caius back two sacks full of corne , which had each of them 6 foot high & 1 foot broad : who beleeved that if the sackes were full he was repaid , and it seems to have an appearance of truth barely looked on . but it is most evident in demonstration , that the 2 sacks of corn paid by sempronius to caius , is but halfe of that one sack which he lent him : for a cylinder or sack having one foot of diameter , and 6 foot of length , is but the 4 part of another cylinder , whose length is 6 foot , and his diameter is 2 foot : therefore two of the lesser cylinders or sackes , is but halfe of the greater ; and so caius was deceived in halfe his corne. 3. incident . some one from a common fountaine of a city hath a pipe of water of an inch diameter ; to have it more commodious , he hath leave to take as much more water , whereupon he gives order that a pipe be made of two inches diameter . now you will say presently that it is reason to be so bigge , to have just twice as 〈…〉 before : but if the magistrate of the citie understood geometricall proportions , he would soon cause it to be amended , & shew that he hath not only taken twice as much water as he had before , but foure times as much : for a circular hole which is two inches diameter is foure times greater than that of one inch , and therefore vvill cast out four times as much vvater as that of one inch , and so the deceit is double also in this . moreover , if there vvere a heap of corne of 20 foot every vvay , vvhich vvas borrovved to be paid next yeare ▪ the party having his corne in heapes of 12 foot every vvay , and of 10 foot every vvay , proffers him 4 heapes of the greater or 7 heaps of the lesser , for his ovvne heap of 20 every vvay , vvhich vvas lent : here it seems that the proffer is faire , nay vvith advantage , yet the losse vvould be neare 1000 foot . infinite of such causes do arise from geometricall figures , vvhich are able to deceive a judge or magistrate , vvhich is not somevvhat seene in mathematicall documents . problem . lxxxviii . containing sundry questions in matter of cosmography . first , it may be demanded , vvhere is the middle of the vvorld ? i speak not here mathematically , but as the vulgar people , vvho ask , vvhere is the middle of the vvorld ? in this sence to speak absolutely there is no point vvhich may be said to be the middle of the surface , for the middle of a globe is every vvhere : notvvithstanding the holy scriptures speake respectively , and make mention of the middle of the earth , and the interpreters apply it to the citie of jerusalem placed in the middle of palestina , and the habitable vvorld , that in effect taking a mappe of the vvorld , and placing one foot of the compasses upon jerusalem , and extending the other foot to the extremity of europe , asia , and afric● , you shall see that the citie of jerusalem is as a centre to that circle . 2. secondly , how much is the depth of the earth , the height of the heavens , and the compasse of the world ? from the surface of the earth unto the centre according to ancient traditions , is 3436. miles , so the vvhole thicknesse is 6872 miles , of which the whole compasse or circuit of the earth is 21600 miles . from the centre of the earth to the moone there is neare 56 semidiameters of the earth , which is about 192416 miles . unto the sunne there is 1142 semidiameters of the earth , that is in miles 3924912 ; from the starry firmament to the centre of the earth there is 14000 semidiameters , that is , 48184000 miles , according to the opinion and observation of that learned ticho brahe . from these measures one may collect by arithmeticall supputations , many pleasant propositions in this manner . first , if you imagine there were a hole through the earth , and that a milstone should be let fall down into this hole , and to move a mile in each minute of time , it would be more than two dayes and a halfe before it would come to the centre , and being there it would hang in the aire . secondly , if a man should go every day 20 miles , it would be three yeares wanting but a fortnight , before he could go once about the earth ; and if a bird should fly round about it in two dayes , then must the motion be 450 miles in an houre . thirdly , the moone runnes a greater compasse each houre , than if in the same time she should runne twice rhe circumference of the whole earth . fourthly , admit it be supposed that one should go 20 miles in ascending towards the heavens every day , he should be above 15 years before he could attaine to the orbe of the moone . fifthly , the sunne makes a greater way in one day than the moone doth in 20 dayes , because that the orbe of the sunnes circumference is at the least 20 times greater than the orbe of the moone . sixthly , if a milstone should descend from the p●ace of the sunne a thousand miles every houre , ( which is above 15 miles in a minute , farre beyond the proportion of motion ) it would be above 163 dayes before it would fall dovvne to the earth . seventhly , the sunne in his proper sphere moves more than seven thousand five hundred and seventy miles in one minute of time : novv there is no bullet of a cannon , arrovv , thunderbolt , or tempest of vvinde that moves vvith such quicknesse . eightly , it is of a farre higher nature to consider the exceeding and unmoveable quicknesse of the starry firmament , for a starre being in the aequator , ( which is just between the poles of the world ) makes 12598666 miles in one houre which is two hundred nine thousand nine hundred and seventy foure miles in one minute of time : & if a horseman should ride every day 40 miles , he could not ride such a compasse in a thousand yeares as the starry firmament moves in one houre , which is more than if one should move about the earth a thousand times in one houre , and quicker than possible thought can be imagined : and if a starre should flye in the aire about the earth with such a prodigious quicknesse , it would burne and consume all the world here below . behold therefore how time passeth , and death hasteth on : this made copernicus , not unadvisedly to attribute this motion of primum mobile to the earth , and not to the starry firmament ; for it is beyond humane sense to apprehend or conceive the rapture and violence of that motion being quicker than thought ; and the word of god testifieth that the lord made all things in number , measure , weight , and time . problem . xcii . to finde the bissextile yeare , the dominicall letter , and the letters of the moneth . let 123 , or 124 , or 125 , or 26 , or 27 , ( which is the remainder of 1500 , or 1600 ) be divided by 4 , which is the number of the leape-yeare , and that which remaines of the division shewes the leap-yeare , as if one remaine , it shewes that it is the first yeare since the bissextile or leap-year , if two , it is the second year &c. and if nothing remaine , then it is the bissextile or leap-yeare , and the quotient shews you how many bissextiles or leap-yeares there are conteined in so many yeares . to finde the circle of the sun by the fingers . let 123 , 24 , 25 , 26 , or 27 , be divided by 28 , ( which is the circle of the sunne or whole revolution of the dominicall letters ) and that which remaines is the number of joynts , which is to be accounted upon the fingers by filius esto dei , coelum bonus accipe gratis : and where the number ends , that finger it sheweth the yeare which is present , and the words of the verse shew the dominicall letter . example . divide 123 by 28 for the yeare ( and so of other yeares ) and the quotient is 4 , and there remaineth 11 , for which you must account 11 words : filius esto dei , &c. upon the joynts beginning from the first joynt of the index , and you shall have the answer . for the present to know the dominicall letter for each moneth , account from january unto the moneth required , including january , and if there be 8 , 9 , 7 , or 5 , you must begin upon the end of the finger from the thumbe and account , adam degebat , &c. as many words as there are moneths , for then one shall have the letter which begins the moneth ; then to know what day of the moneth it is , see how many times 7 is comprehended in the number of dayes , and take the rest , suppose 4 , account upon the first finger within & without by the joynts , unto the number of 4 , which ends at the end of the finger : from whence it may be inferred that the day required was wednesday , sunday being attributed to the first joynt of the first finger or index : and so you have the present yeare , the dominicall letter , the letter which begins the moneth , and all the dayes of the moneth . problem . xciii . to finde the new and full moone in each moneth . adde to t●e epact for the yeare , the moneth from march , then subtract that surplus from 30 , and the rest is the day of the moneth that it vvill be new moone , and adding unto it 14 , you shall have that full moone . note that the epact is made alwayes by adding 11 unto 30 , and if it passe 30 , subtract 30 , and adde 11 to the remainder , and so ad infinitum : as if the epact were 12 , adde 11 to it makes 23 for the epact next year , to vvhich adde 11 makes 34 , subtract 30 , rests 4 the epact for the yeare after , and 15 for the yeare follovving that , and 26 for the next , and 7 for the next , &c. problem . xciv . to finde the latitude of ● countrey . those that dwell between the north-pole and the tropicke of cancer , have their spring and summer between the 10 of march , and the 13 of september : and therefore in any day between that time , get the sunnes distance by instrumentall observation from the zenith at noone , and adde the declination of the sun for that day to it : so the aggragate sheweth such is the latitude , or poles height of that countrey . now the declination of the sunne for any day is found out by tables calculated to that end : or mechanically by the globe , or by instrument it may be indifferently had : and here note that if the day be between the 13 of september and the 10 of march , then the sunnes declination for that day must be taken out of the distance of the sunne from the zenith at noone : so shall you have the latitude , as before . prbolem xcv . of the climates of countreys , and to finde in what c●imate any countrey is under . climates as they are taken geographically signifie nothing else but when the l●ngt● of the longest day of any place , is half an houre longer , or shorter than it is in another place ( and so of the sh●rtest day ) and this account to begin from the equinoctia●l circle , seeing all countreys under it have the shortest and longest day that can be but 12 houres ; but all other countreys that are from the equinoctiall circle either towards the north or south of it unto the poles themselves , are said to be in some one climate or other , from the equinoctiall to either of the poles circles , ( which are in the latitude of 66 degr . 