The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath.
         Brown, John, philomath.
      
       
         
           1661
        
      
       Approx. 191 KB of XML-encoded text transcribed from 105 1-bit group-IV TIFF page images.
       
         Text Creation Partnership,
         Ann Arbor, MI ; Oxford (UK) :
         2004-08 (EEBO-TCP Phase 1).
         A29756
         Wing B5038
         ESTC R33265
         13117505
         ocm 13117505
         97766
         
           
            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. A29756)
         Transcribed from: (Early English Books Online ; image set 97766)
         Images scanned from microfilm: (Early English books, 1641-1700 ; 1545:10)
      
       
         
           
             The description and use of a joynt-rule fitted with lines for the finding the hour of the day and azimuth of the sun, to any particular latitude, or, to apply the same generally to any latitude : together with all the uses of Gunters quadrant applyed thereunto ... / contriv'd & written by J. Brown, philomath.
             Brown, John, philomath.
          
           [24], 168 p., [8] leaves of plates : ill., charts.
           
             Printed by T.J. for J. Brown and H. Sutton, and sold at their houses,
             London :
             1661.
          
           
             Woodcut illustration of man sighting with sextant: T.p. verso.
             Errata: p. 168.
             Imperfect: pages stained and tightly bound with slight loss of print.
             Reproduction of original in the British Library.
          
        
      
    
     
       
         Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.
         Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors.
      
       
         EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.
         EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).
         The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.
         Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.
         Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.
         Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.
         The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.
         Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).
         
          Keying and markup guidelines are available at the
           Text Creation Partnership web site
          .
        
      
       
         
         
      
    
     
       
         eng
      
       
         
           Quadrant.
           Dialing.
           Mathematical instruments.
        
      
    
     
        2004-01 TCP
        Assigned for keying and markup
      
        2004-02 Aptara
        Keyed and coded from ProQuest page images
      
        2004-04 Mona Logarbo
        Sampled and proofread
      
        2004-04 Mona Logarbo
        Text and markup reviewed and edited
      
        2004-07 pfs
        Batch review (QC) and XML conversion
      
    
  
   
     
       
         
         
           The
           Description
           and
           Use
           OF
           A
           JOYNT-RULE
           :
        
         
           Fitted
           with
           Lines
           for
           the
           finding
           the
           Hour
           of
           the
           Day
           ,
           and
           Azimuth
           of
           the
           Sun
           ,
           to
           any
           particular
           Latitude
           ;
           Or
           to
           apply
           the
           same
           generally
           to
           any
           latitude
        
         
           Together
           with
           all
           the
           uses
           of
           Gunters
           quadrant
           applyed
           thereunto
           ,
           as
           Sun-rising
           ,
           Declination
           ,
           Amplitude
           ,
           true
           place
           ,
           right
           Ascension
           ,
           and
           the
           hour
           of
           the
           Night
           by
           the
           Moon
           ,
           or
           fixed
           Stars
           ;
        
         
           A
           speedy
           and
           easie
           way
           of
           finding
           of
           Altitudes
           at
           one
           or
           two
           stations
           ;
           Also
           the
           way
           of
           making
           any
           kinde
           of
           erect
           Sun-Dial
           to
           any
           Latitude
           or
           Declination
           ,
           by
           the
           same
           Rule
           :
           With
           the
           Description
           and
           Use
           of
           several
           Lines
           for
           the
           mensuration
           of
           Superficies
           ,
           and
           Solids
           ,
           and
           of
           other
           Lines
           usually
           put
           on
           Carpenters
           Rules
           :
        
         
           Also
           the
           use
           of
           Mr.
           Whites
           Rule
           for
           measuring
           of
           Board
           and
           Timber
           ,
           round
           and
           square
           ;
           With
           the
           manner
           of
           using
           the
           Serpentine-line
           of
           Numbers
           ,
           Sines
           ,
           Tangents
           ,
           and
           Versed
           Sines
           .
        
         
           Contriv'd
           &
           written
           by
           
             J.
             Brown
          
           ,
           Philom
           .
        
         
           London
           ,
           Printed
           by
           
             T.
             I.
          
           for
           
             I.
             Brown
          
           ,
           and
           
             H.
             Sutton
          
           ,
           and
           sold
           at
           their
           houses
           in
           the
           Minories
           ,
           &
           Thredneedle-street
           .
           1661.
           
        
         
         
      
       
         
         
         
           TO
           THE
           READER
           .
        
         
           
             Courteous
             Reader
             ,
          
        
         
           AMong
           the
           multitude
           of
           Books
           which
           are
           printed
           and
           published
           ,
           in
           this
           scribling
           Age
           ,
           some
           serious
           ,
           some
           seditious
           ;
           some
           discovering
           or
           savouring
           of
           Art
           ,
           others
           of
           ignorance
           ,
           possibly
           every
           one
           endeavoring
           to
           bring
           their
           Male
           :
           Among
           the
           rest
           of
           the
           crowd
           ,
           I
           ,
           like
           the
           widow
           ,
           throw
           in
           my
           mite
           .
           If
           it
           be
           (
           or
           seem
           to
           be
           )
           little
           ,
           it
           .
           is
           like
           the
           Giver
           ,
           and
           therefore
           I
           presume
           will
           of
           some
           be
           accepted
           ,
           as
           little
           as
           it
           is
           ;
           and
           as
           little
           
           worth
           as
           it
           is
           ,
           it
           is
           like
           enough
           to
           be
           challenged
           :
           but
           I
           shall
           endeavor
           to
           prevent
           prejudice
           ,
           by
           the
           following
           Discourse
           .
        
         
           Having
           for
           some
           time
           been
           enquiring
           to
           find
           out
           a
           way
           ,
           whereby
           Work-men
           might
           on
           their
           Rules
           (
           their
           constant
           Companions
           )
           have
           a
           way
           easily
           and
           exactly
           to
           finde
           the
           hour
           of
           the
           Day
           ,
           and
           Suns
           Altitude
           and
           Azimuth
           ,
           and
           the
           like
           ;
           and
           have
           at
           several
           times
           for
           several
           men
           ,
           at
           their
           request
           ,
           used
           one
           and
           the
           other
           contrivance
           ,
           to
           finde
           the
           Hour
           :
           as
           that
           of
           the
           Cillender
           ,
           Quadrant
           ,
           or
           the
           like
           ,
           as
           the
           Altitude
           by
           a
           Tangent
           on
           the
           inside
           of
           a
           Square
           ,
           or
           Joynt-Rule
           ,
           and
           the
           line
           of
           Sines
           on
           the
           flat
           side
           ;
           but
           still
           one
           inconvenience
           or
           other
           of
           trouble
           in
           adding
           of
           complements
           ,
           
           or
           difficulty
           of
           taking
           of
           Aititudes
           ,
           or
           trouble
           to
           the
           memory
           ,
           did
           accrue
           to
           the
           work
           ;
           or
           else
           the
           Radius
           was
           small
           ,
           and
           so
           much
           the
           more
           short
           of
           exactness
           :
           at
           last
           there
           came
           to
           my
           sight
           a
           Quadrant
           made
           by
           Mr.
           Thomson
           ,
           and
           as
           I
           was
           informed
           ,
           was
           first
           drawn
           or
           contrived
           to
           that
           form
           ,
           by
           Mr
           
             Samuel
             Foster
          
           ,
           that
           ingenious
           ●rtist
           ,
           and
           laborious
           Student
           ,
           and
           Reader
           of
           the
           Mathematicks
           in
           Gresham
           Colledge
           :
           And
           considering
           of
           the
           ease
           and
           speed
           in
           the
           using
           thereof
           ,
           I
           set
           my self
           to
           the
           contriving
           thereof
           to
           a
           more
           portable
           form
           ,
           at
           last
           took
           some
           pains
           in
           delineating
           one
           ,
           and
           another
           in
           several
           forms
           ,
           and
           enquired
           after
           the
           uses
           thereof
           ,
           and
           in
           effect
           have
           done
           ,
           as
           Mr.
           Gunter
           with
           
           
             Stofflers
             Astrolabe
          
           ,
           and
           
             Nepeirs
             Logarithms
          
           ,
           and
           as
           Mr.
           Oughtred
           with
           
             Gunters
             Rule
          
           ,
           to
           a
           sliding
           and
           circular
           form
           ;
           and
           as
           my
           father
           
             Thomas
             Brown
          
           into
           a
           Serpentine
           form
           ;
           or
           as
           Mr.
           Windgate
           in
           his
           
             Rule
             of
             Froportion
          
           ,
           and
           as
           of
           late
           Mr.
           Collins
           with
           Mr.
           
             Gunters
             Sector
             on
             a
             Quadrant
          
           ,
           so
           may
           this
           not
           unfitly
           be
           called
           ,
           
             The
             Quadrant
             on
             a
             Sector
          
           .
           And
           in
           fine
           ,
           the
           Invention
           will
           be
           valued
           for
           the
           learned
           Authors
           sake
           ,
           and
           never
           a
           whit
           the
           worse
           for
           the
           new
           Contrivers
           sake
           :
           For
           first
           ,
           hereby
           it
           is
           made
           large
           in
           little
           room
           ,
           and
           as
           well
           on
           wood
           as
           on
           brass
           ,
           which
           is
           an
           incommunicable
           property
           to
           broad
           Quadrants
           ,
           though
           of
           never
           so
           good
           matter
           ,
           as
           experienced
           Workmen
           know
           right
           well
           ;
           and
           
           by
           a
           Tangent
           of
           30
           degrees
           laid
           together
           ,
           is
           gotten
           all
           kinde
           of
           Angles
           or
           Altitudes
           under
           90
           degrees
           ,
           and
           to
           be
           afforded
           for
           a
           low
           price
           ,
           in
           comparison
           of
           other
           Instruments
           ,
           which
           will
           not
           perform
           the
           same
           operations
           any
           better
           .
        
         
           Having
           made
           some
           Joynt-Rules
           in
           the
           manner
           following
           ,
           and
           exposing
           them
           to
           sale
           ,
           I
           have
           been
           many
           times
           solicited
           to
           write
           somewhat
           of
           the
           use
           ,
           and
           now
           at
           last
           after
           near
           a
           years
           suspence
           ,
           have
           committed
           the
           following
           Discourse
           to
           publique
           view
           ,
           partly
           to
           save
           the
           labor
           of
           tedious
           transcribings
           ,
           and
           also
           to
           make
           so
           useful
           ,
           cheap
           ,
           and
           exact
           an
           Instrument
           ,
           (
           if
           it
           be
           truly
           made
           )
           to
           be
           more
           known
           or
           occupied
           .
        
         
         
           In
           which
           business
           ,
           I
           desire
           to
           disclaim
           all
           vain-glorious
           os●entation
           ,
           and
           therefore
           have
           nakedly
           and
           plainly
           asserted
           the
           manner
           how
           ,
           and
           why
           it
           comes
           to
           be
           published
           to
           the
           world
           by
           me
           .
           It
           is
           a
           mechanical
           thing
           ,
           and
           mechanically
           applied
           ,
           and
           of
           mechanical
           men
           will
           be
           humanically
           accepted
           ,
           I
           doubt
           not
           .
           Having
           begun
           to
           write
           ,
           I
           could
           not
           break
           off
           so
           short
           and
           abruptly
           ,
           as
           at
           first
           I
           did
           intend
           to
           do
           ;
           therefore
           have
           added
           this
           short
           Discourse
           of
           ordinary
           Dyalling
           ,
           the
           exact
           Method
           of
           which
           I
           finde
           in
           no
           other
           Author
           that
           ever
           I
           met
           with
           ,
           (
           and
           indeed
           I
           have
           not
           time
           to
           read
           many
           )
           yet
           I
           dare
           presume
           ,
           that
           for
           speed
           ,
           ease
           ,
           convenience
           ,
           and
           exactness
           ,
           inferior
           to
           none
           ,
           especially
           the
           
           way
           of
           making
           far
           declining
           Dyals
           ;
           As
           for
           other
           declining
           reclining
           Dyals
           ,
           I
           referre
           you
           to
           other
           Authors
           ,
           or
           to
           a
           Discourse
           thereof
           by
           it self
           :
           if
           I
           finde
           encouragement
           ,
           and
           ability
           to
           perform
           the
           same
           ,
           a
           Copy
           whereof
           I
           have
           had
           a
           long
           time
           by
           me
           ,
           written
           by
           a
           very
           ingenious
           Artist
           )
           the
           demonstration
           of
           which
           Dyals
           is
           most
           excellently
           and
           easily
           shewed
           by
           the
           Figure
           inserted
           ,
           page
           77.
           
        
         
           As
           for
           the
           other
           part
           for
           taking
           of
           Altitudes
           and
           Angles
           ,
           it
           may
           also
           be
           very
           conveniently
           done
           ,
           if
           the
           Rule
           be
           fitted
           to
           a
           three-leg
           staff
           ,
           with
           a
           small
           Ball-socket
           to
           set
           it
           level
           ,
           or
           upright
           ,
           as
           other
           Surveighing-instruments
           be
           ,
           as
           will
           be
           amply
           found
           ,
           if
           a
           tryal
           be
           made
           thereof
           .
        
         
         
           That
           of
           Ma●er
           White
           's
           Rule
           is
           a
           thing
           that
           hath
           given
           very
           good
           content
           to
           several
           Gentlemen
           in
           the
           Counties
           of
           
             Essex
             ,
             Suffolk
          
           ,
           and
           Norfolk
           ,
           and
           indeed
           is
           a
           very
           neat
           and
           accurate
           way
           of
           operation
           ,
           well
           becoming
           a
           Gentleman
           ;
           for
           while
           a
           Workman
           shall
           take
           measure
           ,
           his
           Rule
           keeps
           the
           count
           of
           length
           ,
           or
           breadth
           ;
           and
           having
           the
           length
           first
           given
           ,
           the
           girt
           or
           squareness
           is
           no
           sooner
           agreed
           on
           ,
           but
           you
           have
           the
           content
           without
           Pen
           or
           Compasses
           .
        
         
           As
           for
           the
           other
           lines
           ,
           as
           Decimal-board
           ,
           and
           Timber-measure
           ,
           Inches
           ,
           and
           Foot
           ,
           in
           the
           way
           of
           Reduction
           ,
           Girt-measure
           ,
           Circles
           ,
           Diameter
           ,
           Circumference
           ,
           Squares
           inscribed
           and
           equal
           :
           The
           Use
           of
           them
           
           will
           be
           very
           grateful
           to
           many
           a
           learner
           .
           Lastly
           ,
           this
           brief
           touch
           of
           the
           Serpentine-line
           I
           made
           bold
           to
           assert
           ,
           to
           see
           if
           I
           could
           draw
           out
           a
           performance
           of
           that
           promise
           ,
           that
           hath
           been
           so
           long
           unperformed
           by
           the
           promisers
           thereof
           .
        
         
           These
           Collections
           ,
           courteous
           Reader
           ,
           I
           have
           printed
           at
           my
           friends
           and
           my
           own
           proper
           charges
           ,
           and
           if
           they
           prove
           to
           be
           (
           as
           I
           do
           hope
           they
           will
           )
           of
           publique
           benefit
           ,
           I
           shall
           enjoy
           my
           expectation
           ,
           and
           be
           ready
           at
           all
           times
           to
           serve
           you
           further
           ,
           as
           I
           may
           ,
           in
           these
           or
           other
           Mathematical
           instruments
           ,
           at
           my
           house
           at
           the
           Sun-dyal
           in
           the
           Minories
           ,
           and
           remain
           to
           you
           much
           obliged
           ,
        
         
           
             
               February
               8
               ▪
               1660.
               
            
          
           
             Iohn
             Brown.
             
          
        
      
       
         
         
           A
           TABLE
           of
           the
           things
           contained
           in
           this
           BOOK
           .
        
         
           
             CHAP.
             I.
             
          
           
             Page
          
           
             THe
             description
             of
             the
             rule
             for
             the
             hour
             onely
             1.
             2.
             3
          
        
         
           
             CHAP.
             II.
             
          
           
             To
             rectifie
             or
             set
             the
             Rule
             to
             his
             true
             angle
             for
             observation
             4
          
           
             To
             finde
             the
             Suns
             altitude
             5
          
           
             To
             finde
             the
             hour
             of
             the
             day
             6
          
           
             To
             finde
             the
             Suns
             rising
             and
             setting
             7
          
           
             To
             finde
             a
             level
             or
             perpendicular
             7
          
        
         
           
             CHAP.
             III.
             
          
           
             A
             further
             description
             of
             the
             rule
             for
             hour
             and
             Azimuth
             generally
             10
             ,
             11
             ,
             12
          
        
         
           
           
             CHAP.
             IIII.
             
          
           
             To
             finde
             the
             Suns
             declination
             13
          
           
             To
             finde
             the
             Suns
             true
             place
             ,
             and
             right
             ascention
             14
          
           
             To
             finde
             the
             Suns
             amplitude
             14
          
        
         
           
             CHAP.
             V.
             
          
           
             To
             finde
             the
             Suns
             Azimuth
             at
             any
             altitude
             and
             declination
             ,
             in
             this
             particular
             latitude
             in
             Summer
             16
          
           
             To
             finde
             the
             Azimuth
             in
             Winter
             ,
             and
             Equinoctial
             17
             ,
             18
          
        
         
           
             CHAP.
             VI.
             
          
           
             To
             finde
             the
             hour
             of
             the
             night
             by
             the
             Moon
             .
             19
          
           
             To
             finde
             the
             Moons
             Age
             20
          
           
             To
             finde
             the
             Moons
             place
             21
          
           
             To
             finde
             the
             Moons
             hour
             by
             the
             11
             chap.
             2
             and
             3
             Proposition
             23
          
           
             To
             finde
             the
             true
             hour
             of
             the
             night
             thereby
             24
          
        
         
           
           
             CHAP.
             VII
             .
          
           
             To
             finde
             the
             hour
             of
             the
             night
             by
             the
             sixed
             Stars
             25
          
           
             Three
             examples
             thereof
             26
             ,
             27
             ,
             28
          
        
         
           
             CHAP.
             VIII
          
           
             To
             finde
             the
             Amplitude
             ,
             Azimuth
             ,
             rising
             and
             so●●hing
             of
             the
             fixed
             Stars
             ,
             and
             Examples
             thereof
             in
             page
             29
             ,
             30
             ,
             31
             ,
             32
          
        
         
           
             CHAP.
             IX
             .
          
           
             To
             finde
             the
             hour
             and
             Azimuth
             ,
             &c.
             in
             any
             latitude
             33
          
           
             To
             finde
             the
             Suns
             rising
             ,
             setting
             ,
             and
             ascentional
             difference
             ibid.
             
          
           
             To
             finde
             his
             amplitude
             in
             any
             latitude
             34
          
           
             To
             finde
             the
             Suns
             altitude
             at
             six
             in
             any
             latitude
             .
             35
          
           
             To
             finde
             the
             hour
             when
             the
             Sun
             is
             in
             the
             Equinoctial
             36
          
           
             To
             find
             the
             hour
             in
             any
             l●titude
             ,
             altitude
             ,
             and
             declination
             37
          
           
             
             To
             finde
             the
             Suns
             Azimuth
             in
             any
             latitude
             ,
             at
             any
             declination
             ,
             and
             altitude
             in
             summer
             38
          
           
             To
             finde
             the
             same
             in
             winter
             39
          
        
         
           
             CHAP.
             X.
             
          
           
             To
             finde
             the
             inclination
             of
             Meridians
             ,
             substile
             ,
             stile
             ,
             and
             angle
             ,
             between
             12
             and
             6
             for
             erect
             decliners
             three
             ways
             ,
             one
             particular
             ,
             and
             two
             general
             40
          
           
             To
             finde
             the
             substile
             ,
             stiles
             elevation
             ,
             inclination
             of
             Meridians
             41
          
           
             To
             finde
             the
             angle
             between
             12
             and
             6
             for
             a
             particular
             latitude
             42
          
           
             To
             perform
             the
             same
             in
             general
             for
             any
             latitude
             by
             the
             general
             scale
             of
             altitudes
             43
             ,
             44
          
           
             Five
             canons
             to
             finde
             the
             same
             by
             the
             artificial
             sines
             and
             ta●gents
             ,
             and
             how
             to
             work
             them
             on
             the
             rule
             45
             ,
             46
          
        
         
           
             CHAP.
             XI
             .
          
           
             To
             draw
             a
             Horizontal
             Dyal
             to
             any
             latitude
             47
          
           
             To
             draw
             a
             vertical
             direct
             North
             or
             
             South
             Dial
             to
             any
             latitude
             48
             ,
             49
          
           
             To
             draw
             a
             direct
             erect
             East
             or
             West
             Dial
             50
             ,
             51
          
        
         
           
             CHAP.
             XIII
             .
          
           
             To
             finde
             the
             declination
             of
             a
             Plain
             52
          
           
             To
             do
             it
             by
             the
             needle
             53
             ,
             54
          
           
             To
             finde
             the
             quantity
             of
             an
             angle
             the
             sector
             or
             rule
             stands
             at
             55
          
           
             An
             example
             of
             the
             work
             56
          
           
             To
             finde
             a
             declination
             by
             the
             Sun
             at
             any
             time
             57
             ,
             58
          
           
             Some
             precepts
             and
             examples
             for
             the
             same
             59
             ,
             60
             ,
             61
          
           
             To
             finde
             a
             declination
             at
             two
             choice
             particular
             times
             ,
             viz.
             when
             the
             Sun
             is
             in
             the
             Meridian
             of
             place
             ,
             or
             Plain
             62
             ,
             63
          
           
             How
             to
             supply
             a
             deficiency
             in
             one
             line
             of
             the
             rule
             ,
             by
             another
             line
             on
             the
             rule
             64
          
        
         
           
             CHAP.
             XIIII
             .
          
           
             To
             draw
             a
             vertical
             declining
             Dial
             to
             any
             declination
             and
             latitude
             .
             65
          
           
             
             To
             perform
             it
             another
             way
             66
             ,
             67
          
           
             To
             supply
             a
             defect
             on
             the
             parallel
             contingent
             68
          
        
         
           
             CHAP.
             XV.
             
          
           
             To
             draw
             the
             hour-lines
             on
             an
             upright
             declining
             Dial
             ,
             declining
             above
             60
             degrees
             96
          
           
             To
             make
             the
             table
             for
             the
             hours
             70
          
           
             To
             finde
             the
             substile
             ,
             stiles
             augmentation
             ,
             and
             Radius
             ,
             to
             fit
             and
             fill
             the
             Plain
             with
             any
             certain
             number
             of
             hours
             71
             ,
             72
             ,
             73
             ,
             74
             ,
             75
          
           
             An
             advertisement
             relating
             to
             declining
             reclining
             Plains
             76
             ,
             77
             ,
             78
             ,
             79
          
        
         
           
             CHAP.
             XVI
             .
          
           
             To
             finde
             a
             perpendicular
             altitude
             at
             one
             or
             two
             stationi
             80
          
           
             To
             do
             it
             at
             one
             station
             81
          
           
             To
             perform
             it
             at
             two
             stations
             82
             ,
             83
             ,
             84
          
           
             To
             work
             it
             by
             the
             sector
             lines
             85
             ,
             86
          
        
         
           
             CHAP.
             XVII
             .
          
           
             The
             use
             of
             several
             lines
             inserted
             on
             rules
             for
             the
             use
             of
             several
             workmen
             ,
             for
             the
             mensuration
             of
             superficial
             ,
             and
             solid
             measure
             ,
             and
             reduction
             ,
             &c.
             87
          
           
             
             The
             use
             of
             inches
             ,
             and
             foot
             measure
             laid
             together
             ,
             in
             giving
             the
             price
             of
             one
             ,
             to
             know
             the
             price
             of
             a
             hundred
             ,
             or
             the
             contrary
             88
          
           
             The
             use
             in
             buying
             of
             timber
             ,
             knowing
             the
             price
             of
             a
             load
             ,
             or
             50
             foot
             ,
             to
             know
             the
             price
             of
             one
             foot
             ,
             or
             the
             contrary
             89
             ,
             90.
             
          
           
             The
             like
             work
             for
             the
             great
             hundred
             or
             112
             l.
             to
             the
             C.
             90
             ,
             91
          
           
             The
             use
             of
             the
             line
             of
             decimal
             board
             measure
             92
          
           
             The
             use
             of
             the
             line
             of
             decimal
             Timber
             measure
             93
          
           
             The
             use
             of
             the
             line
             of
             decimal
             yard
             measure
             94
          
           
             Or
             that
             which
             agreeth
             with
             feet
             and
             inches
             95
          
           
             The
             use
             of
             the
             line
             of
             decimal
             round
             or
             Girt
             measure
             96
          
           
             The
             use
             of
             the
             line
             of
             decimal
             sollid
             measure
             by
             the
             diameter
             97
          
           
             The
             table
             of
             decimal
             superficial
             under
             measure
             98
             ,
             99
          
           
             The
             table
             for
             decimal
             sollid
             under
             measure
             100
             ,
             101
          
           
             
             The
             table
             of
             under-yard
             measure
             for
             foot
             measure
             or
             inches
             102
             ,
             103
          
           
             The
             table
             of
             under
             girt
             measure
             to
             inches
             104
          
           
             The
             table
             of
             under
             diameter
             measure
             to
             inches
             and
             quarters
             105
          
           
             A
             table
             of
             brick
             measure
             106
          
           
             The
             use
             of
             the
             lines
             of
             Circumference
             ,
             diameter
             ,
             and
             squares
             equal
             and
             inscribed
             108
          
        
         
           
             CHAP.
             XVIII
             .
          
           
             The
             use
             of
             
               Mr.
               Whites
            
             rule
             or
             the
             sliding
             rule
             in
             Arithmetique
             ,
             and
             measuring
             superficial
             and
             sollid
             measure
             from
             110
             to
             118
          
        
         
           
             CHAP
             XIX
             .
          
           
             To
             finde
             hour
             and
             Azimnth
             by
             that
             sector
          
           
             To
             finde
             the
             Azimuth
             by
             having
             latitude
             ,
             Suns
             declination
             and
             altitudes
             ,
             cemplements
             ,
             and
             the
             hour
             from
             noon
             119
          
           
             To
             finde
             the
             hour
             ,
             by
             having
             the
             same
             complements
             ,
             and
             the
             Azimuth
             from
             south
             120
          
           
             
             Having
             the
             complements
             of
             the
             latitude
             and
             altitude
             ,
             and
             the
             Suns
             distance
             from
             the
             pole
             ,
             to
             finde
             the
             Suns
             Azimuth
             121
          
           
             Having
             the
             same
             complements
             ,
             to
             finde
             the
             hour
             by
             the
             sector
             122
          
           
             Having
             latitude
             ,
             declination
             ,
             meridian
             ,
             and
             present
             altitude
             ,
             to
             finde
             the
             hour
             of
             the
             day
             .
             123
          
           
             Having
             latitude
             ,
             declination
             ,
             and
             altitude
             ,
             to
             finde
             the
             Suns
             Azimuth
             at
             one
             operation
             by
             the
             sector
             124
          
           
             Having
             the
             length
             of
             the
             shadow
             ,
             to
             find
             the
             altitude
             ,
             and
             the
             contrary
             126
          
           
             Having
             latitude
             and
             declination
             ,
             to
             find
             the
             Suns
             rising
             and
             setting
             127
          
           
             To
             finde
             the
             Suns
             altitade
             at
             any
             hour
             generally
             129
          
           
             To
             finde
             when
             the
             Sun
             shall
             be
             due
             East
             or
             West
             130
          
           
             To
             finde
             the
             quesita
             in
             erect
             Dials
             131
             ,
             132.
             
          
        
         
           
             CHAP.
             XX.
             
          
           
             The
             use
             of
             the
             serpentine
             line
             .
          
           
             The
             description
             thereof
             .
             134
             ,
             135
             ,
             136
             ,
             137
          
           
             
             Some
             observations
             in
             the
             use
             thereof
             138.
             139
             ,
             140
          
           
             To
             finde
             the
             hour
             of
             the
             day
             thereby
             ,
             according
             to
             
               Mr.
               Gunter
            
             142
          
           
             To
             finde
             the
             hour
             thereby
             .
             according
             to
             
               Mr.
               Collins
            
             145
          
           
             To
             finde
             the
             Azimuth
             thereby
             
               Mr.
               Collins
            
             his
             way
             146
          
           
             To
             finde
             the
             Azimuth
             
               Mr.
               Gunters
            
             way
             148
          
           
             To
             finde
             the
             hour
             and
             Azimnth
             at
             one
             operation
             ,
             by
             help
             of
             the
             natural
             sines
             and
             versed
             sines
             149
             to
             154
          
           
             To
             finde
             the
             Suns
             altitude
             at
             any
             hour
             or
             Azimuth
             155
             ,
             156
          
           
             To
             finde
             the
             hour
             by
             having
             the
             Azimuth
             ,
             and
             the
             contrary
             157
          
           
             Five
             other
             useful
             propositions
             159
          
           
             To
             square
             and
             cube
             a
             number
             ,
             and
             to
             finde
             the
             square
             ,
             and
             cube
             root
             of
             a
             number
             162
             ,
             163
          
           
             To
             work
             questions
             of
             interest
             and
             annuities
             164
          
           
             The
             use
             of
             the
             everlasting
             Almanack
             167
          
        
      
       
         
         
           The
           right
           Ascension
           and
           Declination
           of
           12
           principal
           fixed
           Stars
           in
           the
           heavens
           ;
           most
           of
           which
           are
           inserted
           on
           the
           Rule
           :
           or
           if
           room
           will
           allow
           ,
           all
           of
           them
           .
        
         
           
             
               
                  
              
               
                 R.
                 Asc.
                 
              
               
                 Declina
                 .
              
            
             
               
                 Stars
                 Names
              
               
                 H.
                 M.
                 
              
               
                 Deg.
                 M.
                 
              
            
             
               
                 Pleiades
                 ,
                 or
                 7
                 Stars
              
               
                 03
                 24
              
               
                 23
                 20
              
            
             
               
                 Bulls-Eye
              
               
                 04
                 16
              
               
                 15
                 48
              
            
             
               
                 Orions
                 Girdle
              
               
                 05
                 18
              
               
                 01
                 195
              
            
             
               
                 Little
                 Dog
              
               
                 07
                 20
              
               
                 06
                 08
              
            
             
               
                 Lyons
                 Heart
              
               
                 09
                 50
              
               
                 13
                 40
              
            
             
               
                 Lyons
                 Tayl
              
               
                 11
                 30
              
               
                 16
                 30
              
            
             
               
                 Arcturus
              
               
                 14
                 00
              
               
                 21
                 04
              
            
             
               
                 Vultures
                 Heart
              
               
                 19
                 33
              
               
                 08
                 00
              
            
             
               
                 Dolphins
                 Head
              
               
                 20
                 30
              
               
                 14
                 52
              
            
             
               
                 Pegasus
                 Mouth
              
               
                 21
                 27
              
               
                 08
                 19
              
            
             
               
                 Fomahant
              
               
                 22
                 39
              
               
                 31
                 17s
              
            
             
               
                 Pegasus
                 lower
                 wing
              
               
                 23
                 55
              
               
                 13
                 19
              
            
          
           
             
               
                  
              
               
                 1
              
               
                  
              
               
                 3
              
               
                  
              
               
                 5
              
               
                 7
              
               
                 4
              
            
             
               
                  
              
               
                  
              
               
                 6
              
               
                  
              
               
                 8
              
               
                  
              
               
                  
              
               
                  
              
            
             
               
                 Moneths
              
               
                 9
              
               
                  
              
               
                 11
              
               
                  
              
               
                 2
              
               
                 10
              
               
                 12
              
            
             
               
                  
              
               
                 1
              
               
                 2
              
               
                 3
              
               
                 4
              
               
                 5
              
               
                 6
              
               
                 7
              
            
             
               
                  
              
               
                 8
              
               
                 9
              
               
                 10
              
               
                 11
              
               
                 12
              
               
                 13
              
               
                 14
              
            
             
               
                 Days
              
               
                 15
              
               
                 16
              
               
                 17
              
               
                 18
              
               
                 19
              
               
                 20
              
               
                 21
              
            
             
               
                  
              
               
                 22
              
               
                 23
              
               
                 24
              
               
                 25
              
               
                 26
              
               
                 27
              
               
                 28
              
            
             
               
                  
              
               
                 29
              
               
                 30
              
               
                 31
              
               
                  
              
               
                  
              
               
                  
              
               
                  
              
            
             
               
                 Week-days
              
               
                 S
              
               
                 M
              
               
                 T
              
               
                 W
              
               
                 T
              
               
                 F
              
               
                 Sat
              
            
             
               
                 Dom.
                 Letter
              
               
                 d
              
               
                 c
              
               
                 b
              
               
                 a
              
               
                 g
              
               
                 f
              
               
                 e
              
            
             
               
                 Leap
                 years
              
               
                 68
              
               
                 80
              
               
                 64
              
               
                 76
              
               
                 60
              
               
                 72
              
               
                 84
              
            
             
               
                 E●acts
              
               
                 26
              
               
                 9
              
               
                 12
              
               
                 25
              
               
                 28
              
               
                 11
              
               
                 23
              
            
          
        
      
    
     
       
         
         
           THE
           Description
           and
           Use
           OF
           A
           JOYNT-RULE
           .
        
         
           
             CHAP.
             I.
             
          
           
             The
             Description
             of
             the
             Lines
             on
             the
             Rule
             ,
             as
             it
             is
             made
             onely
             for
             one
             Latitude
             ,
             and
             for
             the
             finding
             the
             hour
             of
             the
             day
             onely
             .
          
           
             FIrst
             open
             the
             (
             Joynt
             of
             the
             )
             Rule
             ,
             then
             upon
             the
             head-leg
             ,
             being
             next
             to
             your
             right
             hand
             ,
             you
             have
             a
             line
             beginning
             at
             the
             hole
             ,
             which
             is
             the
             Center
             of
             the
             quadrantal
             lines
             ,
             and
             divided
             from
             thence
             downward
             toward
             the
             head
             ,
             into
             as
             many
             degrees
             as
             the
             Suns
             greatest
             
             altitude
             in
             that
             latitude
             will
             be
             ,
             which
             with
             us
             at
             London
             is
             to
             62
             degrees
             ;
             which
             line
             I
             call
             the
             
               Scale
               of
               Altitudes
            
             ,
             divided
             to
             whole
             ,
             halfs
             ,
             and
             sometime
             quarters
             of
             degrees
             .
          
           
             2.
             
             Secondly
             ,
             On
             the
             other
             leg
             ,
             and
             next
             to
             the
             inside
             is
             the
             line
             of
             hours
             ,
             usually
             divided
             into
             hours
             ,
             quarters
             ,
             and
             every
             fifth
             minute
             ,
             beginning
             at
             the
             head
             with
             4
             ,
             and
             so
             proceeding
             to
             5
             ,
             6
             ,
             7
             ,
             8
             ,
             9
             ,
             10
             ,
             11
             ,
             and
             12
             at
             the
             end
             ,
             and
             then
             back
             again
             with
             1
             ,
             2
             ,
             3
             ,
             4
             ,
             5
             ,
             6
             ,
             7
             ,
             8
             ,
             for
             the
             morning
             and
             afternoon
             hours
             .
          
           
             3.
             
             Next
             to
             this
             is
             a
             Kalendar
             of
             Moneths
             and
             Days
             in
             two
             lines
             ;
             the
             uppermost
             contains
             that
             half
             year
             the
             days
             lengthen
             in
             ,
             and
             the
             lowermost
             the
             shortning
             days
             ,
             as
             by
             the
             names
             of
             the
             moneths
             may
             appear
             ;
             the
             name
             of
             every
             moneth
             standing
             in
             the
             moneth
             ,
             and
             at
             the
             beginning
             of
             the
             moneth
             :
             and
             all
             but
             the
             two
             moneths
             that
             have
             the
             longest
             ,
             and
             the
             shortest
             days
             ,
             
               viz.
               Iune
            
             and
             December
             ,
             are
             divided
             into
             single
             days
             ,
             the
             tenth
             day
             having
             a
             figure
             10
             ,
             or
             a
             point
             or
             prick
             on
             the
             head
             of
             the
             
             stroke
             ,
             and
             the
             fifth
             onely
             a
             longer
             stroke
             without
             a
             prick
             ,
             and
             the
             beginning
             of
             every
             moneth
             a
             long
             stroke
             ,
             and
             every
             single
             day
             all
             alike
             of
             one
             shortness
             ,
             according
             to
             the
             usual
             manner
             of
             distinguishing
             on
             lines
             .
          
           
             4.
             
             And
             lastly
             you
             have
             a
             line
             of
             degrees
             ,
             for
             so
             they
             be
             most
             properly
             called
             ,
             and
             they
             are
             the
             same
             with
             the
             equal
             limb
             on
             quadrants
             ,
             and
             serve
             for
             the
             same
             use
             ,
             viz.
             for
             taking
             of
             Altitudes
             ,
             or
             Horizontal
             Angles
             ,
             and
             are
             divided
             usually
             to
             whole
             ,
             and
             half
             degrees
             of
             the
             quadrant
             ,
             and
             figured
             with
             30
             ,
             40
             ,
             50
             ,
             600
             ,
             7010
             ,
             8020
             ,
             and
             90
             ,
             just
             on
             the
             head
             ,
             cutting
             the
             center
             or
             point
             ,
             where
             the
             Scale
             of
             Altitudes
             and
             the
             Line
             of
             Hours
             meet
             ;
             which
             point
             ,
             for
             distinction
             sake
             ,
             I
             call
             
               The
               rectifying
               point
            
             .
             And
             the
             reckoning
             on
             this
             line
             ,
             as
             to
             taking
             of
             Altitudes
             ,
             is
             thus
             :
             At
             the
             number
             600
             is
             the
             beginning
             ,
             then
             towards
             the
             head
             count
             10
             ,
             20
             ,
             30
             ,
             where
             the
             90
             is
             ;
             then
             begin
             at
             the
             end
             again
             ,
             &
             count
             as
             the
             figures
             shew
             you
             to
             90
             at
             the
             head
             ,
             as
             before
             .
          
        
         
           
           
             CHAP
             II.
             
          
           
             The
             Uses
             of
             the
             Rule
             follow
             .
          
           
             
               1.
               
               To
               Rectifie
               or
               set
               the
               Rule
               to
               his
               true
               Angle
               .
            