30 m. ) between each of which polar circles and the equinoctial circle there is accounted 24 climates , which differ one from another by halfe an hours time : then from each polar circle , to each pole there are reckoned 6. other climates which differ one from another by a moneths time : so the whole earth is divided into 60 climates , 30 being allotted to the northerne hemisphere , and 30● to the southerne hemispheare . and here note , that though these climats which are betweene the equinoctiall and the polar circles are equall one unto the other in respect of time , to wit , by halfe an houre ; yet the latitude , breadth , or internall , conteined between climate and climate , is not equall : and by how much any climate is farther from the equinoctiall than another climate , by so much the lesser is the intervall between that climate and the next : so those that are nearest the equinoctial are largest , and those which are farthest off most contracted : and to finde what climate any countrey is under : subtract the length of an equinoctiall day , to wit , 12 houres from the length of the longest day of that countrey ; the remainder being doubled shews the climate : so at london the longest day is neare 16 houres and a halfe ; 12 taken from it there remaines 4 houres and a halfe , which doubled makes 9 halfe houres , that is , 9 climates ; so london is in the 9 climate . problem . xcvi . of longitude and latitude of the earth and of the starres . longitude of a countrey , or place , is an arcke of the aequator conteined between the meridian of the azores , and the meridian of the place , and the greatest longitude that can be is 360 degrees . note . that the first meridian may be taken at pleasure upon the terrestriall globe or mappe , for that some of the ancient astronomers would have it at hercules pillars , which is at the straights at gibraltar : ptolomy placed it at the canary isl●nds , but now in these latter times it is held to be neare the azores . but why it was first placed by ptolomy at the canary islands , were because that in his time these islands were the farthest westerne parts of the world that vvas then discovered . and vvhy it reteines his place novv at saint michaels neare the azores , is that because of many accurate observations made of late by many expert navigators and mathematicians , they have found the needle there to have no variation , but to point north and south : that , is to each pole of the world : and why the longitude from thence is accounted eastwards , is from the motion of the sunne eastward , or that ptolomy and others did hold it more convenient to begin from the westerne part of the world and so account the longitude eastward from countrey to countrey that was then knowne ; till they came to the easterne part of asia , rather than to make a beginning upon that which was unknowne : and having made up their account of reckoning the longitude from the westerne part to the eastern part of the world knowne , they supposed the rest to be all sea , which since their deaths hath been found almost to be another habitable world . to finde the longitude of a countrey . if it be upon the globe , bring the countrey to the brasen meridian , and whatsoever degree that meridian cuts in the equinoctiall , that degree is the longitude of that place : if it be in a mappe , then mark what meridian passeth over it , so have you the longitude thereof , if no meridian passe over it , then take a paire of compasses , and measure the distance betweene the place and the next meridian , and apply it to the divided parallel or aequator , so have you the longitude required . of the latitude of countreys . latitude of a countrey is the distance of a countrey from the equinoctiall , or it is an arke of the meridian conteined between the zenith of the place and the aequator ; which is two-fold , viz. either north-latitude or south-latitude , either of which extendeth from the equinoctiall to either pole , so the greatest latitude that can be is but 90 degrees : if any northern countrey have the artick circle verticall , which is in the latitude of 66. gr . 30. m. the sun will touch the horizon in the north part thereof , and the longest day will be there then 24 houres , if the countrey have lesse latitude than 66. degrees 30. m. the sun will rise and set , but if it have more latitude than 66. gr . 30 m. it will be visible for many dayes , and if the countrey be under the pole , the sun will make a circular motion above the earth , and be visible for a half yeare : so under the pole there will be but one day , and one night in the whole yeare . to finde the latitude of countreys . if it be upon a globe , bring the place to the brasen meridian , and the number of degrees which it meeteth therewith , is the latitude of the place . or with a paire of compasses take the distance between the countrey and the equinoctiall , which applied unto the equinoctiall will shew the latitude of that countrey ; which is equall to the poles height ; if it be upon a mappe . then mark what parallel passeth over the countrey and where it crosseth the meridian , that shall be the latitude : but if ●o parallel passeth over it , then take the distance betweene the place and the next parallel , which applied to the divided meridian from that parallel will shew the latitude of that place . to finde the distance of places . if it be upon a globe : then with a paire of compasses take the distance betweene the two places , and apply it to the divided meridian or aequator , and the number of degrees shall shew ●e distance ; each degree being 60. miles . ●f it be in a mappe ( according to wrights pro●ection ) take the distance with a paire of com●asses between the two places , and apply this distance to the divided meridian on the mappe right against the two places ; so as many degrees as is conteined between the feet of the compasses so much is the distance between the two places . if the distance of two places be required in a particular map then with the compasses take the distance between the two places , and apply it to the scale of miles , so have you the distance , if the scale be too short , take the scale between the compasses , and apply that to the two places as often as you can , so have you the distance required . of the longitude , latitude , declination , and distance of the starres . the declination of a starre is the nearest distance of a star from the aequator ; the latitude of a starre is the nearest distance of a sarre from the ecliptick : the longitude of a starre is an ark of the ecliptick conteined between the beginning of aries , and the circle of the starres latitude , which is a circle drawne from the pole of the ecliptick unto the starre , and so to the ecliptick . the distance between two sarres in heaven is taken by a crosse-staffe or other instrument , and upon a globe it is done by taking between the feet of the compasses the two starres , and applying it to the aequator , so have you the distance betweene those two starre● . how is it that two horses or other creatures being foaled or brought forth into the world at one and the same time , that after certaine dayes travell the one lived more dayes than the other , notwithstanding they dyed together in one and the sam● moment also ? this is easie to be answered : let one of them travell toward the west and the other towards the east : then that which goes towards the west followeth the sunne : and therefore shall have the day somewhat longer than if there had been no travell made , and that which goes east by going against the sunne , shall have the day shorter , and so in respect of travell though they dye at one and the selfe same houre and moment of time , the one shall be older than the other . from which consideration may be inferred that a christian , a jew , and a saracen , may have their sabbaths all upon one and the same day though notwithstanding the saracen holds his sabath upon the friday , the jew upon the saturday , and the christian upon the sunday : for being all three resident in one place , if the saracen and the christian begin their travell upon the saturday , the christian going west , and the saracen eastwards , shall compasse the globe of the earth , thus the christian at the conclusion shall gaine a day , and the saracen shall lose a day , and so meet with the jew every one upon his owne sabbath . certaine fine observations . 1 under the equinoctiall the needle hangs in equilibrio , but in these parts it inclines under the horizon , and being under the pole it is thought it will hang verticall . 2 in these countreys which are without the tropicall circles , the sunne comes east and west every day for a halfe yeare , but being under the equinoctiall the sun is never east , nor west ▪ but twice in the yeare , to wit , the 10. of march and the 13 of september . 3 if a ship be in the latitude of 23 gr . 30 m. that is , if it have either of the tropicks verticall : then at what time the sunnes altitude is equall to his distan●e from any of the equinoctiall points , then t●e sunne is due east or west . 4 if a ship be betweene the equinoctiall and either of the tropicks , the sunne will come twice to one point of the compasse in the forenoone , that is , in one and the same position . 5 vnder the equinoctiall neare guinea there is but two sorts of windes all the year , 6 moneths a northerly winde , and 6 moneths a southerly winde , and the flux of the sea is accordingly . 6 if two ships under the equinoctiall be 100. leagues asunder , and should sayle northerly untill they were come under the articke circle , they should then be but 50 leagues asunder . 7 those which have the artick circle , verticall : when the sunne is in the tropick of cancer , the sun setteth not , but toucheth the western part of the horizon . 8 if the complement of the sunnes height at noon be found equall to the sunnes declination for that day , then the ●quinoctiall is verticall : or a shippe making such an observation , the equinoctiall is in the zenith , or direct over them , by which navigators know when they crosse the line , in their travels to the indies , or other parts . 9 the sunne being in the equinoctiall , the extremity of the stile in any sunne-dyall upon a plaine , maketh a right line , otherwise it is eclipticall , hyperbolicall , &c. 10 when the shadow of a man , or other thing upon a horizontall 〈◊〉 is equall unto it in length , then is the sunne in the middle point between the horizon and the zenith , that is , 45 degrees high . problem . xcvii . to make a triangle that shall have three right angles . open the c●passes at p●easure : and upon a , describe an arke bc. then at the same opening , place one of the feet in b , and describe the ark ac . lastly , place one of the feet of the compasses in c. and describe the arke ab· so shall you have the sphericall aequilaterall triangle abc , right angled at a , at b , and at c. that is , each angle comprehended 9● . degrees : which can never be in any plaine triangle , whether it be equilaterall , isocelse , scaleve , orthogonall , or opigonall . problem . xcviii . to divide a line in as many equall parts as one will , without compasses , or without seeing of it . this proposition hath a fallacie in it , & cannot be practised but upon a maincordion : for the mathematicall line which proceeds from the flux of a point , cannot be divided in that wise : one may have therefore an instrument which is called maincordion , because there is but one cord : and if you desire to divide your line into 3 parts , run your finger upon the frets untill you sound a third in musick : if you would have the fourth part of the line , then finde the fourth sound , a fifth , &c. so shall you have the answer . problem . xcix . to draw a line which shall incline to another line , yet never meet : against the axiome of parallels . this is done by help of a conoeide line , produced by a right line upon one & the same plaine , held in great account amongst the ancients , and it is drawne after this manner . draw a right line infinitely , and upon some end of it , as at i , draw a perpendicular line i a. augment it to h. then from a. draw lines at pleasure to intersect the line i. m. in each of which lines from the right line , im . transferre ih . viz. kb . lc.od.pe.qf.mg . then from those points draw the line h.b.c.d.e.f.g. which will not meet with the line im . and yet incline nearer and nearer unto it . problem . c. to observe the variation of the compasses , or needle in any places . first describe a circle upon a plaine , so that the sun may shine on it both before noone and afternoone : in the centre of which circle place a gn●●on or wire perpendicular as ab , and an houre before noone marke the extremitie of the shadow of ab , which suppose it be at c. describe a circle at that semidiamiter cdf . then after noone mark when the top of the shadow of ab . toucheth the circle , which admit in d ; divide the distance cd into two equall parts , which suppose at e. draw the line eaf . which is the meridian line , or line of north & south : now if the arke of the circle cd . be divided into degrees . place a needle gh , upon a plaine set up in the centre , and marke how many degrees the point of the needle g , is from e. so much doth the needle vary from the north in that place . problem . ci. how to finde at any time which way the wind is in ones chamber , without going abroad ? vpon the plancking or floore of a chamber , parlor , or hall , that you intend to have this device , let there come downe from the top of the house a hollow post , in which place an iron rod that it ascend above the house 10 , or 6 foot with a vane or a scouchen at it to shew the winds without : and at the lower end of this rod of iron , place a dart which may by the moving of the vane with the wind without , turne this dart which is within : about which upon the plaister must be described a circle divided into the 32 points of the mariners compasse pointed and distinguished to that end , then may it be marked by placi● to compasse by it ; for having noted the north point , the east , &c ▪ it is easie to note all the