             
               OPen
               the
               Rule
               to
               60
               degrees
               ,
               which
               is
               done
               thus
               ,
               (
               indifferently
               :
               )
               make
               the
               lines
               on
               the
               head
               ,
               and
               the
               lines
               on
               the
               other
               leg
               ,
               meet
               in
               a
               streight
               line
               ;
               then
               is
               the
               Scale
               of
               Altitudes
               and
               the
               line
               of
               Hours
               set
               to
               an
               Angle
               of
               60
               degrees
               ,
               the
               rectifying
               point
               ,
               being
               the
               center
               of
               that
               Angle
               ;
               Or
               to
               do
               it
               more
               exactly
               ,
               do
               thus
               :
               put
               one
               point
               of
               a
               pair
               of
               Compasses
               into
               the
               rectifying
               point
               ,
               then
               open
               the
               other
               to
               10
               ,
               20
               ,
               30
               ,
               or
               40
               ,
               on
               the
               Scale
               of
               Altitudes
               ,
               the
               Compasses
               so
               opened
               ,
               and
               the
               point
               yet
               remaining
               in
               the
               rectifying
               point
               ,
               turn
               the
               other
               to
               that
               margenal
               line
               in
               the
               line
               of
               hours
               ,
               that
               cuts
               the
               rectifying
               point
               ,
               and
               there
               stay
               it
               ;
               then
               remove
               the
               point
               that
               was
               fixed
               
               in
               the
               rectifying
               point
               ,
               and
               open
               or
               shut
               the
               Rule
               ,
               till
               the
               point
               of
               the
               Compasses
               will
               touch
               10
               ,
               20
               ,
               30
               ,
               or
               40
               ,
               being
               the
               point
               you
               set
               the
               Compasses
               too
               in
               the
               Scale
               of
               Altitudes
               ,
               in
               the
               innermost
               line
               that
               cuts
               the
               center
               ,
               and
               the
               rectifying
               point
               ,
               then
               is
               it
               set
               exactly
               to
               60
               degrees
               ,
               and
               fitted
               for
               observation
               .
            
          
           
             
               2.
               
               To
               finde
               the
               Suns
               Altitude
               at
               any
               time
               .
            
             
               Put
               a
               pin
               in
               the
               center
               hole
               ,
               at
               the
               upper
               end
               of
               the
               Scale
               of
               Altitudes
               ,
               and
               on
               the
               pin
               hang
               a
               thread
               and
               plummet
               ;
               then
               if
               the
               Sun
               be
               low
               ,
               that
               is
               to
               say
               ,
               under
               25
               degrees
               high
               ,
               as
               in
               the
               winter
               it
               will
               always
               be
               ,
               then
               lift
               up
               the
               moveable
               leg
               ,
               where
               the
               moneths
               and
               the
               degrees
               be
               ,
               till
               the
               shadow
               of
               the
               end
               fall
               just
               on
               the
               meeting
               of
               that
               leg
               with
               the
               head
               ,
               then
               the
               thread
               shall
               shew
               the
               Suns
               altitude
               ,
               counting
               from
               600
               towards
               the
               head
               ,
               either
               10
               ,
               20
               ,
               25
               ,
               or
               any
               degree
               between
               .
            
             
               But
               if
               the
               Sun
               be
               above
               25
               or
               30
               degrees
               
               high
               ,
               lift
               up
               the
               head
               leg
               till
               the
               shadow
               of
               that
               play
               as
               before
               ,
               or
               make
               the
               shadow
               of
               the
               pin
               in
               the
               center
               hole
               play
               on
               the
               innermost
               line
               of
               the
               Scale
               of
               Altitudes
               where
               the
               pin
               standeth
               ,
               then
               the
               thread
               will
               fall
               on
               the
               degree
               ,
               and
               part
               of
               a
               degree
               that
               his
               true
               altitude
               shall
               be
               .
            
             
               But
               if
               the
               Sun
               be
               in
               a
               cloud
               ,
               and
               can
               not
               be
               seen
               so
               as
               to
               give
               a
               shadow
               ,
               then
               look
               up
               along
               by
               the
               head-leg
               ,
               or
               moveable
               leg
               ,
               just
               against
               the
               middle
               of
               the
               round
               body
               of
               the
               Sun
               ,
               and
               the
               thread
               playing
               evenly
               by
               the
               degrees
               ,
               shall
               show
               the
               true
               altitude
               required
               .
               The
               like
               must
               you
               do
               for
               a
               Star
               ,
               or
               any
               other
               object
               ,
               whose
               altitude
               you
               would
               find
               .
            
          
           
             
               3.
               
               Having
               found
               the
               Suns
               altitude
               ,
               and
               the
               day
               of
               the
               moneth
               ,
               to
               finde
               the
               hour
               of
               the
               day
               .
            
             
               Whatsoever
               you
               finde
               the
               altitude
               to
               be
               ,
               take
               the
               same
               off
               from
               the
               Line
               of
               Altitudes
               ,
               from
               the
               center
               downwards
               with
               a
               pair
               of
               Compasses
               ,
               then
               lay
               the
               thread
               (
               being
               put
               over
               the
               pin
               )
               on
               the
               
               day
               of
               the
               moneth
               ,
               then
               put
               one
               foot
               of
               the
               Compasses
               in
               the
               line
               of
               hours
               ,
               in
               that
               line
               that
               cuts
               the
               rectifying
               point
               ,
               and
               carry
               it
               further
               off
               ,
               or
               nigher
               ,
               till
               the
               other
               foot
               of
               the
               Compass
               being
               turned
               about
               ,
               will
               just
               touch
               the
               thred
               ,
               at
               the
               nearest
               distance
               ,
               then
               the
               point
               of
               the
               Compasses
               on
               the
               line
               of
               hours
               ,
               shall
               shew
               the
               true
               hour
               and
               minute
               of
               the
               day
               required
               .
            
             
               
                 Example
                 on
                 the
                 2.
                 of
                 July
                 .
              
               
                 1.
                 
                 I
                 observe
                 the
                 altitude
                 in
                 the
                 morning
                 ,
                 and
                 I
                 finde
                 it
                 to
                 be
                 30
                 degrees
                 high
                 ,
                 then
                 laying
                 the
                 thread
                 on
                 the
                 day
                 of
                 the
                 moneth
                 ,
                 and
                 taking
                 30
                 degrees
                 from
                 the
                 Scale
                 of
                 Altitudes
                 ,
                 and
                 putting
                 one
                 point
                 in
                 the
                 line
                 of
                 hours
                 ,
                 till
                 the
                 other
                 point
                 turned
                 about
                 ,
                 will
                 but
                 just
                 touch
                 the
                 th●ead
                 ,
                 and
                 I
                 finde
                 it
                 to
                 23
                 minutes
                 past
                 7
                 ,
                 but
                 if
                 it
                 had
                 been
                 in
                 the
                 afternoon
                 ,
                 it
                 would
                 have
                 been
                 37
                 minutes
                 past
                 4.
                 
              
               
                 2.
                 
                 Again
                 ,
                 on
                 the
                 tenth
                 of
                 August
                 in
                 the
                 Afternoon
                 ,
                 at
                 20
                 degrees
                 high
                 ,
                 I
                 take
                 20
                 degrees
                 from
                 the
                 Scale
                 of
                 Altudes
                 ,
                 
                 and
                 laying
                 the
                 thread
                 on
                 the
                 day
                 of
                 the
                 moneth
                 ,
                 viz.
                 the
                 tenth
                 aforesaid
                 ,
                 counting
                 from
                 the
                 name
                 at
                 the
                 beginning
                 of
                 August
                 ,
                 toward
                 September
                 ,
                 and
                 carrying
                 the
                 Compasses
                 in
                 the
                 line
                 of
                 hours
                 ,
                 till
                 the
                 other
                 point
                 doth
                 but
                 just
                 touch
                 the
                 thread
                 ,
                 and
                 you
                 shall
                 finde
                 it
                 to
                 be
                 54
                 minutes
                 past
                 4
                 a
                 clock
                 .
              
               
                 3.
                 
                 Again
                 ,
                 on
                 the
                 11.
                 of
                 December
                 at
                 15
                 degrees
                 high
                 ,
                 work
                 as
                 before
                 ,
                 and
                 you
                 shall
                 finde
                 it
                 to
                 be
                 just
                 12
                 a
                 clock
                 ;
                 but
                 to
                 work
                 this
                 ,
                 you
                 must
                 lay
                 the
                 Rule
                 down
                 on
                 something
                 ,
                 and
                 extend
                 the
                 thread
                 beyond
                 the
                 Rule
                 ,
                 for
                 the
                 nighest
                 distance
                 will
                 happen
                 on
                 the
                 out-side
                 of
                 the
                 Rule
                 .
              
               
                 4.
                 
                 Again
                 ,
                 on
                 the
                 11
                 of
                 Iune
                 at
                 noon
                 I
                 finde
                 the
                 altitude
                 to
                 be
                 62
                 degrees
                 high
                 ,
                 then
                 laying
                 the
                 thread
                 on
                 the
                 10
                 th
                 or
                 11
                 th
                 of
                 Iune
                 ,
                 for
                 then
                 a
                 day
                 is
                 unsensible
                 ,
                 and
                 working
                 as
                 before
                 ,
                 you
                 shal
                 finde
                 the
                 point
                 of
                 the
                 Compasses
                 to
                 stay
                 at
                 just
                 12
                 a
                 clock
                 ,
                 the
                 time
                 required
                 for
                 that
                 altitude
                 .
              
            
          
           
             
             
               4.
               
               To
               finde
               the
               Suns
               rising
               any
               day
               in
               the
               year
               .
            
             
               Lay
               the
               thread
               on
               the
               day
               of
               the
               month
               ,
               and
               in
               the
               line
               of
               hours
               it
               sheweth
               the
               true
               hour
               and
               minute
               of
               the
               Suns
               rising
               or
               setting
               ;
               for
               the
               rising
               ,
               count
               the
               morning
               hours
               ;
               and
               for
               the
               setting
               ,
               count
               the
               evening
               hours
               .
            
          
           
             
               5.
               
               To
               finde
               if
               any
               place
               lye
               level
               ,
               or
               nor
               .
            
             
               Open
               the
               rule
               to
               his
               true
               angle
               of
               60
               degrees
               ,
               then
               set
               the
               moveable
               leg
               upon
               the
               place
               you
               would
               make
               level
               ,
               and
               if
               the
               thread
               play
               just
               on
               60
               degrees
               ,
               it
               is
               a
               true
               level
               place
               ,
               or
               else
               not
               .
            
          
           
             
               6.
               
               To
               try
               if
               any
               thing
               be
               upright
               or
               not
               .
            
             
               Hang
               a
               thread
               and
               plummet
               on
               the
               center
               ,
               then
               aply
               the
               head
               leg
               of
               the
               rule
               to
               the
               wall
               or
               post
               ,
               and
               if
               it
               be
               upright
               ,
               the
               thread
               will
               play
               just
               on
               the
               innermost
               line
               of
               the
               scale
               of
               altitudes
               ,
               or
               else
               not
               .
            
          
        
         
           
           
             CHAP.
             III.
             
          
           
             A
             further
             description
             of
             the
             Rule
             ,
             to
             make
             it
             to
             shew
             the
             Suns
             Azimuth
             ,
             Declination
             ;
             True
             place
             ,
             right
             Ascention
             ,
             and
             the
             hour
             of
             day
             or
             night
             ,
             in
             this
             ,
             or
             any
             other
             Lattitude
             .
          
           
             1.
             
             FIrst
             in
             stead
             of
             the
             scale
             of
             Altitudes
             to
             62
             degrees
             ,
             there
             is
             one
             put
             to
             90
             degrees
             in
             that
             place
             ,
             and
             that
             of
             62
             is
             put
             by
             in
             some
             other
             place
             where
             it
             may
             serve
             as
             well
          
           
             2.
             
             The
             line
             of
             hours
             hath
             a
             double
             margent
             ,
             viz
             ,
             one
             for
             hours
             ,
             and
             the
             other
             for
             Azimuths
             ,
             &
             then
             every
             5
             th
             minute
             is
             more
             properly
             made
             4
             ,
             or
             else
             every
             2
             minutes
             ,
             and
             in
             a
             large
             rule
             to
             every
             quarter
             of
             a
             degree
             of
             Azimuth
             ,
             or
             to
             every
             single
             minute
             of
             time
             .
          
           
             3.
             
             The
             degrees
             ought
             to
             be
             reckoned
             after
             3
             maner
             of
             wayes
             :
             first
             as
             before
             is
             exprest
             ;
             secondly
             from
             60
             toward
             the
             end
             ,
             with
             10
             ,
             20
             ,
             30
             ,
             40
             ,
             50
             ,
             60
             ,
             &c.
             to
             be
             so
             accounted
             in
             finding
             the
             Azimuch
             for
             a
             particular
             latitude
             ;
             and
             
             and
             thirdly
             from
             the
             head
             or
             90
             ,
             toward
             the
             end
             ,
             with
             10
             ,
             20
             ,
             30
             ,
             40
             ,
             50
             ,
             60
             ,
             70
             ,
             80
             ,
             &c.
             for
             the
             general
             finding
             of
             Hour
             and
             Azimuth
             in
             any
             latitude
             ,
             and
             many
             other
             problems
             of
             the
             Sphere
             besides
             ;
             to
             which
             may
             be
             added
             ,
             where
             room
             will
             alow
             ,
             a
             line
             of
             hours
             ,
             beginning
             at
             6
             at
             the
             head
             ,
             and
             12
             at
             the
             end
             ,
             but
             reckoning
             15
             degrees
             for
             an
             hour
             ,
             and
             4
             minutes
             for
             every
             degree
             ,
             it
             may
             do
             as
             well
             without
             it
             .
          
           
             4.
             
             To
             the
             Kalender
             of
             moneths
             and
             days
             ,
             is
             added
             a
             line
             of
             the
             Suns
             true
             place
             in
             the
             Zodiack
             ,
             or
             where
             room
             fails
             ,
             the
             Characters
             of
             the
             twelve
             Signs
             put
             on
             that
             day
             of
             the
             moneth
             the
             Sun
             enters
             into
             it
             ,
             and
             counting
             every
             day
             for
             a
             degree
             ,
             may
             indifferently
             serve
             for
             the
             use
             it
             is
             chiefly
             intended
             for
             .
          
           
             5.
             
             Under
             that
             is
             a
             line
             of
             the
             Suns
             right
             Ascension
             ,
             to
             hours
             and
             quarters
             at
             least
             ,
             or
             rather
             every
             fifth
             minute
             ,
             numbred
             thus
             :
             12
             and
             24
             right
             under
             ♈
             and
             ♎
             ,
             or
             the
             tenth
             of
             March
             ,
             and
             so
             forward
             to
             the
             tenth
             of
             Iune
             ,
             or
             ♋
             ,
             where
             stands
             6
             ,
             then
             backwards
             to
             12
             
             where
             you
             began
             ,
             then
             backwards
             still
             to
             the
             eleventh
             of
             December
             ,
             with
             13
             ,
             14
             ,
             15
             ,
             16
             ,
             17
             ,
             18
             ,
             to
             ♑
             ,
             then
             from
             thence
             forward
             to
             24
             where
             you
             first
             began
             :
             but
             when
             you
             are
             streightned
             for
             room
             ,
             as
             on
             most
             ordinary
             Rules
             you
             will
             be
             ,
             then
             it
             may
             very
             well
             suffice
             to
             have
             a
             point
             or
             stroke
             ,
             shewing
             when
             the
             Sun
             shall
             gradually
             get
             an
             hour
             of
             right
             Ascension
             ,
             and
             from
             that
             for
             every
             day
             count
             four
             minutes
             of
             time
             ,
             till
             it
             hath
             increased
             to
             an
             hour
             more
             ,
             and
             this
             computation
             will
             serve
             very
             well
             ;
             and
             in
             stead
             of
             saying
             13
             ,
             14
             ,
             15
             hours
             of
             right
             Ascension
             ,
             say
             1
             ,
             2
             ,
             3
             ,
             &c.
             which
             will
             perform
             the
             work
             as
             well
             ,
             and
             reduce
             the
             time
             to
             more
             proper
             terms
             .
          
           
             6.
             
             There
             is
             fitted
             two
             lines
             ,
             one
             containing
             24
             houres
             ,
             and
             the
             other
             29
             days
             ,
             and
             about
             13
             hours
             ,
             and
             they
             serve
             to
             finde
             the
             time
             of
             the
             Moons
             coming
             to
             the
             South
             ,
             before
             or
             after
             the
             Sun
             ,
             and
             by
             that
             ,
             the
             time
             of
             high-water
             at
             London-bridge
             ,
             or
             any
             other
             place
             ,
             as
             is
             ordinary
             .
          
        
         
           
           
             CHAP.
             IV.
             
          
           
             The
             Uses
             follow
             in
             order
             .
          
           
             
               1.
               
               To
               finde
               the
               Suns
               Declination
               .
            
             
               LAy
               the
               Thread
               on
               the
               day
               of
               the
               moneth
               ,
               then
               in
               the
               line
               of
               degrees
               you
               have
               the
               declination
               .
               From
               March
               the
               tenth
               toward
               the
               head
               is
               the
               Declination
               Northward
               ,
               the
               other
               way
               is
               Southward
               ,
               as
               by
               the
               time
               of
               the
               year
               is
               discovered
               .
            
             
               
                 Example
                 :
              
               
                 On
                 the
                 tenth
                 of
                 April
                 it
                 is
                 11d
                 48
                 '
                 toward
                 the
                 North
                 ;
                 but
                 on
                 the
                 tenth
                 of
                 October
                 it
                 is
                 10d
                 30
                 '
                 toward
                 the
                 South
                 .
              
               
                 2.
                 
                 As
                 the
                 thread
                 is
                 so
                 laid
                 on
                 the
                 day
                 of
                 the
                 moneth
                 ,
                 in
                 the
                 line
                 of
                 the
                 Suns
                 place
                 ,
                 it
                 sheweth
                 that
                 ;
                 and
                 in
                 the
                 line
                 of
                 the
                 Suns
                 right
                 Ascension
                 ,
                 his
                 right
                 Ascension
                 also
                 ,
                 onely
                 you
                 must
                 give
                 it
                 its
                 due
                 order
                 of
                 reckoning
                 ,
                 as
                 thus
                 :
                 it
                 begins
                 at
                 ♈
                 Aries
                 ,
                 and
                 so
                 proceeds
                 to
                 ♋
                 ,
                 then
                 back
                 again
                 to
                 ♑
                 at
                 the
                 eleventh
                 
                 of
                 December
                 ,
                 then
                 forwards
                 again
                 to
                 ♈
                 Aries
                 ,
                 where
                 you
                 began
                 .
              
            
          
           
             
               3.
               
               To
               finde
               the
               Suns
               right
               Ascension
               in
               hours
               and
               minutes
               .
            
             
               Lay
               the
               thread
               as
               before
               ,
               on
               the
               day
               of
               the
               moneth
               ,
               and
               in
               the
               line
               of
               right
               Ascension
               you
               have
               the
               hour
               and
               minute
               required
               ,
               computing
               right
               according
               to
               the
               time
               of
               the
               year
               ,
               that
               is
               ,
               begin
               at
               the
               tenth
               of
               March
               ,
               or
               ♈
               Aries
               ,
               and
               so
               reckon
               forwards
               and
               backwards
               as
               the
               moneths
               go
               .
            
             
               
                 Example
                 .
              
               
                 On
                 the
                 tenth
                 of
                 April
                 the
                 Suns
                 place
                 is
                 1
                 degree
                 in
                 ♉
                 Taurus
                 ,
                 and
                 the
                 Suns
                 right
                 Ascension
                 1
                 hour
                 55
                 minutes
                 :
                 on
                 the
                 tenth
                 of
                 October
                 27d
                 1
                 /
                 4
                 in
                 ♎
                 Libra
                 ,
                 and
                 his
                 right
                 Ascension
                 is
                 13
                 hours
                 and
                 42
                 minutes
                 .
              
            
          
           
             
               4.
               
               To
               finde
               the
               Suns
               Amplitude
               at
               rising
               or
               setting
               .
            
             
               Take
               the
               Suns
               Declination
               out
               of
               the
               particular
               Scale
               of
               Altitudes
               ,
               and
               lay
               it
               the
               same
               way
               as
               the
               Declination
               is
               ,
               
               from
               90
               in
               the
               Azimuth
               Scale
               ,
               and
               it
               shall
               shew
               the
               Amplitude
               from
               the
               east
               or
               west
               ,
               counting
               from
               90.
               
               Example
               :
               May
               the
               tenth
               it
               is
               33.
               37.
               
            
          
        
         
           
             CHAP.
             V.
             
          
           
             Having
             the
             Suns
             Declination
             ,
             or
             day
             of
             the
             moneth
             ,
             to
             finde
             the
             Azimuth
             at
             any
             Altitude
             required
             for
             that
             day
             .
          
           
             FIrst
             finde
             the
             Suns
             Declination
             ,
             by
             the
             first
             Proposition
             of
             the
             fourth
             Chapter
             ,
             then
             take
             that
             out
             of
             the
             particular
             Scale
             of
             Altitudes
             ,
             or
             scale
             to
             62
             degrees
             ;
             then
             whatsoever
             the
             Altitude
             shall
             happen
             to
             be
             ,
             count
             the
             same
             on
             the
             degrees
             from
             60
             toward
             the
             end
             of
             the
             Rule
             ,
             according
             to
             the
             second
             maner
             of
             counting
             ,
             in
             the
             third
             Proposition
             of
             the
             third
             Chapter
             ,
             and
             thereunto
             lay
             the
             thred
             ,
             then
             the
             Compasses
             set
             to
             the
             Declination
             ,
             carry
             one
             point
             along
             the
             line
             of
             hours
             on
             the
             same
             side
             of
             the
             thread
             the
             Declination
             is
             ;
             that
             is
             to
             say
             ,
             if
             the
             day
             of
             the
             
             moneth
             ,
             or
             Declination
             be
             on
             the
             right
             side
             the
             Aequinoctial
             ,
             then
             carry
             the
             Compasses
             on
             the
             right
             side
             ;
             but
             if
             the
             Declination
             be
             on
             the
             South
             side
             ,
             that
             is
             ,
             toward
             the
             end
             ,
             (
             counting
             from
             the
             tenth
             of
             March
             ,
             or
             Aries
             or
             Libra
             ,
             then
             carry
             the
             Compasses
             along
             the
             line
             of
             hours
             and
             Azimuths
             on
             the
             left
             side
             of
             the
             th●ead
             ,
             as
             all
             win●er
             time
             it
             will
             be
             ,
             and
             having
             set
             the
             Compasses
             to
             the
             least
             distance
             to
             the
             thread
             ,
             it
             sh●ll
             stay
             at
             the
             Suns
             true
             Azimuth
             from
             the
             South
             required
             ,
             counting
             as
             the
             figures
             are
             numbred
             ;
             or
             from
             East
             or
             West
             ,
             counting
             from
             90.
             
          
           
             
               Example
               1.
               
            
             
               On
               the
               tenth
               of
               Iuly
               I
               desire
               to
               finde
               the
               Suns
               Azimuth
               at
               any
               Altitude
               ,
               first
               on
               that
               day
               I
               finde
               the
               Suns
               Declination
               to
               be
               20.
               45
               ,
               which
               number
               count
               from
               the
               beginning
               of
               the
               particular
               Scale
               of
               Altitudes
               toward
               62
               ,
               and
               that
               distance
               take
               between
               your
               Compasses
               ,
               then
               are
               they
               set
               for
               all
               that
               day
               ;
               then
               supposing
               the
               Suns
               height
               to
               be
               ten
               degrees
               ,
               lay
               the
               thread
               on
               10
               ,
               counted
               
               from
               60
               toward
               the
               left
               end
               ,
               then
               carrying
               the
               Compasses
               on
               the
               right
               side
               of
               the
               thread
               ,
               (
               because
               it
               is
               summer
               or
               north
               declination
               )
               on
               the
               line
               of
               Azimuths
               ,
               it
               shall
               shew
               110.
               40
               ,
               the
               Azimuth
               from
               the
               south
               required
               ;
               but
               if
               you
               count
               from
               90
               ,
               it
               is
               but
               20.
               40.
               from
               the
               east
               ,
               or
               west
               point
               northward
               ,
               according
               to
               the
               time
               of
               the
               day
               ,
               either
               morning
               or
               evening
               .
            
          
           
             
               Example
               2.
               
            
             
               Again
               ,
               on
               the
               14.
               of
               November
               ,
               or
               the
               6.
               of
               Ianuary
               ,
               when
               the
               Sun
               hath
               the
               same
               declination
               south-ward
               ,
               and
               the
               same
               Altitude
               ,
               to
               work
               this
               you
               must
               lay
               the
               Rule
               down
               on
               something
               ,
               then
               lay
               the
               thread
               on
               the
               Altitude
               ,
               counted
               from
               60
               toward
               the
               end
               (
               as
               before
               )
               and
               carrying
               the
               Compasses
               on
               the
               south-side
               of
               the
               Aequinoctial
               ,
               along
               the
               Azimuth-line
               ,
               till
               the
               other
               point
               do
               but
               just
               touch
               the
               th●ead
               ,
               and
               it
               shall
               stay
               at
               36.
               45
               ,
               the
               Azimuth
               from
               south
               required
               ;
               if
               it
               be
               morning
               ,
               it
               wants
               of
               coming
               to
               south
               ;
               if
               it
               be
               after-noon
               ,
               it
               is
               past
               the
               south
               .
            
          
           
             
             
               Example
               3.
               
            
             
               But
               if
               the
               Sun
               be
               in
               the
               Aequinoctial
               ,
               and
               have
               no
               declination
               ,
               then
               it
               is
               but
               laying
               the
               thread
               to
               the
               Altitude
               ,
               and
               in
               the
               line
               of
               Azimuths
               the
               thread
               shall
               shew
               the
               true
               Azimuth
               required
               .
               As
               for
               instance
               :
               at
               00
               degrees
               of
               altitude
               ,
               the
               Azimuth
               is
               90
               ,
               at
               10
               degrees
               it
               is
               77.
               15
               ,
               at
               20
               degrees
               62.
               45
               ,
               at
               30
               degrees
               high
               43.
               15
               ,
               at
               35
               degrees
               high
               28.
               10
               ,
               at
               38
               degrees
               28
               '
               high
               ,
               it
               is
               just
               south
               ,
               as
               by
               practice
               may
               plainly
               appear
               .
               But
               if
               the
               Suns
               altitude
               be
               above
               45
               ,
               then
               the
               degrees
               will
               go
               beyond
               the
               end
               of
               the
               Rule
               :
               To
               supply
               this
               defect
               ,
               do
               thus
               :
               Substract
               45
               out
               of
               the
               number
               you
               would
               have
               ,
               and
               double
               the
               remainder
               ,
               then
               lay
               the
               Rule
               down
               with
               some
               piece
               of
               the
               same
               thickness
               ,
               in
               a
               streight
               line
               with
               the
               moveable
               leg
               ;
               then
               take
               the
               distance
               from
               the
               tangent
               of
               the
               remainder
               doubled
               (
               counted
               from
               60
               to
               the
               end
               of
               the
               Rule
               ,
               in
               the
               line
               next
               the
               edge
               )
               to
               the
               Center
               ,
               lay
               that
               distance
               in
               the
               same
               streight
               line
               from
               the
               tangent
               
               doubled
               ,
               and
               that
               shall
               be
               the
               tangent
               of
               the
               Angle
               above
               45
               ,
               whereunto
               you
               must
               lay
               the
               thread
               for
               the
               finding
               the
               Azimuth
               ,
               when
               the
               Sun
               is
               above
               45
               degrees
               high
               .
            
          
        
         
           
             CHAP.
             VI.
             
          
           
             To
             finde
             the
             hour
             of
             the
             Night
             by
             the
             Moon
             .
          
           
             FIrst
             by
             the
             help
             of
             an
             Almanack
             ,
             get
             the
             true
             time
             of
             the
             New
             Moon
             ,
             then
             compute
             her
             true
             place
             at
             that
             time
             ,
             which
             is
             always
             the
             place
             of
             the
             Sun
             (
             very
             nigh
             )
             at
             the
             hour
             and
             minute
             of
             conjunction
             ;
             then
             compute
             how
             many
             days
             old
             the
             Moon
             is
             ,
             then
             by
             the
             line
             of
             Numbers
             say
             :
             If
             29
             dayes
             13
             hours
             ,
             (
             or
             on
             the
             line
             29.
             540
             )
             require
             860
             degrees
             ,
             or
             12
             signs
             ,
             what
             shall
             ●ny
             less
             number
             of
             days
             and
             part
             of
             a
             day
             require
             ?
             The
             answer
             will
             be
             :
             The
             Moons
             true
             place
             at
             that
             age
             .
             Having
             ●ound
             her
             true
             place
             ,
             then
             take
             her
             al●itude
             ,
             and
             lay
             the
             thred
             on
             the
             Moons
             
             place
             found
             ,
             and
             work
             as
             you
             did
             for
             the
             Sun
             ,
             and
             note
             what
             hour
             you
             finde
             ;
             then
             consider
             if
             it
             be
             New
             Moon
             ,
             the
             hour
             you
             finde
             is
             thētrue
             hour
             ,
             likewise
             in
             the
             Full
             ;
             but
             if
             it
             be
             before
             or
             after
             ,
             you
             must
             substract
             by
             the
             Line
             of
             Numbers
             thus
             :
             If
             29
             days
             540
             parts
             require
             24
             hours
             ,
             what
             shall
             any
             number
             of
             days
             and
             parts
             require
             ?
             The
             answer
             is
             :
             What
             you
             must
             take
             away
             from
             the
             Moons
             hour
             found
             ,
             to
             make
             the
             true
             hour
             of
             the
             Night
             which
             was
             required
             .
          
           
             But
             for
             more
             plainness
             sake
             ,
             I
             will
             reduce
             these
             Operations
             to
             so
             many
             Propositions
             ,
             before
             I
             come
             to
             an
             Example
             .
          
           
             
               PROP.
               1.
               
            
             
               To
               finde
               the
               Moons
               Age.
               
            
             
               First
               ,
               it
               is
               most
               readily
               and
               exactly
               done
               by
               an
               Ephemerides
               ,
               such
               a
               one
               as
               you
               finde
               in
               Mr.
               Lilly's
               Alman●ck
               ,
               or
               (
               as
               to
               her
               Age
               onely
               )
               in
               any
               book
               or
               Sheet-Almanack
               ;
               but
               you
               may
               do
               it
               
               indifferently
               by
               the
               Epact
               thus
               ,
               (
               by
               the
               Rules
               of
               the
               152
               page
               in
               the
               Appendix
               to
               the
               
                 Carpenters
                 Rule
                 .
              
               )
               Adde
               the
               Epact
               ,
               the
               moneth
               ,
               and
               the
               day
               of
               the
               mone●h
               together
               ,
               and
               the
               sum
               ,
               if
               under
               30
               ,
               is
               the
               Moons
               age
               ;
               but
               if
               above
               ,
               consider
               if
               the
               moneth
               have
               30
               or
               31
               days
               ,
               then
               substract
               29
               or
               30
               out
               ,
               and
               the
               remainder
               is
               the
               Moons
               age
               in
               days
               .
            
             
               
                 Example
                 .
              
               
                 August
                 2.
                 1660.
                 
                 Epact
                 28.
                 
                 Month
                 6.
                 day
                 2.
                 added
                 makes
                 36.
                 
                 Now
                 August
                 or
                 sixt
                 moneth
                 ,
                 hath
                 31
                 days
                 ,
                 therefore
                 30
                 being
                 taken
                 away
                 ,
                 6
                 days
                 remains
                 for
                 the
                 moons
                 age
                 required
                 .
              
            
          
           
             
               PROP.
               2.
               
            
             
               To
               finde
               the
               Moons
               place
               .
            
             
               By
               the
               Ephemerides
               aforesaid
               in
               Mr.
               Lilly's
               Almanack
               ,
               you
               have
               it
               ser
               down
               every
               day
               in
               the
               year
               ;
               but
               to
               finde
               it
               by
               the
               Rule
               ,
               do
               thus
               :
               Count
               six
               days
               back
               from
               
                 August
                 2.
                 viz.
              
               to
               Iuly
               27.
               there
               lay
               the
               thread
               ,
               and
               in
               the
               line
               of
               the
               Suns
               place
               ,
               you
               have
               the
               Moons
               
               place
               required
               ,
               being
               then
               near
               alike
               ;
               then
               in
               regard
               the
               Moon
               goes
               faster
               than
               the
               Sun
               ,
               that
               is
               to
               say
               ,
               in
               29
               days
               13
               hours
               ,
               12
               signs
               ,
               or
               360
               degrees
               ;
               in
               3
               days
               1
               sign
               ,
               6
               degr
               .
               34
               min.
               20
               sec.
               in
               one
               day
               o
               signs
               ,
               12
               degr
               .
               11
               min.
               27
               sec.
               in
               one
               hour
               ,
               30
               min.
               29
               sec.
               (
               or
               half
               a
               minute
               :
               )
               adde
               the
               signs
               ,
               and
               degrees
               ,
               and
               minutes
               the
               Moon
               hath
               gone
               in
               so
               many
               days
               and
               hours
               ,
               if
               you
               know
               them
               together
               ,
               and
               the
               Sun
               shall
               be
               the
               Moons
               true
               place
               ,
               being
               added
               to
               what
               she
               had
               on
               the
               day
               of
               her
               change
               ;
               but
               far
               more
               readily
               ,
               and
               as
               exactly
               by
               the
               line
               of
               Numbers
               (
               or
               Rule
               of
               Three
               )
               say
               ,
               if
               29.
               540.
               require
               360.
               what
               6
               facit
               73
               1
               /
               4
               ,
               that
               is
               ,
               2
               signs
               ,
               13
               degrees
               ,
               15
               minutes
               ,
               to
               be
               added
               to
               14
               degrees
               in
               ♌
               ,
               and
               it
               makes
               27
               1
               /
               4
               in
               ♎
               ,
               the
               Moons
               place
               for
               that
               day
               .
               Or
               thus
               ,
               multiply
               the
               Moons
               age
               by
               4
               ,
               divide
               the
               product
               by
               10
               ,
               the
               quotient
               sheweth
               the
               signs
               ,
               and
               the
               remainder
               multiplied
               by
               3
               ,
               sheweth
               the
               degrees
               which
               you
               must
               adde
               to
               the
               Suns
               place
               on
               the
               day
               required
               ,
               and
               it
               shall
               be
               the
               Moons
               
               true
               place
               required
               for
               that
               day
               of
               her
               Age.
               
            
             
               
                 Example
                 .
              
               
                 Iuly
                 27.
                 the
                 Sun
                 and
                 Moon
                 is
                 in
                 Leo
                 14
                 degr
                 .
                 0.
                 
                 August
                 2.
                 being
                 6
                 days
                 ,
                 adde
                 the
                 Moons
                 motion
                 2.
                 13d
                 9
                 '
                 .
                 makes
                 being
                 counted
                 ,
                 
                   Virgo
                   ♍
                   ,
                   Libra
                
                 ♎
                 .
                 27
                 deg
                 .
                 9
                 the
                 place
                 required
                 ;
                 which
                 on
                 the
                 Rule
                 you
                 may
                 count
                 without
                 all
                 this
                 work
                 or
                 trouble
                 :
                 but
                 for
                 plainness
                 sake
                 I
                 am
                 constrained
                 so
                 to
                 do
                 .
                 Or
                 thus
                 .
              
            
          
           
             
               PROP.
               3.
               
            
             
               To
               finde
               the
               Moons
               hour
               .
            
             
               To
               do
               this
               ,
               you
               must
               do
               the
               work
               of
               the
               second
               Chapter
               ,
               and
               second
               and
               third
               Proposition
               ;
               where
               note
               that
               the
               Moons
               place
               found
               ,
               is
               to
               be
               used
               as
               the
               day
               of
               the
               moneth
               ,
               or
               Suns
               declination
               .
            
             
               
                 Example
                 .
              
               
                 The
                 Moon
                 being
                 27
                 degrees
                 in
                 Libra
                 ,
                 and
                 20
                 degrees
                 high
                 ,
                 I
                 finde
                 the
                 our
                 to
                 be
                 31
                 '
                 past
                 9
                 ,
                 if
                 on
                 the
                 east-side
                 ;
                 or
                 29
                 '
                 past
                 2
                 ,
                 if
                 on
                 the
                 west-side
                 ●f
                 the
                 Meridian
                 .
              
            
          
           
             
             
               PROP.
               4.
               
            
             
               To
               finde
               the
               true
               hour
               of
               the
               Night
               .
            
             
               Having
               found
               the
               Moons
               hour
               ,
               as
               before
               ,
               consider
               the
               Moons
               age
               ,
               then
               say
               by
               the
               Line
               of
               Numbers
               ,
               or
               Rule
               of
               Three
               ,
               if
               29.
               540
               part
               require
               24
               ,
               what
               shall
               6
               days
               require
               facit
               4
               hours
               and
               52
               minutes
               ?
               which
               taken
               from
               9.
               31.
               rest
               ●
               .
               29.
               the
               true
               time
               required
               .
            
             
               
                 Example
                 .
              
               
                 Moons
                 hour
                 —
                 9
                 31
              
               
                 Time
                 to
                 be
                 subst
                 .
                 ●
                 52
              
               
                 True
                 time
                 remain
                 3
                 39.
                 the
                 time
                 required
                 .
                 This
                 work
                 is
                 done
                 more
                 readily
                 by
                 the
                 two
                 lines
                 fitted
                 for
                 that
                 purpose
                 ;
                 for
                 look
                 for
                 6
                 the
                 Moons
                 age
                 in
                 one
                 ,
                 and
                 you
                 shall
                 finde
                 ●
                 hours
                 52
                 minutes
                 ,
                 the
                 time
                 to
                 be
                 substracted
                 in
                 the
                 other
                 .
              
            
          
        
         
           
           
             CHAP.
             VII
             .
          
           
             To
             finde
             the
             hour
             of
             the
             Night
             by
             the
             fixed
             Stars
             .
          
           
             FOr
             the
             doing
             of
             this
             ,
             I
             have
             made
             choice
             of
             twelve
             principal
             fixed
             Stars
             ,
             all
             within
             the
             two
             Tropicks
             ;
             many
             more
             might
             be
             added
             ,
             but
             these
             will
             very
             well
             serve
             the
             turn
             :
             The
             names
             of
             them
             ,
             and
             their
             right
             Ascension
             in
             hours
             and
             minutes
             ,
             is
             set
             on
             the
             Rule
             ,
             and
             the
             star
             is
             placed
             in
             his
             true
             Declination
             on
             ,
             or
             among
             the
             moneths
             ;
             and
             for
             to
             know
             the
             stars
             next
             to
             a
             Tutor
             ,
             a
             Celestial
             Globe
             ,
             or
             a
             Nocturnal
             ,
             of
             all
             the
             chief
             stars
             from
             the
             Pole
             to
             the
             Equinoctial
             ,
             and
             to
             be
             had
             at
             the
             Sun-Dial
             in
             the
             Mi●nories
             ,
             is
             the
             best
             ;
             the
             uses
             whereof
             do
             follow
             .
          
           
             First
             know
             the
             star
             you
             observe
             ,
             then
             observe
             his
             Altitude
             ,
             and
             laying
             the
             thread
             on
             the
             star
             by
             the
             second
             Chapter
             ,
             second
             and
             third
             Proposition
             ,
             get
             
             the
             stars
             hour
             ,
             then
             out
             of
             the
             righ●
             Ascension
             of
             the
             star
             ,
             take
             the
             righ●
             Ascension
             of
             the
             Sun
             ,
             (
             found
             by
             the
             fifth
             Proposition
             of
             the
             third
             Chapter
             )
             for
             that
             day
             ,
             and
             note
             the
             difference
             for
             this
             difference
             added
             to
             the
             star
             hours
             found
             ,
             shall
             shew
             the
             hour
             o●
             the
             night
             .
          