rest of the points : and so at any time comming into this roome , you have nothing to do but to look up to the dart , which will point you out what way the winde bloweth at that instant . problem . cii . how to draw a parallel sphericall line with great ease ? first draw an obscure line gf . in the middle of it make two points ab , ( which serves for centres then place one foot of the compasses in b , and extend the other foot to a , and describe the semicircle ac . then place one foot of the compasses in a , and extend the other foot to c , and describe the semicircle cd . now place the compasses in b , and extend the other foot unto d , and describe the semicircle df , and so ad infinitum ; which being done neatly , that there be no right line seene nor where the compasses were placed , will seeme very strange how possibly it could be drawne with such exactnes , to such which are ignorant of that way . problem . ciii . to measure an in accessible distance , as the breadth of a river with the help of ones hat onely . the way of this is easie : for having ones hat upon his head , come neare to the bank of the river , and holding your head upright ( which may be by putting a small stick to some one of your buttons to prop up the chin ) pluck downe the brim or edge of your hat untill you may but see the other side of the water , then turne about the body in the same posture that it was before towards some plaine , and marke where the sight by the brimme of the hat glaunceth on the ground ▪ for the distance from that place to your standing , is the breadth of the river required . problem . ciiii. how to measure a height with two strawes or two small stickes . take two strawes or two stickes which are one as long as another , and place them at right angles one to the other , as ab . and ac . then holding ab . parallel to the ground , place the end a to the eye at a. and looking to the other top bc. at c. by going backward or forward untill you may see the top of the tower or tree , which suppose at e. so the distance from your standing to the tower or tree , is equall to the height thereof above the levell of the eye : to which if you adde your ovvne height you have the whole height . otherwise . take an ordinary square which carpenters or other workemen use , as hkl . and placing h. to the eye so that hk . be levell , go back or come nearer untill that by it you may see the top m. for then the distance from you to the height is equall to the height . problem . cv . how to make statues , letters , bowles , or other things which are placed in the side of a high building , to be seen below of an equall bignesse . let bc. be a pillar 7 yards high , and let it be required that three yards above the levell of the eye a , viz. at b. be placed a globe , and 9 yards above b. be placed another , & 22. yards above that be placed another globe : how much shall the diameter of these globes be , that at the eye , at a , they may all appeare to be of one and the same magnitude : it is thus done , first draw a line as ak . and upon k. erect a perpendicular kx . divide this line into 27 parts ▪ and according to ak . describe an arke ky . then from k ▪ in the perpendicular kx , account● ▪ par●s , viz at l. which shall represent the former three yardes , and draw the line la. from l , in the said perpendicular reckon the diameter of the lesser globe of what magnitude it is intended to be : suppose sl , and draw the line sa . cutting the arke vk . in n. then from k. in the perpendicular account 9 yards , which admit at t. draw ta , cutting yk. in o transferre the arke mn , from a to p. and draw ap. which will cut the perpendicular in v. so a line drawne from the middle of vf . unto the visuall lines ai , and av , shall be the diameter of the next globe : lastly , account from k. in the perpendicular xk . 22 parts , and draw the line wa . cutting yk in q. then take the arke mn , and transferre it from q to r and draw ar ▪ which will cut the perpendicular in x so the line which passeth by the meddle of xw . perpendicular to the visuall line aw , and ax. be the diameter of the third globe , to wit 5 , 6. which measures transferred in the pillar bc. which sheweth the true magnitude of the globes 1 , 2 , 3. from this an architect or doth proportion his images , & the foulding of the robes which are most deformed at the eye below in the making , yet most perfect when it is set in his true height above the eye . problem . cvi. how to disg●is● or disfigure an image , as a head , an arme , a whole body , &c. so that it hath no proportion the eares to become long : the nose as that of a swan , the mouth as a coaches entrance , &c yet the eye placed at a certaine point will be seen in a direct & exact proportion . i will not strive to set a geometricall figure here , for feare it may seeme too difficult to understand , but i will indeavour by discourse how mechanically with a candle you may perceive it sensible : first there must be made a figure upon paper , such as you please , according to his just proportion , and paint it as a picture ( which painters know well enough to do ) afterwards put a candle upon the table , and interpose this figure obliquely , between the said candle and the bookes of paper , where you desire to have the figure disguised in such sort that the height passe athwart the hole of the picture : then will it carry all the forme of the picture upon the paper , but with deformity ; follow these tracts and marke out the light with a coles black head or ink : and you have your desire . to finde now the point where the eye must see it in his naturall forme : it is accustomed according to the order of perspective , to place this point in the line drawne in height , equall to the largenesse of the narrowest side of the deformed square , and it is by this way that it is performed . problem . cvii . how a cannon after that it hath shot , may be covered from the battery of the enemy . let the mouth of a cannon be i , the cannon m. his charge no , the wheele l , the axletree pb . upon which the cannon is placed , at which end towards b , is placed a pillar ae· supported with props d , c , e , f , g ▪ about which the axeltree turneth : now the cannon being to shoot , it retires to h , which cannot be directly because of the axletree , but it make a segment of a circle ▪ and hides himselfe behind the wal qr , and so preserves it selfe from the enemies battery , by which meanes one may avoid many inconveniences which might arise : and moreover , one man may more easily replace it againe for another shot by help of poles tyed to the wall , or other help which may multiply the strength . problem . cviii . how to make a lever , by which one man may alone place a cannon upon his carriage , or raise what other weight he would . first place two thick boards upright , as the figure sheweth , pierced with holes , alike opposite one unto another as cd , and ef : & let l , and m , be the two barres of iron which passeth through the holes gh , and f , k , the two supports , or props , ab . the cannon , op , the lever , rs , the two notches in the lever , and q , the hooke where the burthen or cannon is tyed to . the rest of the operation is ●cill , that the youngest schollers or learners cannot faile to performe it : to teach minerva were in vaine , and it were to mathematicians injury in the succeeding ages . problem . cix . how to make a clock with one onely wheele . make the body of an ordinary dyall , and divide the houre in the circle into 12. parts : make a great wheele in height above the axletree , to the which you shall place the cord of your counterpoize ▪ so that it may descend , that in 1● houres of time your index or needle may make one revolution , which may be knowne by a watch which you may have by you : then put a balance which may stop the course of the wheele , and give it a regular motion , and you shall see an effect as just from this as from a clock with many wheeles . problem . cx . how by help of two wheeles to make a childe to draw up alone a hogshead of water at a time : and being drawne up shall cast out it selfe into another vessell as one would have it . let r be the pit from whence water is to be drawne , p the hook to throw out the water when it is brought up ( this hook must be moveable ) let ab be the axis of the wheele sf , which wheele hath divers forkes of iron made at g , equally fastened at the wheele ; let i , be a card , which is drawne by k , to make the wheele s , to turne , vvhich vvheele s , beares proportion to the vvheele t , as 8 to ● . let n be a chaine of iron to vvhich is tyed the vessel o , and the other vvhich is in the pit : e● is a piece of vvood vvhich hath a mortes in 1 , and ● , by vvhich the cord i , passeth , tyed at the vvall , as kh , and the other piece of timber of the little vvheele as m , mortified in likevvise for the chaine to passe through : draw the cord i , by k , and the wheele will turne , & so consequently the wheele t , which will cause the vessell o to raise ; which being empty , draw the cord againe by y , and the other vessell which is in the pit ●ill come out by the same reason . this is an invention which will save labour if practised ; but here is to be noted that the pit must be large enough , to the end that it conteine two great vessels to passe up and downe one by another ▪ problem . cxi . to make a ladder of cords , which may be carryed in ones pocket : by which one may easily mount up a wall , or tree alone . take two pullies a , & d , unto that of a , let there be fastened a cramp of iron as b ; and at d , let there be fastened a staffe of a foot and a halfe long as f , then the pully a : place a hand of iron , as e , to vvhich tie a cord of an halfe inch thick ( vvhich may be of silk because it is for the pocket : ) then strive to make fast the pully a , by the help of the crampe of iron b , to the place that you intend to scale ; and the staffe f , being tyed at the pully d , put it betvveen your legges as though you vvould sit upon it : then holding the cord c in your hand , you may guide your selfe to the place required ▪ vvhich may be made more facill by the multiplying of pullies . this secret is most excellent in warre , and for lovers , its supportablenesse avoids suspition . problem . cxii . how to make a pumpe whose strength is marvelous by reason of the great weight of water that it is able to bring up at once , and so by continuance . let 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , be the height of the case about two or three foot high , and broader according to discretion : the rest of the case or concavity let be o : let the sucker of the pumpe vvhich is made , be just for the case or pumpes head 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , & may be made of vvood or brasse of 4 inches thick , having a hole at e , vvhich descending raiseth up the cover p , by which issueth forth the water , & ascending or raising up it shuts it or makes it close : rs , is the handle of the sucker tyed to the handle tx , which works in the post vz . let a , b , c , d , be a piece of brasse , g the piece which enters into the hole to f , to keep out the aire . h , i , k , l , the piece tyed at the funnell or pipe : in which playes the iron rod or axis g , so that it passe through the other piece mn , which is tyed with the end of the pipe of brasse . note , that the lower end of the cisterne ought to be rested upon a gridiron or iron grate ▪ which may be tyed in the pit , by which means lifting up and putting downe the handle , you may draw ten times more water than otherwise you could . problem . cxiii . how by meanes of a cisterne , to make water of a pit continually to ascend without strength , or the assistance of any other pumpe . let il , be the pit where one would cause water to ascend continually to ●●ach office of a house or the places which are separated from it : let there be made a receive● as a , well closed up with lead or other matter that aire enter not in , to which fasten a pipe of lead as at e , which may have vent at pleasure , then let there be made a cisterne as b , which may be communicative to a , by helpe of the pipe g , from vvhich cistern b , may issue the vvater of pipe d , vvhich may descend to h , vvhich is a little belovv the levell of the vvater of the pit as much as is gh . to the end of vvhich shall be soldered close a cock vvhich shall cast out the vvater by kh . novv to make use of it , let b be filled full of vvater , and vvhen you vvould have it run turne the cock , for then the vvater in b , vvill descend by k. and for feare that there should be vacuity , nature vvhich abhors it , vvill labour to furnish and supply that emptinesse out of the spring f , and that the pit dry not , the pipe ought to be small of an indifferent capacity according to the greatnesse or smalnesse of the spring . problem . cxiiii . how out of a fountaine to cast the water very high : different from a probleme formerly delivered . let the fountaine be bd , of a round forme ( seeing it is the most capable and most perfect figure ) place into it two pipes conjoyned as ea , and hc , so that no aire may enter in at the place of joyning : let each of the pipes have a cock g , & l : the cocke at g , being closed , open that at i ▪ & so with a squirt force the water through the hole at h , then close the cocke at a , & draw out the squirt , and open the cock at g. the aire being before rarified will extend his dimensions , and force the water with such violence , that it will amount above the height of one or two pipes : and so much the more by how much the machine is great : this violence will last but a little while if the pipe have too great an opening , for as the aire approacheth to his naturall place , so the force will diminish . problem . cxv . how to empty the water of a cisterne by a pipe which shall have a motion of it selfe . let ab , be the vessell ; cde , the pipe : hg , a little vessell under the greater , in which one end of the pipe is , viz. c , and let the other end of the pipe e. passing through the bottome of the vessell at f , then as the vessell filleth so will the pipe , and when the vessell , shall be full as farre as po , the pipe will begin to runne at e , of his owne accord , and never cease untill the vessell be wholly empty . problem cxvi . how to squirt or spout out a great height , so that one pot of water shall last a long time . let there be prepared two vessels of brasse , lead , or of other matter of equal substance as are the two vessels ab , and bd , & let them be joyned together by the two pillars mn , & ef : then let there be a pipe hg . which may passe through the cover of the vessell cd , and passe through ab , into g , making a little bunch or rising in the cover of the vessell ab , so that the pipe touch it not at the bottome : then let there be soldered fast another pipe il , which may be separated from the bottome of the vessell , and may have his bunchie swelling as the former without touching the bottome : as is represented in l , and passing through the bottome of ab , may be continued unto i , that is to say , to make an opening to the cover of the vessell ab , & let it have a little mouth as a trumpet : to that end to receive the water . then there must further be added a very smal pipe which may passe through the bottome of the vessell ab , as let it be op , and let there be a bunch ; or swelling over it as at p , so that it touch not also the bottome : let there be further made to this lesser vessell an edge in forme of a basin to receive the water , which being done poure water into the pipe il , untill the vessell cd , be full , then turne the whole machine upside downe that the vessell cd , may be uppermost , and ab , undermost ; so by helpe of the pipe gh , the water of the vessell cd , will runne into the vessel ab , to have passage by the pipe po. this motion is pleasant at a feast in filling the said vessel with wine , which will spout it out as though it were from a boyling fountaine , in the forme of a threed very pleasant to behold . problem . cxviii . how to practise excellently the reanimation of simples , in case the plants may not be transported to be replanted by reason of distance of places . take what simple you please , burne it and take the ashes of it , and let it be calcinated two houres between two creusets wel luted , and extract the salt : that is , to put water into it in moving of it ; then let it settle : and do it two or three times , afterwards evaporate it , that is , let the water be boyled in some vessel , untill it be all consumed : then there will remaine a salt at the bottome , which you shall afterwards sowe in good ground wel prepared : such as the theatre of husbandry sheweth , and you shall have your desire . problem . cviii . how to make an infalliable perpetuall motion . m●xe 5. or 6. ounces of ☿ with is equall weight of ♃ , grinde it together with 10. or 12 ounces of sublimate dissolved in a celler upon a marble the space of foure dayes , and it will become like oile , olive , which distill with fire of chaffe or driving fire , and it will sublime dry substance , then put water upon the earth ( in forme of lye ) which will be at the bottom of the limbeck , and dissolve that which you can ; filter it , then distill it , and there will be produced very subtill antomes , which put into a bottle close stopped , and keep it dry , and you shall have your desire , with astonishment to all the world , and especially to those which have travelled herein without fruit . problem . cxix . of the admirable invention of making the philosophers tree , which one may see with his eye to grow by little and little . take two ounces of aqua fortis , and dissolve in it halfe an ounce of fine silver refined in a cappell : then take an ounce of aqua fortis , and two drams of quick-silver : which put in it , and mixe these two dissolved things together , then cast it into a viall of halfe a pound of water , which may be well stopped ; for then every day you may see it grow both in the tree and in the branch . this liquid serves to black haire which is red , or white , without fading untill they fall , but here is to be noted that great care ought to be had in anointing the haire , for feare of touching the flesh : for this composition is very corrosive or searching , that as soone as it toucheth the flesh it raiseth blisters , and bladders very painfull . problem . cxx . how to make the representation of the great world ? draw salt niter out of salt earth ▪ which is found along the rivers side , and at the foot of mountaines , where especially are minerals of gold and silver : mix that niter well cleansed with ♃ , then calcinate it hermetically ▪ then put it in a limbeck and let the receiver be of glasse , well luted , and alwayes in which let there be placed leaves of gold at the bottome , then put fire under the limbeck untill vapours arise which will cleave unto the gold ; augment your fire untill there ascend no more , then take away your receiver , and close it hermetically , and make a lampe fire under it untill you may see presented in it that which nature affords us : as flowers , trees , fruits , fountaines , sunne , moone , starres , &c. behold here the forme of the limbeck , and the receiver : a represents the limbeck , b stands for the receiver . problem . cxxi . how to make a cone , or a pyramidall body move upon a table without springs or other artificiall meanes : so that it shall move by the edge of the table without falling ? this proposition is not so thornie and subtile as it seemes to be , for putting under a cone of paper a beetle or such like creature , you shall have pleasure with astonishment & admiration to those which are ignorant in the cause : for this animall will strive alwayes to free herself from the captivity in which she is in by the imprisonment of the cone : for comming neere the edge of the table she will returne to the other side for feare of falling . problem cxxii . to cleave an anvill with the blow of a pistoll . this is proper to a warrier , and to performe it , let the anvill be heated red hot as one can possible , in such sort that all the solidity of the body be softned by the fire : then charge the pistoll with a bullet of silver , and so have you infallibly the experiment . problem . cxxiii . how to r●st a capon carried in a budget at a saddle-bowe , in the space of riding 5 or 6 miles ? having made it ready and larded it , stuffe ●t with butter ; then heat a piece of steele which may be formed round according to the length of the capon , and big enough to fill the belly of it , and then stop it with butter ; then wrap it up well and inclose it in a box in the budget , and you shall have your desire : it is said that count mansfield served himse●fe with no others , but such as were made ready in this kinde , for that it loseth none of its substance , and it is dressed very equally . problem . cxxiv . how to make a candle burne and continue three times as long as otherwise it would ? vnto the end of a candle half●burned stick a farthing lesse or more , to make it hang perpendicular in a vessel of water , so that it swimme above the water ; then light it , and it will susteine it self & float in this manner ; and being placed into a fountaine , pond , or lake that runnes slowly , where many people assemble , it will cause an extreme feare to those which come therein in the night , knowing not what it is . problem . cxxv . how out of a quantitie of wine to extract that which is most windy , and evill , that it hurt not a sick person ? take two vials in such sort that they be of like greatnesse both in th● belly and the neck ; fill one of them of wine , and the other of water : let the mouth of that which hath the water be placed into the mouth of that which hath the wine , so the water shall be uppermost , now because the water is heavier than the wine , it will descend into the other viall , and the wine which is lowest , because it is highest will ascend above to supply the place of the water , and so there will be a mutuall interchange of liquids , and by this penetration the wine wil lose her vapors in passing through the water . problem cxxvi . how to make two marmouzets , one of which shall light a candle , and the other put it out ? upon the side of a wall make the figure of a marmouzet or other animall or forme , and right against it on the other wall make another ; in the mouth of each put a pipe or quill so artificially that it be not perceived ; in one of which place salt peter very fine , and dry and pulverised ; and at the end set a little match of paper , in the other place sulphur beaten smal , then holding a candle lighted in your hand , say to one of these images by way of command , blow out the candle ; then lighting the paper with the candle , the salt-peter wil blow out the candle immediatly , and going to the other image ( before the match of the candle be out ) touch the sulphur with it and say , light the candle , & it will immediatly be lighted , which will cause an admiration to those which see the action , if it be wel done vvith a secret dexterity . problem . xxvii . how to keepe wine fresh as if it were in a celler though it were in the heat of summer , and without ice or snow , yea though it were carried at a saddles bow , and exposed to the sun all the day ? set your wine in a viall of glasse ; and place it in a box made of wood , leather , or such like : about which vial place salt-peeter , and it will preserve it and keep it very fresh : this experiment is not a little commodious for those which are not neare fresh waters , and whose dwellings are much exposed to the sunne . puoblem . cxxviii . to make a cement which indureth or lasteth as marble , which resisteth aire and water without ever disjoyning or uncementing ? take a quantity of strong and gluing morter vvell beaten , mixe vvith this as much nevv slaked lime , and upon it cast oile of olive or linseed-oile , and it vvill become hard as marble being applyed in time . problem . cxxix . how to melt metall very quickly , yea in a shell upon a little fire . make a bed upon a bed of metall with pouder of sulphur , of salt-peeter , and saw-dust alike ; then put fire to the said pouder with a burning charcole , and you shall see that the metall will dissolve incontinent and be in a masse . this secret is most excellent , and hath been practised by the reverend father mercen●● of the order of the minims . problem . cxxx . how to make iron or steele exceeding hard ? qvench your blade or other instrument seven times in the blood of a male hog mixt with goose-grease , and at each time dry it at the fire before you wet it : and it will become exceeding hard , and not brittle , which is not ordinary according to other temperings and quenchings of iron : an experiment of small cost , often proved , and of great consequence for armorie in warlike negotiations . prbolem cxxxi . to preserve fire as long as you will , imitating the inextinguible fire of vestales . after that you have extracted the