           
             
               Example
               .
            
             
               On
               the
               first
               of
               November
               I
               observe
               the
               Altitude
               of
               the
               Bulls-eye
               ,
               and
               find
               it
               to
               be
               30.
               then
               by
               second
               and
               third
               of
               the
               second
               Chapter
               ,
               I
               finde
               the
               hour
               to
               be
               7.
               54
               past
               the
               Meridian
               the
               Suns
               right
               Ascension
               that
               day
               finde
               to
               be
               15
               hours
               8
               minutes
               ,
               the
               stars
               right
               ascension
               is
               4
               hours
               16
               minutes
               ;
               which
               taken
               ,
               the
               greatest
               from
               the
               least
               ,
               by
               adding
               of
               24
               hours
               ,
               re●
               13.
               08.
               then
               7.
               54.
               the
               stars
               hou●
               added
               ,
               makes
               21.
               02.
               from
               which
               taking
               12
               hours
               ,
               rest
               9.
               02.
               the
               hour
               o●
               the
               night
               required
               .
               For
               more
               plainness
               ,
               note
               the
               work
               of
               two
               or
               thre●
               Examples
               .
            
             
               
               
                 Stars
                 right
                 Ascension
                 being
                 set
                 on
                 the
                 Rule
                 —
                 4
                 —
                 16
                 
                 Suns
                 right
                 Asc.
                 Nov.
                 1.
                 —
                 15
                 —
                 08
                 —
                 
                 Substraction
                 being
                 made
                 ,
                 by
                 adding
                 24
                 ,
                 remain
                 —
                 13
                 —
                 08
                 
                 To
                 which
                 you
                 must
                 adde
                 the
                 stars
                 hour
                 found
                 —
                 7
                 —
                 54
                 
                 Then
                 the
                 remainder
                 ,
                 taking
                 away
                 12
                 hours
                 ,
                 is
                 —
                 9
                 —
                 02
                 
                 the
                 true
                 hour
                 .
              
            
             
               
                 Again
                 
                   May
                   15
                
                 by
                 Arcturus
                 ,
                 at
                 50
                 0
                 '
                 high
                 westwards
                 .
              
               
                 The
                 right
                 Asc.
                 of
                 Arcturus
                 ,
                 is
                 14h●
                 —
                 0
                 '
                 
                 Suns
                 right
                 Asc.
                 May
                 15.
                 is
                 —
                 4
                 —
                 10
                 —
                 
                 The
                 right
                 Asc.
                 of
                 the
                 Sun
                 taken
                 from
                 the
                 R.
                 Asc.
                 of
                 the
                 star
                 ,
                 rest
                 ●
                 —
                 50
                 
                 The
                 stars
                 hour
                 at
                 50
                 degr
                 .
                 high
                 ,
                 found
                 to
                 be
                 —
                 02
                 —
                 13
                 —
                 
                 Which
                 being
                 added
                 to
                 the
                 difference
                 before
                 ,
                 makes
                 —
                 1●
                 —
                 03
                 or
                 —
                 11
                 —
                 03.
                 
              
            
             
               
               
                 Again
                 
                   January
                   5.
                   
                
                 by
                 the
                 Great-dog
                 at
                 15
                 degrees
                 high
                 east
                 .
              
               
                 The
                 right
                 Asc.
                 of
                 Great-dog
                 ,
                 is
                 06-29
                 
                 The
                 right
                 Asc.
                 of
                 the
                 Sun
                 ,
                 is
                 19
                 —
                 50
                 —
              
               
                 Substr
                 .
                 made
                 ,
                 by
                 adding
                 24
                 h.
                 is
                 10-39
                 Stars
                 hour
                 at
                 15
                 degr
                 .
                 high
                 ,
                 is
                 9
                 —
                 32
                 Ante
                 M.
                 or
                 P.
                 Septen
                 .
                 Which
                 added
                 to
                 the
                 difference
                 found
                 ,
                 and
                 12
                 substracted
                 ,
                 remains
                 8.
                 11
                 
                 '
                 .
                 for
                 the
                 true
                 Hour
                 of
                 the
                 Night
                 required
                 to
                 be
                 found
                 ,
                 and
                 so
                 of
                 any
                 other
                 star
                 se●
                 down
                 in
                 the
                 Rule
                 ,
                 as
                 by
                 the
                 trial
                 and
                 pr●ctice
                 ,
                 will
                 prove
                 easie
                 and
                 ready
                 to
                 the
                 ingenuous
                 practitioner
                 .
              
               
                 But
                 by
                 the
                 line
                 of
                 24
                 ,
                 or
                 twice
                 12
                 hours
                 ,
                 and
                 the
                 help
                 of
                 a
                 pair
                 of
                 Compasses
                 ,
                 you
                 may
                 perform
                 it
                 withou●
                 writing
                 it
                 down
                 ,
                 thus
                 :
                 Take
                 the
                 righ●
                 Ascension
                 of
                 the
                 Sun
                 ,
                 out
                 of
                 that
                 lin●
                 of
                 hours
                 between
                 your
                 Compasses
                 (
                 being
                 always
                 counted
                 under
                 12
                 )
                 an●
                 set
                 the
                 same
                 from
                 the
                 right
                 Ascensio●
                 of
                 the
                 star
                 ,
                 toward
                 a
                 lesser
                 number
                 ,
                 o●
                 the
                 beginning
                 of
                 the
                 hours
                 ,
                 and
                 th●
                 
                 point
                 shall
                 stay
                 at
                 the
                 remainder
                 that
                 is
                 to
                 be
                 added
                 to
                 the
                 stars
                 hour
                 found
                 ,
                 then
                 open
                 the
                 Compasses
                 from
                 thence
                 to
                 the
                 beginning
                 of
                 the
                 hours
                 ,
                 and
                 adde
                 that
                 to
                 the
                 stars
                 hour
                 found
                 ,
                 and
                 it
                 shall
                 reach
                 to
                 the
                 hour
                 of
                 the
                 night
                 required
                 .
              
            
          
           
             
               Example
               .
            
             
               Feb.
               6.
               1660.
               by
               Arcturus
               20
               degrees
               high
               :
               Take
               10
               hours
               1
               minute
               between
               your
               Compasses
               ,
               and
               set
               it
               from
               14
               hours
               ,
               or
               2
               hours
               beyond
               12
               ,
               and
               it
               shall
               stay
               at
               3.
               59
               :
               then
               take
               3.
               59
               ,
               and
               adde
               it
               to
               6.
               24
               ,
               the
               stars
               hour
               at
               20
               degrees
               high
               ,
               and
               it
               shall
               be
               10.
               23
               ,
               the
               hour
               of
               the
               night
               required
               .
            
          
        
         
           
             CHAP.
             VIII
             .
          
           
             
               To
               finde
               the
               Amplitude
               or
               Azimuth
               of
               the
               fixed
               Stars
               ;
               also
               their
               rising
               ,
               setting
               ,
               and
               southing
               .
            
             
               1.
               
               FIrst
               for
               the
               Amplitude
               ,
               take
               the
               stars
               declination
               from
               the
               particular
               
               Scale
               of
               Altitudes
               ,
               and
               lay
               it
               from
               90
               in
               the
               Azimuth-line
               ,
               and
               it
               shall
               shew
               his
               amplitude
               from
               East
               or
               West
               toward
               South
               or
               North
               ,
               according
               to
               the
               declination
               ,
               and
               time
               of
               day
               ,
               morning
               or
               evening
               .
               The
               same
               work
               is
               for
               the
               sun
               .
            
             
               
                 Example
                 .
              
               
                 The
                 Bulls-eye
                 hath
                 25.
                 54
                 degrees
                 of
                 amplitude
                 ,
                 so
                 hath
                 the
                 Sun
                 at
                 15
                 degrees
                 48
                 minutes
                 of
                 declination
                 .
              
            
          
           
             
               2.
               
               To
               finde
               the
               Azimuth
               ,
               work
               as
               you
               did
               for
               the
               Sun
               at
               the
               same
               declination
               the
               star
               hath
               ,
               by
               Chapter
               5.
               and
               you
               shall
               have
               your
               desire
               .
            
             
               
                 Example
                 .
              
               
                 December
                 24.
                 at
                 6
                 degrees
                 high
                 ,
                 by
                 the
                 Bulls-eye
                 I
                 finde
                 the
                 Azimuth
                 to
                 be
                 107.
                 53.
                 from
                 the
                 south
                 .
              
            
          
           
             
               3
               To
               finde
               the
               stars
               rising
               and
               setting
               ,
               lay
               the
               thread
               on
               the
               star
               among
               the
               moneths
               ,
               and
               in
               the
               line
               of
               hours
               it
               shews
               the
               stars
               rising
               and
               setting
               ,
               as
               you
               counted
               for
               the
               Sun
               ;
               but
               yet
               note
               this
               is
               not
               the
               time
               in
               common
               hours
               ,
               but
               is
               thus
               found
               :
               Adde
               the
               
               complement
               of
               the
               Suns
               Ascension
               ,
               and
               the
               stars
               right
               Ascension
               ,
               and
               the
               stars
               hour
               last
               found
               together
               ,
               and
               the
               Sun
               ,
               if
               less
               than
               12
               ;
               or
               the
               remain
               12
               being
               substracted
               ,
               shall
               be
               the
               time
               of
               his
               rising
               in
               common
               hours
               ;
               but
               for
               his
               setting
               ,
               adde
               the
               stars
               setting
               last
               found
               to
               the
               other
               numbers
               ,
               and
               the
               sum
               or
               difference
               shall
               be
               the
               setting
               .
            
             
               
                 Example
                 .
              
               
                 For
                 the
                 Bulls-eye
                 on
                 the
                 23
                 of
                 December
                 ,
                 it
                 riseth
                 at
                 2
                 in
                 the
                 afternoon
                 ,
                 and
                 sets
                 at
                 4.
                 46
                 in
                 the
                 morning
                 .
              
            
          
           
             
               4.
               
               To
               finde
               the
               time
               of
               the
               southing
               of
               any
               star
               on
               the
               Rule
               ,
               or
               any
               other
               whose
               right
               ascension
               and
               declination
               is
               known
               ,
               Substract
               the
               Suns
               right
               ascension
               from
               the
               stars
               ,
               increased
               by
               24
               ,
               when
               you
               cannot
               do
               without
               ,
               and
               the
               remainder
               ,
               if
               less
               than
               12
               ,
               is
               the
               time
               required
               ,
               in
               the
               afternoon
               or
               night
               before
               12
               ;
               but
               if
               there
               remain
               more
               than
               12
               ,
               substract
               12
               ,
               and
               the
               residue
               is
               the
               time
               from
               mid-night
               to
               mid-day
               following
               .
            
             
               
               
                 Example
                 .
              
               
                 Lyons-heart
                 on
                 the
                 tenth
                 of
                 March
                 ,
                 the
                 Suns
                 Ascension
                 is
                 0
                 2
                 '
                 .
              
               
                 Lyons-heart
                 whole
                 right
                 asc
                 .
                 is
                 9
                 50
                 '
                 Time
                 of
                 southing
                 is
                 9
                 48
                 '
                 at
                 night
                 .
              
            
          
           
             
               5.
               
               To
               finde
               how
               long
               any
               Star
               will
               be
               above
               the
               Horizon
               .
            
             
               Lay
               the
               thread
               to
               the
               star
               ,
               and
               in
               the
               hour-line
               it
               sheweth
               the
               ascensional
               difference
               ,
               counting
               from
               90
               ;
               then
               note
               if
               the
               star
               have
               North
               declination
               ,
               adde
               that
               to
               6
               hours
               ,
               and
               the
               sum
               is
               half
               the
               time
               ;
               if
               south
               ,
               substract
               it
               from
               6
               ,
               and
               the
               residue
               is
               half
               the
               time
               ;
               and
               the
               complement
               of
               each
               to
               24
               being
               doubled
               ,
               is
               the
               whole
               Nocturnal
               Arch
               under
               the
               Horizon
               .
            
             
               
                 Example
                 .
              
               
                 For
                 the
                 Bulls-eye
                 ,
                 his
                 Ascensional
                 difference
                 will
                 be
                 found
                 to
                 be
                 one
                 hour
                 ,
                 23
                 minutes
                 ,
                 which
                 added
                 to
                 6
                 hours
                 and
                 doubled
                 ,
                 makes
                 14.
                 46
                 ,
                 the
                 Diurnal
                 Ark
                 of
                 the
                 Star
                 ,
                 and
                 the
                 residue
                 from
                 24
                 is
                 9.
                 14.
                 for
                 the
                 
                 Nocturnal
                 Ark
                 ,
                 or
                 the
                 time
                 of
                 its
                 being
                 under
                 the
                 Horizon
                 .
              
            
          
        
         
           
             CHAP.
             IX
             .
          
           
             
               To
               perform
               the
               fore-going
               work
               in
               any
               latitude
               ,
               as
               rising
               ,
               amplitude
               ,
               ascensional
               difference
               ,
               latitude
               ,
               hour
               ,
               and
               azimuth
               ,
               wherein
               I
               shall
               give
               onely
               the
               rule
               ,
               and
               leave
               out
               the
               examples
               for
               brevity
               sake
               .
            
             
               1.
               
               FOr
               the
               rising
               ,
               and
               setting
               ,
               and
               ascensional
               difference
               ,
               being
               all
               one
               ,
               do
               thus
               :
               Take
               the
               Suns
               declination
               out
               of
               the
               general
               Scale
               of
               Altitudes
               ,
               then
               set
               one
               foot
               of
               the
               Compasses
               in
               the
               colatitude
               on
               the
               same
               scale
               ,
               and
               with
               the
               other
               lay
               the
               thred
               to
               the
               nighest
               distance
               ;
               then
               the
               thred
               so
               laid
               ,
               take
               the
               nighest
               distance
               from
               the
               latitude
               to
               the
               thread
               ,
               with
               that
               distance
               set
               one
               foot
               in
               the
               Suns
               declination
               ,
               counted
               from
               90
               toward
               the
               center
               ,
               and
               the
               thread
               laid
               to
               the
               nearest
               distance
               ,
               shall
               in
               the
               degrees
               
               shew
               the
               ascensional
               difference
               required
               ,
               counting
               from
               90
               at
               the
               head
               toward
               the
               end
               of
               the
               Rule
               ;
               and
               if
               you
               reduce
               those
               degrees
               and
               minutes
               to
               time
               ,
               you
               have
               the
               rising
               and
               setting
               before
               and
               after
               6
               ,
               according
               to
               the
               declination
               and
               time
               of
               the
               year
               .
            
          
           
             
               2.
               
               To
               finde
               the
               Suns
               amplitude
               .
            
             
               Take
               the
               Suns
               declination
               ,
               and
               setting
               one
               foot
               in
               the
               colatitude
               ,
               with
               the
               other
               lay
               the
               thread
               to
               the
               nearest
               distance
               ,
               and
               on
               the
               degrees
               it
               sheweth
               the
               Suns
               amplitude
               at
               rising
               or
               setting
               ,
               counting
               as
               be●ore
               from
               90
               to
               the
               left
               end
               of
               the
               Rule
               .
            
          
           
             
               3.
               
               Having
               amplitude
               and
               declination
               ,
               to
               finde
               the
               latitude
               .
            
             
               Take
               the
               declination
               from
               the
               general
               scale
               ,
               and
               set
               one
               foot
               in
               the
               amplitude
               ,
               the
               thread
               laid
               to
               the
               nearest
               distance
               in
               the
               line
               of
               degrees
               ,
               it
               sheweth
               the
               complement
               of
               the
               latitude
               required
               ,
               or
               the
               converse
               .
            
          
           
             
             
               4.
               
               Having
               latitude
               ,
               Suns
               declination
               and
               altitude
               ,
               to
               find
               the
               height
               at
               6
               ,
               and
               then
               at
               any
               other
               time
               of
               the
               day
               and
               year
               .
            
             
               Count
               the
               declination
               in
               the
               degrees
               from
               90
               toward
               the
               end
               ,
               thereto
               lay
               the
               thread
               ,
               the
               least
               distance
               from
               which
               to
               the
               latitude
               in
               the
               general
               Scale
               ,
               shall
               be
               the
               Suns
               height
               at
               6
               in
               the
               summer
               ,
               or
               his
               depression
               in
               the
               winter
               .
               The
               Compasses
               standing
               at
               this
               distance
               ,
               take
               measure
               on
               the
               general
               Scale
               of
               altitudes
               ,
               from
               the
               beginning
               at
               the
               pin
               towards
               90
               ,
               keeping
               one
               point
               there
               ,
               open
               the
               other
               to
               the
               Suns
               altitude
               ,
               thus
               have
               you
               substracted
               the
               height
               at
               6
               ,
               out
               of
               the
               Suns
               altitude
               ;
               but
               in
               winter
               you
               must
               adde
               the
               depression
               at
               6
               ,
               which
               is
               all
               one
               at
               the
               same
               declination
               with
               his
               height
               at
               6
               in
               summer
               ,
               and
               that
               is
               done
               thus
               :
               Put
               one
               point
               of
               the
               Compasses
               so
               set
               in
               the
               general
               Scale
               to
               the
               Suns
               Altitude
               ,
               then
               turn
               the
               other
               outwards
               toward
               90
               ,
               
               there
               keep
               it
               ,
               then
               open
               the
               Compasses
               to
               the
               beginning
               of
               the
               Scale
               ,
               then
               have
               you
               added
               it
               to
               the
               Suns
               altitude
               ;
               having
               this
               distance
               ,
               set
               one
               foot
               in
               the
               colatitude
               on
               the
               general
               Scale
               ,
               lay
               the
               thread
               to
               the
               nearest
               distance
               ;
               the
               thread
               so
               laid
               ,
               take
               the
               nearest
               distance
               from
               90
               to
               the
               thred
               ,
               then
               set
               one
               foot
               in
               the
               declination
               ,
               counted
               from
               90
               ,
               and
               on
               the
               degrees
               it
               sheweth
               the
               hour
               from
               6
               ,
               reckoning
               from
               the
               head
               ,
               or
               from
               12
               ,
               counting
               from
               the
               end
               of
               the
               Rule
               .
            
             
               I
               shall
               make
               all
               more
               plain
               ,
               by
               making
               three
               Propositions
               of
               it
               ,
               thus
               :
            
             
               
                 Prop.
                 1.
                 
                 To
                 finde
                 the
                 hour
                 in
                 the
                 Aequinoctial
                 .
              
               
                 Take
                 the
                 Altitude
                 from
                 the
                 beginning
                 of
                 the
                 general
                 Scale
                 of
                 altitudes
                 ,
                 and
                 set
                 one
                 foot
                 in
                 the
                 colatitude
                 ,
                 the
                 thread
                 laid
                 to
                 the
                 nearest
                 distance
                 (
                 with
                 the
                 other
                 foot
                 )
                 in
                 the
                 degrees
                 ,
                 shall
                 shew
                 the
                 hour
                 from
                 6
                 ,
                 counting
                 from
                 90
                 ,
                 and
                 allowing
                 for
                 every
                 15d
                 1
                 hour
                 ,
                 and
                 4
                 min
                 :
                 for
                 every
                 degree
                 .
              
            
             
               
               
                 Prop.
                 2.
                 
                 To
                 finde
                 it
                 at
                 just
                 6.
                 
              
               
                 Is
                 before
                 exprest
                 by
                 the
                 converse
                 of
                 the
                 first
                 part
                 of
                 the
                 fourth
                 ,
                 which
                 I
                 shall
                 again
                 repeat
                 .
              
            
             
               
                 Prop.
                 3.
                 
                 To
                 finde
                 it
                 at
                 any
                 time
                 do
                 thus
                 .
              