burning spirit of the salt of ♃ , by the degrees of fire , as is required according to the art of chymistrie , the fire being kindled of it selfe , break the limbeck , and the irons which are found at the bottome will flame and appeare as burning coles as soone as they feele the aire ; which if you promptly inclose in a viall of glasse , and that you stop it exactly with some good lute : or to be more assured it may be closed up with hermes wax for feare that the aire get not in . then will it keep more than a thousand yeares ( as a man may say ) yea at the bottome of the sea ; and opening it at the end of the time , as soone as it feeles the aire 〈◊〉 takes fi●e ▪ with which you may light a match . this secret merits to be travelled after and put in practice , for that it is not common , and full of astonishment , seeing that all kinde of fire lasteth but as long as his matter lasteth , and that there is no matter to be found that will so long in●●●e . artificiall fire-workes : or the manner of making of rockets and balls of fire , as well for the water , as for the aire ; with the composition of starres , golden-rain , serpen●s , lances , whee●s of fire and such like , pleasant and recreative . of the composition for rockets . in the making of rockets , the chiefest thing to be regarded is the composition that they ought to be filled with ; forasmuch as that which is proper to rockets which are of a lesse sort is very improper to those which are of a more greater forme ; for the fire being lighted in a great concave , which is filled with a quick composition , burnes with great violence ; contrarily , a weak composition being in a small concave , makes no effect : therefore we shall here deliver in the first place rules and directions , which may serve for the true composition , or matter with which you may charge any rocket , from rockets which are charged but with one ounce of powder unto great rockets which requireth for their charge 10 pound of powder , as followeth . for rockets of one ounce . vnto each pound of good musket powder smal beaten , put two ounces of smal cole dust , and with this composition charge the rocket . for rockets of 2 or 3 ounces . vnto every foure ounces and a halfe of powder dust , adde an ounce of salt-peter , or to every 4 ounces of powder dust , adde an ounce of cole dust . for rockets of 4 ounces . vnto every pound of powder dust adde 4 ounces of salt peter & one ounce of cole dust : but to have it more slow , unto every 10. ounces of good dust powder adde 3 ounces of salt-peter , and 3 ounces of cole dust . for rockets of 5 or 6 ounces . vnto every pound of powder dust , adde 3 ounces and a halfe of salt peter , and 2 ounces and a halfe of coledust , as also an ounce of sulphur , and an ounce of fyle dust . for rockets of 7 or 8 ounces . vnto every pound of powder dust adde 4 ounces of salt peter , and 3 ounces of sulphur . of rockets of 10 or 12 ounces . vnto the precedent composition adde halfe an ounce of sulphur , and it will be sufficient . for rockets of 14 or 15 ounces . vnto every pound of powder dust adde 4 ounces of salt peter , or cole dust 2 ¼ ounces of sulphur and file dust of 1 ¼ ounce . for rockets of 1 , pound . vnto every pound of powder dust adde 3 ounces of cole dust , and one ounce of sulphur . of rockets of 2 , pound . vnto every pound of powder dust adde 9 ½ ounces of salt peter , of cole dust 2 1 / ● ounces , filedust 1 ● / 2 ounce , and of sulphur ¾ of ounce . for rockets of 3 , pound . vnto every pound of salt peter adde 6 ounces of cole dust , and of sulpher 4 , ounces . for rockets of 4 , 5 , 6 , or 7 , pound . vnto every pound of salt peter adde 5 ounces of cole dust , and 2 ½ ounces of sulphur . for rockets of 8 , 9 , or 10 pound . vnto every pound of salt peter , adde 5 ½ ounces of cole dust , and of sulphur 2 ½ ounces . here note that in all great rockets , there is no powder put , because of the greatnesse of the fire which is lighted at once , which causeth too great a violence , therefore ought to be filled with a more weaker composition . of the making of rockets and other fireworkes . for the making of rockets of sundry kindes , divers moulds are to be made , with their rolling pins , breaths , chargers , &c. as may be seen here in the figure . and having rolled a case of paper upon the rolling pin for your mould , fill it with the composition belonging to that mould as before is delivered : now may you load it on the top , with serpents , reports , stars , or golden raine : the serpents are made about the bignesse of ones little finger , by rolling a little paper upon a small stick , and then tying one end of it , and filling it with the mixt composition somewhat close , and then tying the other end . the reports are made in their paper-cases as the serpents , but the paper somewhat thicker to give the greater report . these are filled with graine-powder or halfe powder and halfe composition , and tying both ends close , they are finished . the best kinde of starres are made with this mixture following ; unto every 4 ounces of salt-peter , adde 2 ounces of sulphur , and to it put 1. ounce of powder-dust , and of this composition make your starres , by putting a little of it within a small quantity of towe ; and then tying it up in the form of a ball as great as an hasel-nut or a little wal-nut , through which there must be drawne a little primer to make it take fire . touching the making of the golden raine , that is nothing but filling of quilles with the composition of your rockets somewhat hard . now if the head of a rocket be loaded with a thousand of those quilles , it s a goodly sight to see how pleasantly they ●pread themselves in the aire , and come downe like streames of gold much like the falling downe of snow being agitated by some turbulent winde . of recreative fires . 1 phil●strates saith , that if wine in a platter be placed upon a receiver of burning coles , to exhale the spirit of it , and be inclosed within a cupboard or such like place , so that the aire may not go in , nor out , and so being shut up for 30 yeares , he that shall open it , having a wax candle lighted , and shall put it into the cubboard there will appeare unto him the figure of many cleare starres . 2 if aquavitae have camphire dissolved in it ; and be evaporated in a close chamber , where there is but a charcole fire , the first that enters into the chamber with a candle lighted , will be extremely astonished , for all the chamber will seeme to be full of fire very subtile , but it will be of little continuance . 3 candles which are deceitful are made of halfe powder , covered over with tallow , and the other halfe is made of cleane tallow , or waxe , with an ordinary week ; this candle being lighted , and the upper halfe consumed , the powder will take fire , not without great noise and astonishment to those which are ignorant of the cause . 4 a dozen or twenty smal serpents placed secretly under a candlestick that is indifferent big , which may have a hole passe through the socket of it to the candle , through which a piece of primer may be placed , and setting a smal c●ndle in the socket to burne according to a time limited : which candlestick may be set on a side table without suspition to any ; then when the candle is burned , that it fires the primer , that immediately will fire all the serpents , which overthrowing the candlestick will flye here and there , intermixing themselves , sometimes in the aire , sometimes in the planching , one amongst another , like the crawling of serpents , continuing for a pretty while in this posture , and in extinguishing every one will give his report like a pistoll ; this will not a little astonish some , thinking the house will be fired , though the whole powder together makes not an ounce , and hath no strength to do such an effect . how to make fire run up and downe , forward and backward take small rockets , and place the taile of one to the head of the other upon a cord according to your fancie , as admit the cord to be abcdefg . give fire to the rocket at a , which will flye to b , which will come back againe to a , and fire another at c , that will flie at d , which will fire another there , and fl●e to e , and that to f , and so from f , to g , and at g , may be placed a pot of fire , viz. gh . which fired will make good sport ▪ bec●u●e the serpents which are in it will variously ●ntermix themselves in the aire , and upon the ground , and every one will extinguish with a report : and here may you note that upon the rockets may be placed fierie dragons , combatants , or such like to meet one another , having lights placed in the concavity of their bodies which will give great grace to the action . how to make wheels of fire . take a hoop , and place two lath● acrosse one the other ; upon the crossing of which make a hole , so that it may be placed upon a pin to turne easily , as the figure q. sheweth upon the sides of which hoope or round circle place your rockets , to which you may place lances of fire between each rocket : let this wheele be placed upon a standard as here is represented , and place a piece of primer from one lance to another , then give fire at g , which will fire f , that b , that will fire d , that c , and that will fire the rocket at a ▪ then immediatly the wheel will begin to move , and represent unto the spectators a circle of changeable fire , and if pots of fire be tied to it , you will have fine sport in the turning of the wheele and casting out of the serpents . of night-combatants . clubbes , targets , faulchons , and maces charged with severall fires , do make your night-combatants , or are used to make place amongst a throng of people . the clubbes at the ends are made like a round panier with small sticks , filled with little rockets in a spirall forme glu●d and so placed that they fire but one after another ; the ma●es are of divers fashions , some made oblong at the end , some made of a sp●rall forme , but all made hollow to put in several composition , and are boared in divers places , which are for sundry rockets , and lances of weak composition to be fired at pleasure : the faulchons are made of wood in a bowing forme like the figure a , having their backes large to receive many rockets , the head of one neare the neck of another , glued and fastned well together , so that one being spent another may be fired . 〈◊〉 targets are made of wooden thinne boards , which are channeled in spiral lines to containe primer to fire the rockets one after another , which is all covered with thinne covering of wood , or pastboard , boared with holes spirally also ; which rockets must be glued and made fast to the place of the channels : now if two men , the one having a target in his hand , and the other a falchon , or mace of fire , shall begin to fight , it will appeare very pleasant to the spectators : for by the motion of fighting , the place will seem to be ful of streames of fire : and there may be adjoyned to each target a sunne or a burning comet with lances of fire , which will make them more beautifull and resplendent in that acti●n . of standing fires . svch as are used for recreation , are collossus , statues , arches , pyramides , chariots , chaires of triumph and such like , which may be accommodated with rockets of fire , and beautified with sundry other artificiall fires , as pots of fire for the aire which may cast forth several figures , scutchions , rockets of divers sorts , starres , crownes , leaters , and such like , the borders of which may be armed with sundry lances of fire , of small flying rockets with reports , flames , of small birds of cypres , lan●hornes of fire , candles of divers uses , and colours in burning : and whatsoever the fancie of an ingenious head may allude unto . of pots of fire for the aire , which are throwne out of one case one after another of a long continuance . make a long trunk as ag , and by the side ah , let there be a channel which may be fiered with slow primer or composition ; then having charged the trunk ag , with the pots of fire for the aire at igec , and make the trunk ag , very fast unto a post as ik , give fire at the top as at a , which burning downewards will give fire to c , and so throw out that pot in the aire , vvhich being spent , in the meane time the fire vvil-burne from b to d , and so fire e , and throvv it out also into the ayre , and so all the rest one after another vvill be throvvne out : and if the pots of fire for the aire vvhich are cast out , be filled vvith diverse fire-vvorkes , they vvill be so much the more pleasant to the beholders . these trunks of fire doe greatly adorne a firevvorke , and may conveniently be placed at each angle of the vvhole vvorke . of pots of fire for the ground . many pots of fire being fired together do give a fine representation , and recreation to the spectators , and cause a vvonderfull shout amongst the common people vv ch are standers by ; for those pots being filled vvith balles of fire and flying serpents for the aire , they vvill so intermix one vvithin another , in flying here and there a little above the ground , and giving such a volley of reports that the aire vvill rebound vvith their noise , and the vvhole place be filled vvith sundry streames of pleasant fire ; which serpents will much occupie these about the place to defend themselves in their upper parts , when they will no lesse be busied by the balls of fire , which seeme to annoy their feet . of balles of f●re . these are very various according to a mans fancy ; some of which are made with very small rockets , the head of one tyed to the neck of another : the ball being made may be covered over with pitch except the hole to give fire to it ; this ball will make fine sport amongst the standers by , which will take all a fire , and rolle sometimes this way , sometimes that way , between the legs of those that are standers by ▪ if they take not heed , for the motion will be very irregular , and in the motion will cast forth severall fires with reports . in the second kind there may be a channell of iron placed in divers places in spirall manner , against which may be placed as many small petards of paper as possible may be , the channell must be full of slow comp●sition , and may be covered a● the former , and made fit with his rockets in the middle : this ball may be shot out of a morter peece , or charged on the top of a roc●et : for in its motion it will flye here and there , and give many reports in the aire : because of the discharge of the petards . of fire upon the water . places which are 〈◊〉 upon rivers or great ponds , are proper to make recreative fres on : and if it be required to make some of consequence , such may conveniently be made upon two bo●ts , upon which may be built two beasts , turrets , pagins , castles , or such like , to receive or hold the diversity of fire workes that may be made within it , in which may play 〈◊〉 fires , petards , &c. and cast out many simple granadoes , balls of fire to burne in the water-serpents and other things , and often times these boates in their incounters may hang one in another , that so the combatants with the targets , and maces may fight ; which will give great ▪ content to the eyes of those which are lookers on , and in the conclusion fire one another , ( for which end they were made : ) by which the dexterity of the one may be knowne in respect of the other , and the triumph and victory of the fight gotten . of balles of fire which move upon the water . these may be made in forme of a ball stuffed with other little balls , glued round about and filled with composition for the water , which fiered , will produce marvellous and admirable effects , for which there must be had little cannons of white iron , as the ends of small funnels ; these iron cannons may be pierced in sundry places , to which holes , may be set small balles ful of composition for the water which small balls must be peirced deep and large , and covered with pitch , except the hole : in which hole must be first placed a little quantitie of grain-powder ; and the rest of the hole filled up with composition ; and note further that these iron cannons , must be filled with a slow composition ; but such which is proper to burne in the water : then must these cannons with their small balls be put so together that it may make a globe , and the holes in the cannons be answerable to the hollow balls , and all covered over with pitch and tallow ; afterwards pierce this ball against the greatest cannon ( to which all the lesser should answer ) unto the composition , then fire it , and when it begins to blow , throw it into the water , so the fire comming to the holes will fire the graine powder , the which will cause the balls to separate and fly here and there , sometimes two at a time , sometimes three , sometimes more , which will burne within the water with great astonishment and content to those which see it . of lances of fire . standing lances of fire , are made commonly with hollow wood , to containe sundry petards , or rockets , as the figure here sheweth , by which is easie to invent others occording to ones fancy . these lances have wooden handles , that so they may be fastned at some post , so that they be not overthrowne in the flying out of the rockets or petards : there are lesser sorts of lances whose cases are of three or foure fouldings of paper of a foote long , and about the bignesse of ones finger , which are filled with a composition for lances . but if these lances be filled with a composition , then ( unto every 4 ouncs of powder add● 2 ounces of salt-peter , and unto that adde 1 ounce of sulphur ) it will make a brick fire red before it be halfe spent , if the lance be fiered and held to it : and if 20 such lances were placed about a great rocket and shot to a house or ship , it would produce a mischievous effect . how to shoot a rocket horizontall , or otherwise . vnto the end of the rocket place an arrow which may not be too heavy , but in stead of the feathers let that be of thinne white tinne plate , and place it upon a rest , as here you may see by the figure , then give fire unto it , and you may see how serviceable it may be . to the head of such rockets , may be placed petards , balls of fire , granadoes , &c. and so may be applyed to warlike affaires . how a rocket burning in the water for a certaine time , at last shall fly up in the aire with an exceeding quickness . to do this , take two rockets , the one equall to the other , and joyne them one unto another in the middle at c. in such sort that the fire may easily passe from one to another : it being thus done , tye the two rockets at a stick in d , and let it be so long and great that it may make the rockets in the water hang , or lye upright : then take a pack-thread and tye it at g. and let it come double about the stick dm . at 〈◊〉 and at that point hang a bullet of some weight as k. for then giving fire at a. it will burne to b. by a small serpent filled there and tyed at the end , and covered so that the water injure it hot , which will fire the rocket bd , and so mounting quick out of the water by the loose tying at c. and the bullet at the pack-thread , will leave the other rocket in the vvater : and so ascend like a rocket in the aire , to the admiration of such as knovv not the secrecie . of the framing of the parts of a fire-worke , together , that the severall workes may fire one after another . cause a frame to be made as abcd. of tvvo foot square every vvay , or thereabouts ( according to the quantity of your severall vvorkes ) then may you at each angle have a great lance of fire to stand , vvhich may cast out pots of fire as they consume : upon the ledges ab.bc. and cd . may be placed small lances of fire about the number of 30 or 60 , some sidevvise , and others upright , betvveen these lances may be placed pots of fire sloping outvvards , but made very fast , and covered very close , that they chance not to fire before they should ; then upon the ledges re. fg.hi . and ad may be placed your soucisons , and behinde all the vvork may be set your boxes of rockets , in each of vvhich you may place 6 , 9 , ●2 . or 20 small rockets : novv give fire at a. ( by help of a piece of primer going from one lance to another ) all the lances vvill instantly at once be lighted , and as soone as the lance at a is consumed , it vvill fire the channell vvhich is made in the ledge of the frame vvhich runnes under the pots of fire , and as the fire goes along burning , the pots vvill be cast forth , and so the rank of pots upon the sides of the frame ab.bc. and cd . being spent , the soucisons vvill begin to play being fiered also by a channel vvhich runnes under them , upon the ledges ad , hi●g , and re. then when the soucisons are spent upon the last ledge re. there may be a secret channel in the ledge cd which may fire the box of rockets at k. and may fire all the rest one after another , which boxes may be all charged with severall fire-workes : for the rockets of the first box may be loaden with serpents , the second with stars , the third with reports , the fourth with golden raine , and the fifth with small flying serpents ; these mounting one after another and flying to and fro will much inlighten the aire in their ascending , but when these rockets discharge themselves above , then will there be a most pleasant representation , for these fires will dilate themselves in divers beautifull formes , some like the branching of trees , others like fountaines of water gliding in the aire , others like flashes of lightning , others like the glittering of starres , giving great contentment , and delight to those which behold them ; but if the worke be furnished also with balons ( which is the chiefest in recreative fire-works ) then shall you see ascending in the aire but as it were onely a quill of fire , but once the balon taking fire , the aire will seeme more than 100. foot square full of crawling , and flying serpents , which will extinguish with a volley of more than 500 reports : and so fill the aire and firmament with their rebounding clamour . the making of which with many other rare and excellent fire-workes , and other practises , not onely for recreation , but also for service : you may finde in a book intituled artificiall fire-workes , made by mr. malthas ( a master of his knowledge ) and are to be sold by vvilliam leake , at the crowne in fleet-street , between the two temple-gates . conclusion . in this booke we have nothing omitted what was materiall in the originall , but have abundantly augmented it in sundry experiments : and though the examinations are not so full , and manifold ; yet ( by way of brevitie ) we have expressed fully their substance , to avoid prolixitie , and so past by things reiterated . finis . printed or sold by william leak , at the crovvne in fleetstreet neere the temple , these books following . york's heraldry , folio a bible of a very fair large roman letter , 4● orlando f●rios● folio . callu learned readings on the scat. 21. hen. 80. cap 5 of sewer● perkins on the laws of england . wi●kinsons office of she●●●fs . vade mecum , of a justice of peace . the book of fees. peasons law. mirrour of just●ce . topicks in the laws of england . sken de significatione verborum . delaman's use of the horizontal quadrant . wilby's 2d set of musique , 345 and 6 parts . corderius in english. d●ctor fulk's meteors . malthus fire-workes . nyes gunnery & fire-workes c●to ma●or with annotations , by wil. austin esquire . mel helliconium , by alex. ross● nosce teipsum , by sr john davis animadversions on lil●i●s grammer . the history of vienna , & paris lazarillo de tormes . hero and l●ander , by g. chapman and christoph. marlow . al●ilia or philotas loving folly . bishop andrews sermons . adams on ●eter . posing of the accidence . am●dis de gaule . guillieliam's heraldry . herberts travels . bacc●s tales . man become guilty , by john francis sen●●t , and englished by henry earl of monmouth . the ideot in 4 books ; the first and second of wisdom ; the third of the mind , the fourth of s●●tick experiments of the ballance . the life and reign of hen. the eighth , written by the l. herbet cornwallis essays , & paradoxes . clenards greek g●ammar 80 a●laluci● , or the house of light : a discourse written in the year 1651 , by sn . a modern speculator . a tragedy written by the most learned hugo grotius called , christus patience , and translated into engl. by george sand ▪ the mount of olives : or sollitary devotions , by henry vaughan silurist vvith an excellent discourse of man in glory , written by the reverend anselm arch bishop of canterbury . the fort royall of holy scriptures by i. h. playes . hen. the fourth . philaster . the wedding . the hollander . maids tragedie . king & no k. the gratefull servant . the strange discovery . othello ; the moor of venice . the merchant of venice . the description and use of the dovble horizontall dyall . whereby not onely the houre of the day is shewn ; but also the meridian line is found : and most astronomicall questions , which may be done by the globe : are resolved . invented and written by w. o. whereunto is added , the description of the generall horologicall ring . london , printed for william leake , and are to be sold at his shop at the signe of the crown in fleetstreet , between the two temple gates . 