               
                 Count
                 the
                 Suns
                 declination
                 in
                 the
                 degrees
                 ,
                 thereunto
                 lay
                 the
                 thred
                 ,
                 the
                 least
                 distance
                 ,
                 to
                 which
                 from
                 latitude
                 in
                 the
                 general
                 Scale
                 ,
                 shall
                 be
                 the
                 Suns
                 altitude
                 at
                 6
                 ;
                 which
                 distance
                 in
                 summer
                 you
                 must
                 substract
                 from
                 ,
                 but
                 in
                 winter
                 you
                 must
                 add
                 to
                 the
                 Suns
                 present
                 altitude
                 ;
                 having
                 that
                 distance
                 ,
                 set
                 one
                 foot
                 in
                 the
                 coaltitude
                 ,
                 with
                 the
                 other
                 lay
                 the
                 thread
                 to
                 the
                 neerest
                 distance
                 ,
                 take
                 again
                 the
                 neerest
                 distance
                 from
                 90
                 to
                 the
                 thread
                 ,
                 then
                 set
                 one
                 foot
                 in
                 the
                 Suns
                 diclination
                 counted
                 from
                 90
                 ,
                 and
                 lay
                 the
                 thread
                 to
                 the
                 neerest
                 distance
                 ,
                 and
                 in
                 the
                 degrees
                 it
                 shall
                 shew
                 the
                 hour
                 required
                 .
              
               
                 
                   Example
                   .
                
                 
                   At
                   10
                   declination
                   north
                   ,
                   and
                   30
                   high
                   ,
                   latitude
                   51.
                   32
                   ,
                   the
                   hour
                   is
                   found
                   
                   to
                   be
                   8.
                   25
                   ,
                   counting
                   90
                   for
                   6
                   ,
                   and
                   so
                   forward
                   .
                   Again
                   at
                   20
                   degrees
                   of
                   declination
                   South
                   ,
                   and
                   10
                   degrees
                   of
                   altitude
                   ,
                   I
                   finde
                   the
                   hour
                   in
                   the
                   same
                   latitude
                   to
                   be
                   17
                   minutes
                   past
                   9.
                   
                
              
            
             
               
                 Having
                 latitude
                 ,
                 delination
                 ,
                 and
                 altitude
                 ,
                 to
                 finde
                 the
                 Suns
                 Azimuth
                 .
              
               
                 Take
                 the
                 sine
                 of
                 the
                 declination
                 ,
                 put
                 one
                 foot
                 in
                 the
                 latitude
                 ,
                 the
                 thread
                 laid
                 to
                 the
                 neerest
                 distance
                 :
                 in
                 the
                 degrees
                 ,
                 it
                 sheweth
                 the
                 Suns
                 height
                 at
                 due
                 East
                 or
                 West
                 ,
                 which
                 you
                 must
                 in
                 summer
                 substract
                 from
                 the
                 Suns
                 altitude
                 ,
                 as
                 before
                 on
                 the
                 general
                 Scale
                 of
                 Altitudes
                 ,
                 with
                 which
                 distance
                 put
                 one
                 foot
                 in
                 the
                 colatitude
                 ,
                 and
                 lay
                 the
                 thread
                 to
                 the
                 neerest
                 distance
                 ,
                 then
                 take
                 the
                 neerest
                 distance
                 from
                 the
                 sine
                 of
                 the
                 latitude
                 ,
                 fit
                 that
                 again
                 in
                 the
                 colatitude
                 ,
                 and
                 the
                 thread
                 laid
                 to
                 the
                 heerest
                 distance
                 ,
                 in
                 the
                 degrees
                 shall
                 shew
                 the
                 Suns
                 Azimuth
                 required
                 .
              
            
          
           
             
               6.
               
               But
               in
               winter
               you
               must
               do
               thus
               :
            
             
               By
               the
               second
               Proposition
               of
               the
               
               ninth
               Chapter
               ,
               finde
               the
               Suns
               Amplitude
               for
               that
               day
               ,
               then
               take
               the
               altitude
               from
               the
               general
               Scale
               of
               altitudes
               ,
               and
               putting
               one
               point
               in
               colatitude
               ,
               lay
               the
               thread
               to
               the
               neerest
               distance
               ,
               then
               the
               neerest
               distance
               from
               the
               latitude
               must
               be
               added
               to
               the
               Suns
               Amplitude
               ;
               this
               distance
               so
               added
               must
               be
               set
               from
               the
               coaltitude
               ,
               and
               the
               thread
               laid
               to
               the
               highest
               distance
               ,
               and
               in
               the
               line
               of
               degrees
               ,
               it
               gives
               the
               Azimuth
               from
               south
               ,
               counting
               from
               the
               end
               of
               the
               rule
               ,
               or
               from
               the
               East
               or
               West
               ,
               counting
               from
               the
               head
               or
               90
               degrees
               .
            
             
               
                 Example
                 .
              
               
                 At
                 15
                 degrees
                 of
                 declination
                 and
                 10
                 altitude
                 ,
                 latitude
                 51.
                 32.
                 the
                 Azimuth
                 is
                 49.
                 20.
                 from
                 the
                 South
                 ,
                 or
                 40
                 degrees
                 and
                 40
                 '
                 from
                 East
                 or
                 West
                 .
              
            
          
        
         
           
           
             CHAP.
             X.
             
          
           
             To
             finde
             all
             the
             necessary
             quesita
             for
             any
             erect
             declining
             Sun-dial
             both
             ,
             particularly
             and
             general
             ,
             by
             the
             lines
             on
             the
             Dial
             side
             ,
             also
             by
             numbers
             ,
             sines
             ,
             and
             tangents
             artificial
             ,
             being
             Logarithms
             on
             a
             Rule
             .
          
           
             
               1.
               
               First
               a
               particular
               for
               the
               Substile
               .
            
             
               COunt
               the
               plains
               declination
               on
               ●he
               Azimuth
               scale
               ,
               from
               90
               toward
               the
               end
               ,
               and
               thereunto
               lay
               the
               thread
               ,
               in
               the
               line
               of
               degrees
               it
               shews
               the
               distance
               of
               the
               substile
               from
               12.
               
            
             
               
                 Example
                 .
              
               
                 At
                 10
                 degrees
                 declination
                 ,
                 I
                 find
                 7.
                 51.
                 for
                 the
                 substile
                 .
              
            
          
           
             
               2.
               
               For
               the
               height
               of
               the
               stile
               above
               the
               substile
               .
            
             
               Take
               the
               Plaines
               Declination
               from
               90
               in
               the
               Azimuth
               line
               ,
               but
               counted
               from
               the
               South
               end
               ,
               between
               your
               compasses
               :
               and
               measure
               it
               in
               the
               particular
               scale
               of
               altitudes
               ,
               and
               
               it
               shall
               give
               the
               height
               of
               the
               stile
               required
               .
            
             
               
                 Example
                 .
              
               
                 At
                 30
                 declination
                 is
                 32.
                 35.
                 
              
            
          
           
             
               3.
               
               For
               the
               inclination
               of
               Meridians
               :
            
             
               Count
               the
               substile
               on
               the
               particuler
               scale
               of
               Altitudes
               ,
               and
               take
               that
               distance
               between
               your
               compasses
               ,
               measure
               this
               distance
               on
               the
               Azimuth
               line
               from
               90
               toward
               the
               end
               ,
               and
               counting
               that
               way
               it
               sheweth
               the
               inclination
               of
               Meridians
               required
               .
            
             
               
                 Example
                 .
              
               
                 At
                 15
                 the
                 substile
                 ,
                 the
                 inclination
                 of
                 Meridians
                 will
                 be
                 found
                 to
                 be
                 24.
                 36.
                 
              
            
          
           
             
               4.
               
               To
               finde
               the
               Angle
               of
               6
               from
               12.
               
            
             
               Take
               the
               plaines
               declination
               from
               the
               particular
               scale
               of
               altitudes
               ,
               and
               lay
               it
               from
               90
               on
               the
               Azimuth
               scale
               ,
               and
               to
               the
               Compasses
               point
               lay
               the
               thread
               :
               then
               on
               the
               line
               of
               degrees
               you
               have
               the
               complement
               of
               6
               from
               12
               ,
               counting
               from
               60
               toward
               the
               end
               .
            
             
             
               Note
               this
               Rule
               (
               as
               this
               line
               is
               drawn
               )
               doth
               not
               give
               this
               Angle
               exactly
               ,
               neither
               will
               it
               be
               worth
               the
               while
               to
               delineate
               another
               line
               for
               this
               purpose
               .
               But
               if
               it
               be
               required
               it
               may
               be
               done
               ,
               but
               I
               rather
               prefer
               this
               help
               ,
               the
               greatest
               error
               is
               about
               the
               space
               of
               45
               minutes
               of
               the
               first
               degree
               in
               the
               particular
               scale
               of
               altitudes
               ;
               so
               that
               if
               you
               conceive
               those
               45
               minutes
               to
               be
               divided
               as
               the
               particular
               scale
               of
               altitudes
               is
               ,
               like
               a
               natural
               sine
               ,
               and
               if
               your
               declination
               be
               30
               ,
               then
               take
               half
               the
               space
               of
               the
               45
               minutes
               less
               ,
               and
               that
               shall
               be
               the
               true
               distance
               to
               lay
               on
               the
               Azimuth
               line
               from
               90
               whereunto
               to
               lay
               the
               thread
               .
            
             
               
                 Example
                 .
              
               
                 A
                 plaine
                 declining
                 30
                 degrees
                 ,
                 the
                 angle
                 will
                 be
                 found
                 to
                 be
                 32.
                 21.
                 whose
                 complement
                 57.
                 49.
                 is
                 the
                 angle
                 required
                 .
              
            
          
           
             
               5.
               
               To
               perform
               the
               same
               generally
               by
               the
               general
               scale
               of
               altitudes
               ;
               and
               first
               for
               the
               stile
               .
            
             
             
               Lay
               the
               thred
               to
               the
               complement
               of
               the
               latitude
               ,
               counted
               in
               the
               degrees
               from
               the
               head
               toward
               the
               end
               ,
               then
               the
               nightest
               distance
               from
               the
               complement
               of
               the
               plaines
               declination
               to
               the
               thread
               ,
               taken
               and
               measured
               on
               the
               general
               scale
               ,
               from
               the
               center
               ,
               shall
               be
               the
               stiles
               height
               required
               .
            
          
           
             
               6.
               
               To
               finde
               the
               inclination
               of
               Meridians
            
             
               Take
               the
               plaines
               declination
               ,
               from
               the
               general
               scale
               ,
               and
               fit
               it
               in
               the
               complement
               of
               the
               stiles
               elevation
               ,
               and
               lay
               the
               thread
               to
               the
               neerest
               distance
               ,
               and
               on
               the
               degrees
               it
               sheweth
               the
               inclination
               of
               Meridians
               required
               .
            
          
           
             
               7.
               
               For
               the
               substile
               ,
            
             
               Count
               in
               the
               inclination
               of
               Meridians
               on
               the
               degrees
               from
               90
               ,
               and
               thereto
               lay
               the
               thread
               ,
               then
               take
               the
               least
               distance
               from
               the
               latitudes
               complement
               to
               the
               thread
               ,
               set
               one
               foot
               of
               that
               distance
               in
               90
               ,
               and
               lay
               the
               thread
               to
               the
               neerest
               distance
               ,
               and
               in
               the
               
               degrees
               it
               shall
               shew
               the
               substile
               from
               12
               required
               .
            
          
           
             
               8.
               
               For
               the
               angle
               of
               6
               from
               12.
               
            
             
               Take
               the
               side
               of
               the
               square
               ,
               or
               the
               measure
               of
               the
               parallel
               from
               12
               ,
               and
               fit
               it
               in
               the
               cosine
               of
               the
               latitude
               ,
               and
               lay
               the
               thread
               to
               the
               nighest
               distance
               ,
               then
               take
               out
               the
               nearest
               distance
               from
               the
               sine
               of
               the
               latitude
               to
               the
               thread
               ,
               then
               fit
               that
               over
               in
               the
               sine
               of
               90
               ,
               and
               to
               the
               nearest
               distance
               lay
               the
               thread
               ,
               then
               take
               the
               nearest
               distance
               from
               the
               sine
               of
               the
               plains
               declination
               to
               the
               thread
               ,
               and
               it
               shall
               reach
               on
               the
               parallel
               line
               ,
               or
               side
               of
               the
               square
               ,
               from
               the
               Horizon
               to
               6
               a
               clock
               line
               required
               .
            
             
               
                 Four
                 Canons
                 to
                 work
                 the
                 same
                 by
                 the
                 artificial
                 sines
                 &
                 tangents
                 .
              
               
                 Inclination
                 of
                 Meridians
                 .
              
               
                 As
                 the
                 Sine
                 of
                 the
                 latitude
                 ,
                 
                 To
                 the
                 Sine
                 of
                 90
                 :
                 
                 So
                 the
                 Tangent
                 of
                 the
                 Declination
                 ,
                 
                 
                 To
                 the
                 Tangent
                 of
                 inclination
                 of
                 Meridians
                 .
              
            
             
               
                 Stiles
                 Elevation
                 .
              
               
                 As
                 the
                 Sine
                 of
                 90
                 ,
                 
                 To
                 the
                 Cosine
                 of
                 the
                 Declination
                 :
                 
                 So
                 the
                 Cosine
                 of
                 the
                 latitude
                 ,
                 
                 To
                 the
                 Sine
                 of
                 the
                 Stiles
                 elevation
                 ,
              
            
             
               
                 Substile
                 from
                 12.
                 
              
               
                 As
                 the
                 Sine
                 of
                 90
                 ,
                 
                 To
                 the
                 Sine
                 of
                 the
                 Declination
                 :
                 
                 So
                 the
                 Cotangent
                 of
                 Latitude
                 ,
                 
                 To
                 Tangent
                 of
                 the
                 Substile
                 from
                 12
                 ▪
              
            
             
               
                 For
                 6
                 and
                 12.
                 
              
               
                 As
                 the
                 Co●tangent
                 of
                 the
                 Latitude
                 ,
                 
                 To
                 the
                 Sine
                 of
                 90
                 :
                 
                 So
                 is
                 the
                 sine
                 of
                 Declination
                 ,
                 
                 To
                 the
                 Cotangent
                 of
                 6
                 from
                 12.
                 
              
            
             
               
                 For
                 the
                 hours
                 .
              
               
                 As
                 the
                 Sine
                 of
                 90
                 ,
                 
                 To
                 the
                 Sine
                 of
                 the
                 Stiles
                 height
                 :
                 
                 So
                 the
                 Tangent
                 of
                 the
                 hour
                 from
                 the
                 proper
                 Meridian
                 ,
                 
                 
                 To
                 the
                 Tangent
                 of
                 the
                 hour
                 from
                 the
                 Substile
                 .
              
               
                 The
                 way
                 to
                 work
                 these
                 Canons
                 on
                 the
                 Sines
                 and
                 Tangents
                 ,
                 is
                 generally
                 thus
                 :
                 As
                 first
                 ,
                 for
                 the
                 inclination
                 of
                 Meridians
                 ,
                 set
                 one
                 point
                 in
                 the
                 Sine
                 of
                 the
                 latitude
                 ,
                 open
                 the
                 other
                 to
                 the
                 Sine
                 of
                 90
                 ,
                 that
                 extent
                 applied
                 the
                 same
                 way
                 ,
                 from
                 the
                 Tangent
                 of
                 the
                 Plains
                 declination
                 ,
                 will
                 reach
                 to
                 the
                 Tangent
                 of
                 the
                 inclination
                 of
                 Meridians
                 required
                 .
              
            
          
        
         
           
             CHAP.
             XI
             .
          
           
             To
             draw
             a
             Horizontal
             Dyal
             to
             any
             latitude
             .
          
           
             FIrst
             draw
             a
             streight
             line
             for
             12
             ,
             as
             the
             line
             A
             B
             ,
             then
             make
             a
             point
             in
             that
             line
             for
             a
             Center
             ,
             as
             at
             C
             ,
             then
             through
             the
             Center
             C
             ,
             raise
             a
             perpendicular
             to
             A
             B
             ,
             for
             the
             two
             six
             a
             clock
             hour-lines
             ,
             as
             the
             line
             D
             E
             ;
             then
             draw
             two
             occult
             lines
             parallel
             to
             A
             B
             ,
             
             as
             large
             as
             the
             Plain
             will
             give
             leave
             ,
             as
             D
             E
             ,
             and
             E
             G
             then
             fit
             C
             D
             in
             the
             Sine
             of
             the
             Latitude
             ,
             in
             the
             general
             Scale
             ,
             and
             lay
             the
             thread
             to
             the
             nighest
             distance
             ,
             then
             take
             the
             nearest
             distance
             from
             90
             to
             the
             thread
             ,
             and
             set
             it
             from
             D
             and
             E
             in
             the
             two
             occult
             lines
             ,
             to
             F
             and
             G
             ,
             and
             draw
             the
             line
             F
             and
             G
             parallel
             to
             the
             two
             sixes
             ,
             (
             or
             make
             use
             of
             the
             Sines
             on
             the
             other
             side
             ,
             thus
             :
             Fit
             A
             D
             ,
             or
             C
             D
             in
             the
             Sine
             of
             the
             latitude
             ,
             and
             take
             out
             the
             Sine
             of
             90
             ,
             and
             lay
             it
             as
             before
             from
             D
             and
             E
             ,
             )
             then
             fit
             D
             F
             ,
             or
             E
             G
             in
             the
             Tangent
             of
             45
             degrees
             ,
             on
             the
             other
             side
             of
             the
             Rule
             ,
             and
             lay
             off
             15
             ,
             30
             ,
             and
             45
             ,
             for
             every
             whole
             hour
             ,
             or
             every
             3
             degrees
             and
             45
             minutes
             for
             every
             quarter
             ,
             from
             D
             and
             E
             ,
             toward
             F
             and
             G
             ,
             for
             7
             ,
             8
             9
             ,
             and
             for
             3
             ,
             4
             ,
             and
             5
             a
             clock
             hour
             points
             .
             Lastly
             ,
             set
             C
             D
             ,
             or
             B
             E
             in
             the
             Tangent
             of
             45
             ,
             and
             lay
             the
             same
             points
             of
             15
             ,
             30
             ,
             45
             ,
             both
             wayes
             ,
             from
             B
             or
             12
             ,
             for
             10
             ,
             11
             ,
             and
             1
             ,
             2
             ,
             and
             to
             all
             those
             points
             draw
             lines
             for
             the
             true
             hour-lines
             
             required
             ,
             for
             laying
             down
             the
             Stiles
             height
             ;
             if
             you
             take
             the
             latitudes
             complement
             ,
             out
             of
             the
             Tangent-line
             as
             the
             Sector
             stood
             ,
             to
             prick
             the
             noon
             hours
             ,
             and
             set
             it
             on
             the
             line
             D
             F
             ,
             or
             E
             G
             ,
             from
             D
             or
             E
             downwards
             from
             D
             to
             H
             ,
             it
             will
             shew
             you
             where
             to
             draw
             C
             H
             for
             the
             Stile
             ,
             then
             to
             those
             lines
             set
             figures
             ,
             and
             plant
             the
             Dial
             Horizontal
             ,
             and
             the
             Stile
             perpendicular
             ,
             and
             right
             north
             and
             south
             ,
             and
             it
             shall
             shew
             when
             the
             sun
             shineth
             ,
             the
             true
             hour
             of
             the
             day
             .
             Note
             well
             the
             figure
             following
             .
          
        
         
           
             CHAP.
             XII
             .
          
           
             To
             draw
             a
             Vertical
             ,
             Direct
             ,
             South
             ,
             or
             North
             Dyal
             .
          