1652. the description , and use of the double horizontall diall . there are upon the plate two severall dyals . that which is outermost , is an ordinary diall , divided into houres and quarters , and every quarter into three parts which are five minutes a piece : so that the whole houre is understood to contein 60 minutes . and for this dyall the shadow of the upper oblique , or slanting edge of the style , or cocke , doth serve . the other diall , which is within , is the projection of the upper hemisphaere , upon the plain of the horizon : the horizon it self is understood to be the innermost circle of the limbe : and is divided on both sides from the points of east and west into degrees , noted with 10.20.30 , &c. as far as need requireth : and the center of the instrument is the zenith , or verticall point . within the horizon the middle straight line pointing north and south upon which the style standeth , is the meridian or twelve a clock line : and the other short arching lines on both sides of it , are the houre lines , distinguished accordingly by their figures : and are divided into quarters by the smaller lines drawn between them : every quarter conteining 15 minutes . the two arches which crosse the houre lines , meeting on both sides in the points of intersection of the sixe a clocke lines with the horizon , are the two semicircles of the ecliptick or annuall circle of the sun : the upper of which arches serveth for the summer halfe yeere ; and the lower for the winter half yeer : and therefore divided into 365 dayes : which are also distinguished into twelve moneths with longer lines , having their names set down : and into tenths and fifts with shorter lines : and the rest of the dayes with pricks as may plainly be seene in the diall . and this is for the ready finding out of the place of the sun every day : and also for the shewing of the suns yeerely motion , because by this motion the sun goeth round about the heavens in the compasse of a yeer , making the four parts , or seasons thereof ▪ namely , the spring in that quarter of the ecliptick which begins at the intersection on the east side of the diall ▪ and is therefore called the vernall intersection . then the summer in that quarter of the ecliptick which begin at the intersection with the meridian in the highest point next the zenith . after that , autumne in that quarter of the ecliptick which beginneth at the intersection on the west side of the diall , and is therefore called the au●umna●l intersection and lastly , the winter in that quarter of the ecliptic● , which beginneth at the intersection , with the meridian i● the lowest point next the horizon . but desides this yeerely motte● , the sun hath a diurnall , or daily motion , whereby it maketh day and night , with all the diversities and inaequalities thereof : which is expressed by those other circles drawn crosse the houre lines ; the middlemost whereof , being grosser then the rest , meeting with the ec●iptick in the points of the vernall , and autumnall intersections ▪ is the equinoctiall : and the rest on both sides of it are called the parallels , or diurnall arch of the sun , the two outermost whereof are the tropicks , because in them the sun hath his furthe●t digression or declination from the aequinoctiall , which is degrees 23 1 / ● ▪ and thence beginneth againe to return towards the equinoctiall . the upper of the two tropicks in this nor northerne hemisphere is the trop●ck of cancer , and the sun being in it , is highest into the north , making the longest day of summer : and the lower next the horizon is the tropick of capricorne ; and the sun being in it , is lowest into the south , making the shortest day of winter . between the two tropicks and the aequinoctiall , infinite such parallel circles are understood to be conteined : for the sun , in what point soever of the ecliptick it is carried ▪ describeth by his lation a circle parallel to the aequinoctiall : yet those parallels which are in the instrument , though drawn but to every second degree of declination , may be sufficient to direct the eye in imagining and tracing out through every day of the whole yeere in the ecliptick , a proper circle which may be the diurnall arch of the sun for that day . for upon the right estimation of that imaginary parallel doth the manifold use of this instument especially rely : because the true place of the sun all that day is in some part or point of that circle . wherefore for the bet●er conceiving and bearing in minde thereof , every fift parallel is herein made a little g●osser then the rest . for this inner diall serveth the shadow of the upright edge of the style ; which i therefore call the upright shadow . and thus by the eye and view onely to behold and comprehend the course of the sun ▪ throughout the whole yeere both for his annuall and diurnall motion , may be the first use of this instrument . ii use. to finde the declination of the sun every day . looke the day of the moneth proposed in the ecliptick , and mark how many degrees the prick shewing that day , is distant from the equinoctiall , either on the summer or winter side , viz. north or south . example 1. what will the declination of the sun be upon the eleven●h day of august ? look the eleventh day of august and you shall finde it in the sixth circle above the ●quinoctiall : now because each parallel standeth ( as hath been said before ) for two degrees , the sun shall that day decline northwards 12. degrees . example 2. what declination hath the sun upon the 24 day of march ? look the 24 day of march , and you shall finde it betweene the second and third northern parallels , as it were an half and one fift part of that di●tance from the second : reckon therefore four degrees for the two circles , and one de●ree for the halfe space : so shall the suns declination be five degrees , and about one fift part of a degree northward that same day . example 3. what declination hath the sun upon the 13 day of november ? look the 13 day of november , and you shall finde it below the equinoctiall ten parallels , and about one quarter which is 20 degrees and an halfe southward . so much is the declination . and according to these examples judge of all the rest . iii. use. to finde the diurnall arch , or circle of the suns course every day . the sun every day by his motion ( as hath been said ) describeth a circle parallel to the ●quinoctiall , which is either one of the circles in the diall , or some-where ●etween two of them . first , theref●re se●k the day of the moneth ; and if it fall upon one of those parallels ; that is the circle of the suns course that same day : but if it fall betweene any two of the parallels , imagine in your mind● , and estimate with your eye , another parallel th●ough that point betweene those two parallels keeping still the same distance from each of them . as in the first of the three former examples , the circle of the suns course upon 11 of august ▪ shal be the very sixt circle above the equinoctiall toward the cente● . in ●xample 2. the circle of the suns cou●se upon the 24 of march shall be an imaginary circle between the second and third parallels still keeping an half of that space , and one fifth part more of the rest , from the second . in example 3. the circle of the suns course upon the 13 of november : shall be an imaginary circle between the tenth and eleventh parallels below the equinoctiall , still keeping one quarter of that space from the tenth . iiii use. to finde the r●sing and setting of the sun eve●yday . 〈…〉 ( as was last shewed ) the imaginary circle or parallel of the suns course for that day , and marke the point where it meeteth with the horizon , both on the east and w●st sides , for that is the very point of the suns r●sing , and setting that same day , and the houre lines which are on both sides of it , by proportioning the distance reasonably , according to 15 minutes for the quarter of the houre , will shew the houre of the suns rising on the east side , and the suns setting on the west side . v use. to know the reason and manner of the increasing and decreasing of the nights●hroughout ●hroughout the whole yeere . when the sun is in the equinoctiall , it riseth and setteth at 6 a clock , for in the instrument the intersection of the equinoctiall , and the ecliptick with the horizon is in the six a clocke circle on both sides . but if the sun be out of the equinoctial , declining toward the north , the intersections of the parallel of the sun with the horizon is before 6 in the morning , and after 6 in the evening : and the diurnall arch greater then 12 houres ; and so much more great , the greater the northerne declination is . againe , if the sun be declining toward the south , the intersections of the parallel of the sun , with the horizon is after 6 in the morning , and before 6 in the evening : and the diurnall arch lesser then 12 houres ; and by so much lesser , the greater the southerne declination is . and in those places of the ecliptick in which the sun most speedily changeth his declination , the length also of the day is most a●tered : and where the ecliptick goeth most parallel to the equinoctiall changing the declination , but little altered . as for example , when the sun is neer unto the equinoctiall on both sides , the dayes increase and also decrease suddenly and apace ; because in those places the ecliptick inclineth to the equinoctiall in a manner like a streight line , making sensible declination . again , when the sun is neere 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 ecliptick is in a manner parallel to the equinoctiall , the length o● the day also is but little , scarce altering the declination : and because in those two times of the yeer , the sun standeth as it were still at one declination , they are called the summer solstice , and winter solstice . and in the mean space the neerer every place is to the equinoctiall , the greater is the diversity of dayes . wherefore , we may hereby plainly see that the common received opinion , that in every moneth the dayes doe equally increase , is erroneous . also we 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 . vi. vse . to finde how far the sun riseth , and setteth from the true east and west points , which is called the suns amp●itude ortive , and occasive . seek out ( as was shewed in iii vse ) the imaginary circle , or parallel of the suns course , and the points of that circle in the horizon , on the east and west sides cutteth the degree of the amplitude ortive , and occasive . vii use. to finde the length of every day and night . double the houre of the sunnes setting , and you shal have the length of the day ; & double the hour of the sunnes rising , and you shal have the length of the right . viii vse . to finde the true place of the sun upon the dyall , that is , the point of the instrument which answereth to the place of the sun in the heavens at any time , which is the very ground of all the questions following . if the dyall be fixed upon a post : look what a clock it is by the outward dyall , that is , look what houre and part of houre the shadow of the slanting edge of the style sheweth in the outward limbe . then behold the shadow of the upright edge , and marke what point thereof is upon that very houre and part in the inner dyall among the parallels , that point is the true place of the sunne at the same instant . if the dyal be not fixed , and you have a meridian line no●ed in any window where the sunne shineth : place the meridian of your dyal upon the meridian line given , so that the top of the stile may point into the north : and so the dyal is as it were fixed , wherefore by the former rule you may finde the place of the sunne upon it . if the dyal be not fixed , neither you have a meridian line , but you know the true houre of the day exactly : hold the dyal even and parallel to the horizon , moving it till the slanting edge of the stile cast his shadow justly upon the time or houre given ; for then the dyal is truly placed , as upon a post . seek therefore what point of the upright shadow falleth upon that very houre , and there is the place of the sun. but if your dyal be loose , and you