           
             FIrst
             draw
             a
             perpendicular
             line
             for
             12
             a
             clock
             ,
             then
             in
             that
             line
             at
             the
             upper
             end
             ,
             in
             the
             south
             plain
             :
             and
             at
             the
             lower
             end
             in
             a
             north
             plain
             ,
             appoint
             a
             place
             for
             the
             center
             ,
             through
             which
             point
             cross
             it
             at
             right
             angles
             ,
             
             
               
                 A
                 Horizontall
                 Diall
              
            
             
               
                 A
                 South
                 Diall
              
            
             
             
             for
             6
             and
             6
             ,
             as
             you
             did
             in
             the
             Horizontal
             Plain
             ,
             as
             the
             lines
             A
             B
             ,
             and
             C
             D
             ,
             on
             each
             side
             12
             make
             two
             parallels
             ,
             as
             in
             the
             Horizontal
             ,
             then
             take
             A
             D
             the
             parallel
             ,
             and
             fit
             it
             in
             the
             sine
             of
             the
             latitudes
             complement
             ,
             and
             take
             out
             the
             sine
             of
             90
             and
             90
             ,
             and
             lay
             it
             in
             the
             parallels
             from
             D
             and
             C
             ,
             to
             E
             and
             F
             ,
             and
             draw
             the
             line
             E
             F
             ,
             then
             make
             D
             E
             ,
             and
             B
             E
             tangents
             of
             45
             ,
             and
             lay
             down
             the
             hours
             as
             you
             did
             in
             the
             horizontal
             ,
             and
             you
             shall
             have
             points
             whereby
             to
             draw
             the
             hour
             lines
             .
          
           
             For
             the
             north
             you
             must
             turn
             the
             hours
             both
             ways
             for
             4
             ,
             5
             ,
             8
             and
             7
             in
             the
             morning
             and
             4
             ,
             5
             ,
             7
             and
             8
             ,
             at
             night
             the
             height
             of
             the
             stile
             must
             be
             the
             tangent
             of
             the
             complement
             of
             the
             of
             the
             latitude
             when
             the
             sector
             is
             set
             to
             lay
             off
             the
             hours
             from
             D
             ,
             as
             here
             it
             is
             laid
             down
             from
             C
             to
             G
             ,
             and
             draw
             the
             line
             A
             G
             for
             the
             stile
             .
             For
             illustration
             sake
             note
             the
             figure
             .
          
        
         
           
           
             CHAP.
             XII
             .
          
           
             To
             draw
             an
             erect
             East
             or
             West
             Dial.
             
          
           
             FIrst
             by
             the
             fifth
             Proposition
             of
             the
             second
             Chapter
             draw
             a
             horizontal
             line
             ,
             as
             the
             line
             A
             B
             at
             the
             upper
             part
             of
             the
             plaine
             .
             Then
             at
             one
             third
             part
             of
             the
             line
             A
             B
             ,
             from
             A
             the
             right
             end
             if
             it
             be
             an
             East
             plaine
             ,
             or
             from
             B
             the
             left
             end
             ,
             if
             it
             be
             a
             West
             Plain
             ,
             appoint
             the
             center
             C
             ,
             from
             which
             point
             C
             draw
             the
             Semicircle
             A
             E
             D
             ,
             and
             fit
             that
             radius
             in
             the
             sine
             of
             30
             degrees
             ,
             (
             which
             in
             the
             Chords
             is
             60
             degrees
             )
             then
             take
             out
             the
             sine
             of
             half
             the
             ,
             latitude
             ,
             and
             lay
             it
             from
             A
             to
             E
             ,
             and
             draw
             the
             line
             C
             E
             for
             six
             in
             the
             morning
             on
             the
             East
             ,
             (
             or
             the
             contrary
             way
             for
             the
             West
             .
             )
             Then
             lay
             the
             sine
             of
             half
             the
             complement
             of
             the
             latitude
             ,
             from
             D
             to
             F
             ,
             and
             draw
             the
             line
             C
             F
             ,
             for
             the
             contingent
             or
             equinoctial
             line
             ,
             to
             which
             line
             you
             must
             draw
             another
             line
             parallel
             ,
             as
             far
             
             
             
               
                 An
                 East
                 Diall
                 .
              
            
             
               
                 A
                 West
                 diall
                 .
              
            
             
             assunder
             as
             the
             plaine
             will
             give
             leave
             ,
             then
             take
             the
             neerest
             distance
             from
             A
             to
             the
             six
             a
             clock
             line
             ,
             or
             more
             or
             less
             as
             you
             best
             fancy
             ,
             and
             fit
             it
             in
             the
             tangent
             of
             45
             degrees
             ,
             and
             prick
             down
             all
             the
             houres
             and
             quarters
             ,
             on
             both
             the
             equinoctial
             lines
             ,
             both
             ways
             from
             six
             ,
             and
             they
             shall
             be
             points
             ,
             whereby
             to
             draw
             the
             hoor
             lines
             by
             ,
             but
             for
             the
             two
             houres
             of
             10
             and
             11
             there
             is
             a
             lesser
             tangent
             beginning
             at
             45
             ,
             and
             proceeding
             to
             75
             ,
             which
             use
             thus
             :
             fit
             the
             space
             from
             six
             to
             three
             ,
             in
             the
             little
             tangent
             of
             45
             ,
             and
             then
             and
             then
             lay
             of
             60
             in
             the
             little
             tangents
             from
             6.
             to
             10
             and
             the
             tangent
             of
             75
             from
             6
             to
             11
             ,
             and
             the
             respective
             quarters
             also
             if
             you
             please
             ,
             so
             have
             you
             all
             the
             hour●s
             in
             the
             East
             ,
             or
             west
             Diall
             ,
             the
             distance
             from
             six
             to
             nine
             or
             from
             six
             to
             three
             ,
             in
             the
             West
             ,
             is
             the
             height
             of
             the
             stile
             ,
             in
             the
             East
             and
             West
             Diall
             ,
             and
             must
             stand
             in
             the
             six
             a
             clock
             line
             ,
             and
             parallel
             to
             the
             plaine
             .
          
        
         
           
           
             CHAP.
             XIII
             .
          
           
             To
             finde
             the
             declination
             of
             any
             Plain
             .
          
           
             FOr
             the
             finding
             of
             the
             declination
             of
             a
             Plain
             ,
             the
             most
             usual
             and
             easie
             way
             ,
             is
             by
             a
             magnetical
             needle
             fitted
             according
             to
             Mr.
             Failes
             way
             ,
             in
             the
             index
             of
             a
             Declinatory
             ;
             or
             in
             a
             square
             box
             with
             the
             90
             degrees
             of
             a
             quadrant
             on
             the
             two
             sides
             ,
             or
             by
             a
             needle
             fitted
             on
             the
             index
             of
             a
             quadrant
             ,
             after
             all
             which
             ways
             ,
             you
             may
             have
             them
             at
             the
             Sign
             of
             the
             Sphere
             and
             Sun-Dial
             in
             the
             Minories
             ,
             made
             by
             
               Iohn
               Brown.
            
             
          
           
             But
             the
             work
             may
             be
             very
             readily
             and
             exactly
             performed
             by
             the
             rule
             ,
             either
             by
             the
             Sun
             or
             needle
             in
             this
             manner
             following
             ,
             of
             which
             two
             ways
             that
             by
             the
             Sun
             is
             always
             the
             best
             ,
             and
             most
             exact
             and
             artificial
             ,
             and
             the
             other
             not
             to
             be
             used
             (
             if
             I
             may
             advise
             )
             but
             when
             the
             other
             failes
             ,
             by
             the
             Suns
             not
             shining
             ,
             or
             as
             a
             proof
             or
             
             confirmation
             of
             the
             other
             .
          
           
             
               And
               first
               by
               the
               needle
               because
               the
               easiest
               .
            
             
               For
               this
               purpose
               you
               must
               have
               a
               needle
               well
               touched
               with
               the
               Loadstone
               of
               about
               three
               or
               four
               inches
               long
               ,
               and
               fitted
               into
               a
               box
               somewhat
               broader
               then
               one
               of
               the
               legs
               of
               the
               sector
               ,
               with
               a
               lid
               to
               open
               and
               shut
               ;
               and
               on
               the
               inside
               of
               the
               lid
               may
               be
               drawn
               a
               South
               erect
               Dial
               ,
               and
               a
               wire
               to
               set
               the
               lid
               upright
               ,
               and
               a
               thread
               to
               be
               the
               Gnomon
               or
               stile
               to
               that
               Dial
               :
               it
               will
               not
               be
               a
               miss
               also
               to
               extend
               the
               lines
               on
               the
               Horizontal
               part
               for
               the
               same
               thread
               is
               a
               stile
               for
               that
               also
               .
               Also
               on
               the
               bottom
               let
               there
               be
               a
               rabbit
               ,
               or
               grove
               ,
               made
               to
               fit
               the
               leg
               of
               the
               rule
               or
               sector
               ;
               so
               as
               being
               pressed
               into
               it
               ,
               it
               may
               not
               fall
               off
               from
               the
               rule
               ,
               if
               your
               hand
               should
               shake
               ,
               or
               you
               cease
               to
               hold
               it
               there
               .
               This
               being
               so
               fitted
               ,
               the
               uses
               follow
               in
               their
               order
               .
            
             
               Put
               your
               box
               and
               needle
               on
               that
               leg
               of
               the
               rule
               ,
               that
               will
               be
               most
               fit
               
               for
               your
               purpose
               ,
               and
               also
               the
               north
               end
               of
               the
               needle
               toward
               the
               wall
               ,
               if
               it
               be
               a
               south
               wall
               ;
               and
               the
               contrary
               ,
               if
               a
               north
               ,
               as
               the
               playing
               of
               the
               needle
               will
               direct
               you
               ,
               better
               then
               the
               way
               how
               in
               a
               thousand
               words
               ,
               then
               open
               or
               close
               the
               Rule
               ,
               till
               the
               needle
               play
               right
               over
               the
               north
               and
               south-line
               ,
               in
               the
               bottom
               of
               the
               Box
               :
               then
               the
               complement
               of
               the
               Angle
               that
               the
               Sector
               standeth
               at
               (
               which
               may
               always
               be
               under
               90
               degrees
               )
               is
               the
               declination
               of
               the
               Plain
               .
               But
               if
               it
               happen
               to
               stand
               at
               any
               Angle
               above
               90
               ,
               then
               the
               quantity
               thereof
               above
               90
               ,
               is
               the
               declination
               of
               the
               wall
               .
            
             
               To
               finde
               the
               quantity
               of
               the
               Angle
               the
               Sector
               stands
               at
               may
               be
               done
               two
               ways
               :
               first
               by
               protraction
               ,
               by
               laying
               down
               the
               Rule
               so
               set
               on
               a
               board
               ,
               and
               draw
               two
               lines
               by
               the
               legs
               of
               the
               Sector
               ,
               and
               finde
               the
               Angle
               by
               a
               line
               of
               Chords
               .
               Secondly
               ,
               more
               speedily
               and
               artificially
               ,
               thus
               :
               By
               the
               lines
               of
               Sines
               being
               drawn
               to
               2.
               4
               ,
               or
               6
               degrees
               asunder
               .
               The
               Sector
               so
               set
               ,
               take
               
               the
               parallel
               sine
               of
               30
               and
               30
               ,
               and
               measure
               it
               on
               the
               lateral
               sines
               from
               the
               center
               ,
               and
               it
               shall
               reach
               to
               the
               sine
               of
               half
               the
               Angle
               the
               line
               of
               sines
               stands
               at
               ,
               being
               more
               by
               2
               ,
               4
               ,
               or
               6
               degrees
               then
               the
               sector
               stands
               at
               ,
               because
               it
               is
               drawn
               one
               ,
               two
               ,
               or
               three
               degrees
               from
               the
               inside
               .
            
             
               Or
               else
               take
               the
               latteral
               sine
               of
               30
               from
               the
               center
               ,
               and
               keeping
               one
               foot
               fixed
               in
               30
               ,
               turn
               the
               other
               till
               it
               cross
               the
               line
               of
               sines
               on
               that
               line
               next
               the
               inside
               ,
               and
               counting
               from
               90
               ,
               it
               shall
               touch
               at
               the
               Angle
               the
               line
               of
               sines
               standsat
               ,
               being
               two
               degrees
               more
               then
               the
               Sector
               stands
               at
               ;
               the
               lines
               being
               drawn
               so
               ,
               will
               be
               (
               as
               I
               conceive
               )
               most
               convenien
               .
            
             
               
                 Take
                 an
                 Example
                 .
              
               
                 I
                 come
                 to
                 a
                 south-east-wall
                 ,
                 and
                 putting
                 my
                 box
                 and
                 needle
                 on
                 my
                 Rule
                 ,
                 with
                 the
                 cross
                 or
                 north-end
                 of
                 the
                 needle
                 toward
                 the
                 wall
                 ,
                 and
                 the
                 Rule
                 being
                 applyed
                 ●lat
                 to
                 the
                 wall
                 ,
                 on
                 the
                 edge
                 thereof
                 ,
                 on
                 the
                 evenest
                 
                 place
                 thereof
                 ,
                 and
                 held
                 level
                 ,
                 so
                 as
                 the
                 needle
                 may
                 play
                 well
                 with
                 the
                 head
                 of
                 the
                 Rule
                 toward
                 your
                 right
                 hand
                 ,
                 as
                 you
                 shall
                 finde
                 it
                 to
                 be
                 in
                 an
                 east-wall
                 most
                 convenient
                 ;
                 then
                 I
                 open
                 or
                 close
                 the
                 Rule
                 ,
                 till
                 the
                 needle
                 play
                 right
                 over
                 the
                 north
                 and
                 south
                 line
                 in
                 the
                 bottom
                 of
                 the
                 box
                 ;
                 then
                 having
                 got
                 the
                 Angle
                 ,
                 (
                 take
                 off
                 the
                 box
                 ,
                 or
                 if
                 you
                 put
                 it
                 on
                 the
                 other
                 side
                 that
                 labor
                 may
                 be
                 saved
                 .
                 )
                 I
                 take
                 the
                 parallel
                 sine
                 of
                 30
                 ,
                 and
                 measuring
                 it
                 from
                 the
                 center
                 it
                 reaches
                 ,
                 suppose
                 to
                 the
                 sine
                 of
                 20
                 ,
                 then
                 is
                 the
                 line
                 of
                 sines
                 at
                 an
                 Angle
                 of
                 40
                 degrees
                 ,
                 but
                 the
                 Sector
                 at
                 two
                 degrees
                 less
                 ,
                 viz.
                 38
                 degrees
                 ,
                 whose
                 complement
                 52
                 is
                 the
                 declination
                 ;
                 then
                 to
                 consider
                 which
                 way
                 ,
                 minde
                 thus
                 :
                 First
                 it
                 is
                 south
                 ,
                 because
                 the
                 sun
                 being
                 in
                 the
                 south
                 ,
                 shines
                 on
                 the
                 wall
                 .
                 Secondly
                 consider
                 ,
                 the
                 sun
                 being
                 in
                 the
                 east
                 ,
                 it
                 shines
                 also
                 on
                 the
                 wal
                 ,
                 therefore
                 it
                 is
                 east
                 plain
                 :
                 thus
                 have
                 you
                 got
                 the
                 denomination
                 which
                 way
                 ,
                 and
                 also
                 the
                 quantity
                 how
                 much
                 that
                 ways
                 .
              
               
               
                 Or
                 if
                 you
                 take
                 the
                 latteral
                 sine
                 of
                 30
                 from
                 the
                 center
                 ,
                 and
                 turn
                 the
                 point
                 of
                 the
                 Compasses
                 from
                 30
                 towards
                 90
                 ,
                 on
                 the
                 other
                 leg
                 you
                 shall
                 finde
                 it
                 to
                 reach
                 to
                 the
                 sine
                 of
                 50
                 degrees
                 ,
                 whose
                 complement
                 ,
                 counting
                 from
                 90
                 ,
                 is
                 40
                 ,
                 or
                 rather
                 38
                 ,
                 for
                 the
                 reason
                 before-said
                 ,
                 or
                 else
                 adde
                 2
                 to
                 50
                 ,
                 and
                 you
                 have
                 the
                 angle
                 required
                 ,
                 without
                 complementing
                 of
                 it
                 ,
                 being
                 the
                 true
                 declination
                 sought
                 for
                 .
                 Thus
                 by
                 the
                 needle
                 you
                 may
                 get
                 the
                 declination
                 of
                 any
                 wall
                 ,
                 which
                 in
                 cloudy
                 weather
                 may
                 stand
                 you
                 in
                 good
                 stead
                 ,
                 or
                 to
                 examine
                 an
                 observation
                 by
                 the
                 Sun
                 ,
                 as
                 to
                 the
                 mis-counting
                 or
                 mistaking
                 therein
                 ;
                 but
                 for
                 exactness
                 the
                 Sun
                 is
                 alwayes
                 the
                 best
                 ,
                 because
                 the
                 needle
                 ,
                 though
                 never
                 so
                 good
                 ▪
                 may
                 be
                 drawn
                 aside
                 by
                 iron
                 in
                 the
                 wall
                 ,
                 and
                 also
                 by
                 some
                 kinde
                 of
                 bricks
                 ,
                 therefore
                 not
                 to
                 be
                 too
                 much
                 trusted
                 unto
                 .
              
            
          
           
             
               To
               finde
               a
               Declination
               by
               the
               Sun.
               
            
             
               First
               open
               the
               Rule
               to
               an
               Angle
               of
               60
               degrees
               ,
               as
               you
               do
               to
               finde
               the
               
               hour
               of
               the
               day
               ,
               and
               put
               a
               pin
               in
               the
               hole
               ,
               and
               hang
               the
               thread
               and
               plummet
               on
               the
               pin
               ;
               also
               you
               must
               have
               another
               thread
               somewhat
               longer
               and
               grosser
               then
               that
               for
               the
               hour
               in
               a
               readiness
               for
               your
               use
               .
            
             
               Then
               apply
               the
               head
               leg
               to
               the
               wall
               ,
               if
               the
               sun
               be
               coming
               on
               the
               Plain
               ,
               and
               hold
               the
               Rule
               horizontal
               or
               level
               ,
               then
               hold
               up
               the
               long
               thread
               till
               the
               shadow
               falls
               right
               over
               the
               pin
               ,
               or
               the
               center
               hole
               ,
               at
               the
               same
               instant
               the
               shadow
               shall
               shew
               on
               the
               degrees
               ,
               how
               much
               the
               sun
               wants
               of
               coming
               to
               be
               just
               against
               the
               Plain
               ,
               which
               I
               call
               the
               Meridian
               or
               Pole
               of
               the
               Plain
               ,
               which
               number
               you
               must
               write
               down
               thus
               ,
               as
               suppose
               it
               fell
               on
               40
               ,
               write
               down
               40-00
               want
               :
               then
               as
               soon
               as
               may
               be
               take
               the
               Suns
               true
               altitude
               ,
               and
               write
               that
               down
               also
               ,
               with
               which
               you
               must
               finde
               the
               Suns
               Azimuth
               ▪
               then
               substract
               the
               greater
               out
               of
               the
               less
               ,
               and
               the
               remainder
               is
               the
               declination
               required
               .
               But
               for
               a
               general
               rule
               ,
               take
               this
               :
               if
               the
               Sun
               do
               
               want
               of
               coming
               to
               the
               Meridian
               of
               the
               place
               ,
               and
               also
               want
               of
               coming
               to
               the
               Meridian
               of
               the
               Plain
               ,
               then
               you
               must
               alway
               substract
               the
               greater
               number
               out
               of
               the
               less
               ▪
               whether
               it
               be
               forenoon
               or
               after-noon
               ,
               so
               likewise
               when
               the
               Sun
               is
               past
               the
               Meridian
               ,
               and
               past
               the
               Plain
               also
               .
               But
               if
               the
               signes
               be
               unlike
               ,
               that
               is
               to
               say
               ,
               one
               past
               the
               Plain
               ,
               or
               Meridian
               ,
               and
               the
               other
               want
               either
               of
               the
               Plain
               ,
               or
               Meridian
               ,
               then
               you
               must
               add
               them
               together
               ,
               and
               the
               sum
               is
               the
               declination
               from
               the
               South
               .
               Which
               rule
               for
               better
               tenaciousness
               sake
               take
               in
               this
               homely
               rime
               .
            