know neither the meridian nor the time of the day . first , by the day of the moneth in the ecliptique , finde the su●s parallel , or d●urnall arch for that day ▪ then holding the dyal level to the horizon , move it every way untill the slanting shadow of the style in the outward limbe , and the upright shadow in the sunnes diurnal arch , both shew the very same houre and minute , for that very point of the sunnes parallel , which the upright shadow cutteth , is the true place of the sun on the dyal at that present . but note that by reason of the thicknes of the style , and the bluntnesse of the angle of the upright edge , the sun cannot come unto that edge for some space before and after noone . and so during the time that the sunne shineth not on that upright edge , the place of the sunne in the dyal cannot be found . wherefore they that make this kinde of double dyal , are to be careful to file the upright edge of the style as thinne and sharpe as possible may be . that which hath here bin taught concerning the finding out the suns true place in the dyal , ought perfectly to be understood , that it may be readily , and dexteriously practised , for upon the true performance thereof dependeth all that followeth . ix vse . to finde the houre of the day . if the dyal be fastned upon a post , the houre by the outward dyal , or limbe , is known of every one , and the upri●ht shadow in the suns parallel , or diurnal arch will also shew the very same houre . but if the dyall be loose , either hold it or set it parallel to the horizon , with the style pointing into the north and move it gently every way untill the houre shewed in both dialls exactly agreeth , or which is all one , finde out the true place of the sun upon the dyall , as was taught in the former question , for that point among the houre lines sheweth the houre of the day . x vse . to finde out the meridian , and other points of the compasse . first , you must seek the tru● houre of the day ( by the last question ) for in that situation the meridian of the dyall standeth direct●y north and south : and the east pointeth into the east , and the west into the west , and the rest of the points may be given by allowing degrees 11. 1 / ● unto every point of the compasse . xi vse . to finde out the azumith of the sun , that is , the distance of the verticall circle , in which the sun is at that present , from the meridian . set your diall upon any plain or flat which is parallel to the horizon , with the meridian pointing directly north or south , as was last shewed : then follow with your eye the upright shadow in a streight line , till it cutteth the horizon : for the degree in which the point of intersection is , shal shew how far the suns azumith is distant from the east and west points , and the complement thereof unto 90 ; shal give the distance thereof from the meridian . xii vse . to finde out the declination of any wall upon which the sun shineth , that is , how far that wall swerveth from the north or south , either eastward or westward . take aboard having one streight edg ▪ & a line stricken perpendicular upon it ; apply the streight edg unto the wall at what time the sun shineth upon it , holding the board parallel to the horizon : set the dyal thereon , and move it gently every way , untill the same hour and minute be shewed in both dyals : and so let it stand : then if the dyal have one of the sides parallel to the meridian strike a line along that side upon the board , crossing the perpendicular , or else with a bodkin make a point upon the board , at each end of the meridian , and taking away the instrument from the board , and the board from the wall , lay a ruler to those two points , and draw a line crossing the perpendicular : for the angle which that line maketh with the perpendicular , is the angle of the decli●nation of the wall . and if it be a right angle , the wall is exactly east or west : but if that line be parallel to the perpendicular , the wall is direct north or south without any declination at all . you may also finde out the declination of a wall , if the dial be fixed on a post not very far from that wall ; in this manner . your board being applyed to the wall , as was shewed , hang up a thred with a plummet , so that the shadow of the thred may upon the board crosse the perpendicular line : make two pricks in the shadow and run instantly to the dyal and look the horizontal distance of the suns azumith , or upright shadow from the meridian . then through the two pricks draw a line crossing the perpendicular : and upon the point of the intersection , make a circle equal to the horizon of your instrument , in which circle you shal from the line through the two pricks measure the horizontal distance of the upright shadow , or azumith from the meridian , that way toward which the meridian is : draw a line out of the center , to the end of that arch measured : and the angle which this last line maketh with the perpendicular , shall be equall to the declination of the wall . xiii vse . how to place the dyall upon a post without any other direction but it selfe . set the diall upon the post , with the stile into the north , as neere as you can guesse : then move it this way and that way , till the same houre and minute be shewed , both in the outward and inward dials by the severall shadowes , as hath been already taught , for then the diall standeth in its truest situation ; wherefore let it be nailed down in that very place . xiiii vse . to finde the height of the sun at high noon everyday . seeke out the diurnall arch or parallel of the suns course for that day , ( by vse iii. ) and with a paire of compasses , setting one foot in the center , and the other in the point of intersection of that parallel with the meridian , apply that same distance unto the semidiameter divided : for that measure shal therein shew the degree of of the suns altitude above the the horizon that day at high noon . xv vse . to finde the height of the sun at any houre or time of the day . seeke out the diurnal arch , or parallel of the suns course for that day : and marke what point of it is in the very houre and minute proposed . and with a paire of compasses , setting one foot in the center , and the other in that point of the parallel , apply the same distance upon the semidiameter divided : for that measure shall shew the degree of the suns altitude above the horizon at that time . and by this meanes you may finde the height of the sun above the horizon at every houre throughout the whole yeere , for the making of rings and cylinders and other instruments which are used to shew the houre of the day . xvi vse . the height of the sun being given , to finde out the houre , or what it is a clocke . this is the converse of the former : seeke therefore in the semidiameter divided , the height of the sun given . and with a paire of compasses , setting one foot in the center , and the other at that height , apply the same distance unto the diurnall arch , or parallel of the sun for that day : for that point of the diurnall arch , upon which that same distance lights , is the true place of the sun upon the dial ; and sheweth among the houre lines , the true time of the day . xvii use. considerations for the use of the instrument in the night . in such questions as concerne the night ▪ or the time before sun rising , and after sun setting , the instrument representeth the lower hemisphaere wherein the southerne pole is elevated . and therefore the parallels which are above the aequinoctiall toward the center shall be for the southerne , or winter parallels : and those beneath the aequinoctiall , for the northerne or summer paral●els ; and the east shall be accounted for west , and the west for east ; altogether contrary to that which was before , when the instrument represented the upper hemisphaere . xviii use. to finde how many degrees the sun is under the horizon at any time of the night . seeke the declination of the sun for the day proposed ( by vse ii. ) and at the same declination the contrary side imagine a parallel for the sun that night ▪ and mark what point of it is in the very houre and minute proposed : and with a pair of compasses , setting one foot in the center , and the other in that point of the parallel , apply that same distance unto the semidiameter divided : for that measure shall shew the degree of the suns depression below the horizon at that time . xix use. to finde out the length of the c●epusculum , or twylight , every day . seek the declination of the sun for the day proposed ( by vse ii. ) and at the same declination on the contrary side imagine a parallel for the sun that night . and with a paire of compasses setting one foot in the center , and the other at 72 degrees upon the semidiameter divided , apply that same distance , unto the suns nocturnall parallel : for that point of the parallel , upon which that same distance shall light , sheweth among the houre lines , the beginning of the twilight in the morning , or the end of the twilight in the evening . xx use. if the day of the moneth be not known , to finde it out by the dyall . for the working of this question , either the diall must be fixed rightly on a post , or else you must have a true meridian line drawn in some window where the sun shineth , wherefore supposing the diall to be justly set either upon the post , or upon the meridian . look what a clock it is by the outward diall , and observe what point of the upright shadow falleth upon the very same minute in the inner diall , and through that same point imagine a parallel circle for the suns course ; that imaginary circle in the ecliptick shall cut the day of the moneth . i the description of it . this instrument serveth as a diall to finde the houre of the day , not in one place onely ( as the most part of dials do ) but generally in all countreys lying north of the aequinoctiall : and therefore i call it the generall h●rologicall ●ing . it consisteth of two br●zen circles : a diameter , and a little ring to hang it by . the two circles are so made , that though they are to be set at right angles , when you use the instrument : yet for more convenient carrying , they may be one folded into the other . the lesser of the two circles is for the aequinoctiall , having in the midst of the inner side or thicknesse , a line round it , which is the true aequinoctiall circle , divided into twice twelue hours , from the two opposite points in which it is fastened within the greater . the greater and outer of the two circles is the meridian : one quarter whereof , beginning at one of the points in which the aequin●cti●ll is hung , is divided into ninety degrees . the diameter is fastened to the meridian in two opposite points or poles , o●e of them being the very end of the quadrant , and is the north pole. wherefore it is perpendicular to the ●quinoctiall , having his due position . the diameter is broad , and slit in the middle : and about the slit on both sides are the moneths and dayes of the yeer . and within this slit is a litt●e sliding plate pierced through with a small hole : which hole in the motion of it , while it is applied to the dayes of the yeer , representeth the axis of the world . the little ring whereby the instrument hangeth , is made to slip up and down along the quadrant : that so by help of a little tooth annexed , the instrument may be rectified to any elevation of the pole. ii. the use of it . in using this instrument , first , the tooth of the little ring must carefully be set to the height of the pole in the quadrant , for the place wherein you are . secondly , the hole of the sliding plate within the slit , must be brought exactly unto the day of the moneth . thirdly , the aeqinoctiall is to be drawn out , and by means of the two studs in the meridian staying it , it is to be set perpendicular thereto . fourthly , guesse as neer as you can at the houre , and turn the hole of the little plate toward it . lastly , hold the instrument up by the little ring , that it may hang freely with the north pole thereof toward the north : and move it gently this way and that way , till the beams of the sun-shining thorow that hole , fall upon that middle line within the aequinoctiall : for there shall be the houre of the day : and the meridan of the instrument shall hang directly north and south . these instrument all dials are made in brasse by elias allen dwelling over against st. clements church without temple barre , at the signe of the horse-shooe neere essex gate . finis