             
               
                 Signs
                 both
                 alike
                 substraction
                 doth
                 require
                 ,
              
               
                 But
                 unlike
                 signs
                 addition
                 doth
                 desire
                 .
              
            
             
               The
               further
               illustration
               by
               two
               or
               three
               Examples
               .
            
             
               Suppose
               on
               the
               first
               of
               May
               ,
               in
               the
               forenoon
               ,
               I
               come
               and
               apply
               the
               Rule
               ,
               being
               opened
               to
               his
               Angle
               of
               60
               degrees
               to
               the
               wall
               ,
               (
               viz.
               the
               head
               leg
               ,
               
               or
               the
               leg
               where
               the
               center
               is
               )
               and
               holding
               up
               my
               thickest
               thread
               and
               plummet
               ,
               so
               as
               the
               shadow
               of
               it
               crosseth
               the
               center
               ,
               and
               at
               the
               same
               instant
               also
               on
               60
               degrees
               ,
               then
               I
               say
               the
               sun
               wants
               60
               deg
               .
               of
               coming
               to
               the
               Meridian
               of
               the
               Plain
               ;
               at
               the
               same
               instant
               ,
               or
               as
               soon
               as
               possible
               I
               can
               ,
               I
               take
               the
               suns
               altitude
               ,
               as
               before
               is
               shewed
               ,
               and
               set
               that
               down
               ,
               which
               suppose
               it
               to
               be
               20
               degrees
               ,
               then
               by
               the
               rules
               before
               ,
               get
               the
               Suns
               azimuth
               for
               that
               day
               ,
               and
               altitude
               ;
               which
               in
               our
               example
               will
               be
               found
               to
               be
               94
               degrees
               from
               the
               (
               south
               or
               )
               Meridian
               ,
               then
               in
               regard
               the
               signs
               are
               both
               alike
               ,
               
                 i.
                 e.
              
               want
               ,
               if
               you
               substract
               one
               out
               of
               the
               other
               ,
               there
               remains
               34
               the
               declination
               required
               ;
               but
               for
               the
               right
               denomination
               which
               way
               ,
               either
               north
               or
               south
               ,
               toward
               either
               east
               or
               west
               ,
               observe
               this
               plain
               rule
               :
               First
               ,
               if
               the
               Sun
               come
               to
               the
               Meridian
               or
               Pole
               of
               the
               Plain
               ,
               before
               it
               come
               to
               the
               Meridian
               or
               Pole
               of
               the
               place
               ,
               then
               it
               is
               always
               an
               East-plain
               ;
               but
               if
               
               the
               contrary
               ,
               it
               is
               a
               West-plain
               ,
               that
               is
               to
               say
               ,
               if
               the
               Sun
               come
               to
               the
               Meridian
               or
               Pole
               of
               the
               place
               ,
               before
               it
               comes
               to
               the
               Meridian
               or
               Pole
               of
               the
               Plain
               ,
               then
               it
               is
               a
               West-plain
               .
               Also
               if
               the
               sum
               or
               remainder
               ,
               after
               addition
               or
               substraction
               ,
               be
               under
               90
               ,
               it
               is
               a
               South-plain
               ;
               but
               if
               it
               be
               above
               90
               ,
               it
               is
               a
               North-plain
               .
               Also
               note
               ,
               that
               when
               the
               sum
               or
               remainder
               is
               above
               90
               ,
               then
               the
               complement
               to
               180
               ,
               is
               always
               the
               declination
               from
               the
               north
               toward
               either
               east
               or
               west
               ;
               So
               that
               according
               to
               these
               rules
               in
               our
               example
               it
               is
               34
               degrees
               South-east
               .
            
             
               Again
               in
               the
               morning
               ,
               Iune
               13.
               
               I
               apply
               my
               rule
               to
               the
               wall
               ,
               and
               I
               finde
               the
               Sun
               is
               past
               the
               Pole
               or
               Meridian
               of
               the
               Plain
               10
               degrees
               ,
               and
               the
               altitude
               at
               the
               same
               time
               15
               degrees
               ,
               the
               Azimuth
               at
               that
               altitude
               ,
               and
               day
               in
               this
               latitude
               ,
               will
               be
               found
               to
               be
               109
               degrees
               want
               of
               south
               or
               pole
               of
               the
               place
               ;
               therefore
               unlike
               signs
               ,
               and
               to
               be
               added
               ,
               and
               they
               make
               up
               119
               degrees
               ,
               whose
               complement
               to
               180
               
               is
               61
               ;
               for
               61
               and
               119
               added
               ,
               make
               up
               180
               ,
               therefore
               this
               Plain
               declineth
               61
               degrees
               from
               the
               north
               toward
               the
               east
               .
            
             
               Again
               the
               same
               day
               in
               the
               afternoon
               ,
               I
               finde
               the
               Azimuth
               past
               the
               south
               or
               meridian
               of
               the
               place
               30
               degrees
               ,
               and
               at
               the
               same
               time
               the
               Sun
               wants
               in
               coming
               to
               the
               meridian
               or
               pole
               of
               the
               Plain
               10
               degrees
               ,
               here
               by
               addition
               I
               finde
               the
               declination
               to
               be
               40
               degrees
               south-west
               .
            
             
               Note
               what
               I
               have
               said
               in
               these
               three
               examples
               ,
               is
               general
               at
               all
               times
               ;
               but
               if
               it
               be
               a
               fair
               day
               ,
               and
               time
               and
               opportunity
               serve
               ,
               to
               come
               either
               just
               at
               12
               a
               clock
               ,
               when
               the
               Sun
               is
               the
               meridian
               or
               pole
               of
               the
               place
               ;
               or
               just
               when
               the
               Sun
               is
               in
               the
               meridian
               or
               pole
               of
               the
               Plain
               ,
               then
               your
               work
               is
               onely
               thus
               :
            
             
               First
               if
               you
               come
               to
               observe
               at
               12
               ,
               then
               applying
               your
               rule
               to
               the
               wall
               ,
               and
               holding
               up
               the
               thread
               and
               plummet
               ,
               how
               much
               so
               ever
               the
               Sun
               wants
               or
               is
               past
               the
               pole
               of
               the
               Plain
               ,
               that
               
               is
               the
               declination
               ,
               if
               it
               be
               past
               it
               is
               east-wards
               ;
               if
               it
               wants
               ,
               it
               is
               south-west-wards
               ;
               if
               neither
               ,
               a
               just
               South
               Plain
               ,
               and
               then
               the
               poles
               ,
               or
               Meridians
               of
               place
               and
               Plain
               ,
               are
               the
               same
               .
            
             
               But
               secondly
               ,
               if
               you
               come
               when
               the
               Sun
               is
               just
               in
               the
               pole
               of
               the
               Plain
               ,
               then
               whatsoever
               you
               finde
               the
               Suns
               Azimuth
               to
               be
               ,
               that
               is
               the
               declination
               ;
               if
               it
               wants
               of
               south
               ,
               it
               declineth
               East-wards
               ;
               if
               it
               be
               past
               ,
               it
               declineth
               West-wards
               .
            
             
               Thus
               I
               have
               copiously
               ,
               (
               and
               yet
               very
               briefly
               )
               shewed
               you
               the
               most
               artificial
               way
               of
               getting
               the
               declination
               of
               any
               wall
               ,
               howsoever
               situated
               .
            
             
               Note
               if
               the
               Sun
               be
               above
               15
               degrees
               wanting
               of
               the
               Meridian
               of
               the
               Plain
               ,
               your
               rule
               will
               prove
               defective
               in
               taking
               the
               Plains
               Meridian
               when
               the
               center
               leg
               is
               next
               the
               wall
               ,
               then
               you
               must
               turn
               the
               other
               leg
               to
               the
               wall
               ,
               and
               then
               you
               finde
               a
               supply
               for
               all
               angles
               to
               45
               degrees
               past
               the
               Plain
               .
               But
               for
               the
               supply
               of
               the
               rest
               which
               is
               45
               degrees
               ,
               do
               thus
               :
               open
               the
               rule
               till
               the
               
               great
               line
               of
               tangents
               &
               the
               outside
               of
               the
               leg
               make
               a
               right
               angle
               ,
               for
               which
               on
               the
               head
               you
               may
               make
               a
               mark
               for
               the
               ready
               setting
               ,
               then
               making
               the
               inside
               of
               the
               leg
               at
               the
               end
               of
               45
               ,
               as
               a
               center
               ,
               the
               tangents
               on
               the
               other
               leg
               supply
               very
               largely
               the
               defect
               of
               the
               othersides
               .
               Or
               if
               you
               set
               on
               the
               box
               and
               needle
               on
               the
               rule
               ,
               and
               open
               or
               shut
               the
               rule
               till
               the
               shadow
               of
               the
               thread
               shew
               just
               12
               ,
               then
               the
               Angle
               the
               Sector
               stands
               at
               ,
               is
               the
               complement
               of
               what
               the
               sum
               wants
               ,
               or
               is
               past
               the
               meridian
               or
               pole
               of
               the
               Plain
               .
            
          
        
         
           
             CHAP.
             XIV
             .
          
           
             To
             draw
             a
             vertical
             declining
             Plain
             to
             any
             declination
             .
          
           
             FIrst
             draw
             a
             perpendicular-line
             for
             12
             ,
             as
             A
             B
             then
             design
             a
             point
             in
             that
             line
             for
             the
             center
             ,
             as
             C
             ,
             at
             the
             upper
             end
             ,
             if
             it
             be
             a
             South
             Plain
             ;
             or
             contrary
             ,
             if
             it
             be
             a
             North
             Plain
             ,
             then
             on
             that
             center
             describe
             an
             Arch
             of
             a
             
             
             
             
             Circle
             on
             that
             side
             of
             12
             ,
             which
             is
             contrary
             to
             the
             Plains
             declination
             ,
             as
             D
             E
             ,
             and
             in
             that
             Arch
             lay
             off
             from
             12
             the
             substile
             ,
             and
             on
             that
             the
             stiles
             height
             ,
             and
             the
             hour
             of
             6
             ,
             being
             found
             by
             the
             tenth
             chapter
             ,
             and
             draw
             those
             lines
             from
             C
             the
             center
             ,
             then
             draw
             two
             parallels
             to
             12
             ,
             as
             in
             the
             direct
             south
             :
             then
             fit
             the
             distance
             of
             the
             parallels
             in
             the
             secant
             of
             the
             declination
             ,
             and
             take
             the
             secant
             of
             the
             latitude
             ,
             and
             set
             it
             from
             the
             center
             C
             on
             the
             line
             of
             12
             ,
             to
             F
             ,
             and
             on
             the
             parallel
             from
             6
             to
             G
             ,
             and
             draw
             a
             line
             by
             those
             two
             points
             F
             and
             G
             ,
             to
             cut
             the
             other
             parallel
             in
             H
             :
             then
             have
             you
             found
             6
             ,
             3
             ,
             and
             9
             ,
             then
             fit
             6
             G
             in
             the
             tangent
             of
             45
             ,
             and
             prick
             off
             15
             ,
             30
             ,
             and
             the
             respective
             quarters
             both
             ways
             from
             6
             ,
             for
             the
             morning
             ,
             and
             afternoon
             hours
             ,
             then
             fit
             F
             G
             in
             45
             ,
             and
             lay
             off
             the
             same
             points
             from
             F
             both
             ways
             for
             10
             ,
             11
             ,
             1
             ,
             and
             2
             ,
             and
             the
             quarters
             also
             ,
             if
             you
             please
             ,
             and
             those
             shall
             be
             points
             to
             draw
             the
             hour-lines
             by
             .
             The
             stile
             must
             be
             set
             perpendicular
             
             over
             the
             substile
             ,
             to
             the
             Angle
             found
             by
             the
             rules
             in
             the
             tenth
             chapter
             ,
             and
             then
             the
             Dial
             shall
             shew
             the
             true
             hour
             of
             the
             day
             ,
             being
             drawn
             fit
             to
             his
             proper
             declination
             .
          
           
             
               Another
               way
               to
               perform
               this
               Geometrically
               for
               all
               erect
               Dyals
               with
               centers
               .
            
             
               When
               you
               have
               drawn
               a
               line
               of
               12
               ,
               and
               appointed
               a
               center
               ,
               make
               a
               Geometrical
               square
               on
               that
               side
               12
               ,
               as
               the
               stile
               must
               stand
               on
               ,
               as
               A
               B
               C
               D
               ,
               the
               perpendicular
               side
               of
               which
               square
               may
               also
               be
               the
               parallel
               as
               before
               .
            
             
             
               Again
               ,
               fit
               the
               side
               of
               the
               square
               in
               the
               cosine
               of
               the
               latitude
               ,
               and
               take
               out
               the
               sine
               of
               the
               latitude
               ,
               and
               fit
               that
               over
               in
               the
               sine
               of
               90
               ,
               then
               take
               out
               the
               〈◊〉
               sine
               of
               the
               declination
               ,
               and
               lay
               it
               from
               D
               to
               G
               for
               the
               hour
               of
               6
               ,
               and
               draw
               the
               line
               A
               G
               for
               the
               6
               a
               clock
               hour
               line
               .
            
             
               Then
               again
               ,
               fit
               the
               side
               of
               the
               square
               (
               or
               the
               distance
               of
               the
               parallel
               in
               the
               other
               way
               ,
               when
               you
               want
               a
               secant
               ,
               or
               your
               secant
               too
               little
               )
               in
               the
               sine
               of
               90
               ,
               and
               take
               out
               the
               cosine
               of
               the
               declination
               ,
               fit
               that
               in
               the
               cosine
               of
               the
               latitude
               ,
               then
               take
               out
               the
               sine
               of
               90
               ,
               and
               lay
               it
               from
               the
               center
               on
               the
               line
               of
               12
               ,
               and
               〈◊〉
               6
               in
               the
               side
               of
               the
               square
               ,
               and
               by
               those
               two
               points
               draw
               the
               contingent
               line
               ▪
               and
               then
               fit
               those
               points
               or
               distances
               in
               the
               tangent
               of
               45
               ,
               and
               lay
               down
               the
               hours
               as
               in
               the
               former
               part
               of
               this
               chapter
               ;
               but
               if
               you
               want
               the
               hours
               before
               9
               in
               west-decliners
               ,
               or
               the
               hours
               after
               3
               in
               east-decliners
               ,
               and
               the
               6
               fall
               too
               high
               above
               
               the
               horizontal
               line
               on
               that
               side
               ,
               you
               may
               supply
               this
               defect
               thus
               :
            
             
               Take
               the
               measure
               with
               your
               Compasses
               from
               3
               to
               4
               and
               5
               ,
               upwards
               in
               the
               west-decliners
               ,
               or
               from
               9
               to
               7
               and
               8
               in
               east-decliners
               ,
               and
               lay
               it
               upwards
               from
               9
               or
               3
               on
               the
               deficient
               side
               ,
               in
               the
               parallel
               as
               it
               should
               have
               been
               from
               6
               downward
               in
               south
               or
               upward
               in
               north
               Plains
               ,
               and
               you
               shall
               see
               all
               your
               defective
               points
               to
               be
               compleatly
               supplied
               ,
               whereunto
               draw
               the
               hour-lines
               accordingly
               .
            
          
        
         
           
             CHAP.
             XV
          
           
             To
             draw
             the
             Hour-lines
             on
             an
             upright
             declining
             Dial
             ,
             declining
             above
             60
             degrees
             ▪
          
           
             IN
             all
             erect
             Decliners
             ,
             the
             way
             of
             finding
             the
             stile
             ,
             substile
             6
             and
             12
             ,
             and
             inclination
             of
             Meridians
             ,
             is
             the
             same
             ;
             but
             when
             you
             come
             to
             protract
             ,
             or
             lay
             down
             the
             hour-lines
             ,
             you
             shall
             finde
             them
             come
             so
             close
             together
             ,
             
             as
             they
             will
             be
             useless
             ,
             unless
             the
             stile
             be
             augmented
             .
          
           
             The
             usual
             way
             for
             doing
             of
             which
             ,
             is
             to
             draw
             the
             Dial
             on
             a
             large
             floor
             ,
             and
             then
             cut
             off
             so
             much
             and
             at
             such
             a
             distance
             ,
             as
             best
             serveth
             your
             turn
             ,
             but
             this
             being
             not
             always
             to
             be
             affected
             ,
             for
             want
             of
             conveniency
             ,
             and
             large
             instruments
             ,
             it
             may
             very
             artificially
             be
             done
             by
             a
             natural
             tangent
             of
             75
             or
             80
             degrees
             ,
             fitted
             on
             the
             legs
             of
             a
             sector
             in
             this
             manner
             .
          
           
             The
             example
             I
             shall
             make
             use
             of
             ,
             let
             be
             a
             Plain
             declination
             75
             degrees
             ,
             South
             ,
             West
             .
          
           
             First
             on
             the
             north
             edge
             of
             the
             Plain
             draw
             a
             perpendicular-line
             A
             B
             ,
             representing
             12
             a
             clock
             line
             ;
             then
             on
             the
             center
             draw
             an
             occult
             Arch
             of
             a
             circle
             as
             large
             as
             you
             can
             ,
             as
             B
             D
             ,
             therein
             lay
             off
             an
             angle
             of
             37.
             30
             for
             the
             substile
             ,
             (
             though
             indeed
             this
             line
             will
             not
             prove
             the
             very
             substile
             ,
             yet
             it
             is
             a
             parallel
             to
             it
             :
             )
             then
             cross
             it
             with
             two
             perpendiculers
             for
             the
             two
             contingent
             lines
             ,
             at
             the
             most
             convenient
             
             places
             ,
             one
             in
             the
             upper
             part
             ,
             and
             the
             other
             in
             the
             lower
             part
             of
             the
             Plain
             ,
             as
             the
             two
             lines
             C
             E
             ,
             and
             F
             G
             do
             shew
             ,
             then
             by
             help
             of
             the
             inclination
             of
             Meridians
             ,
             make
             the
             table
             for
             the
             hours
             in
             this
             manner
             ;
             for
             this
             declination
             ,
             it
             is
             78
             :
             09
             ,
             Now
             every
             hour
             containing
             15
             degrees
             ,
             and
             every
             quarter
             three
             degrees
             45
             '
             I
             find
             that
             the
             substile
             will
             fall
             on
             neer
             a
             quarter
             of
             an
             hour
             past
             five
             at
             night
             ,
             therefore
             if
             you
             take
             75
             ,
             the
             measure
             of
             five
             hours
             out
             of
             78.
             9
             ,
             there
             remaines
             03.
             09.
             for
             the
             first
             quarter
             from
             the
             substile
             next
             12.
             
             Again
             ,
             if
             you
             take
             78.
             09
             ,
             the
             inclination
             of
             Meridians
             out
             of
             78.
             45
             ,
             the
             measure
             of
             5
             hours
             one
             quarter
             ,
             and
             there
             remaines
             00.
             36.
             for
             the
             first
             quarter
             on
             the
             other
             side
             of
             the
             substile
             ,
             then
             by
             continual
             adding
             of
             03.
             45
             ,
             to
             03.
             09
             ,
             and
             to
             00.
             36
             ,
             I
             make
             up
             the
             table
             as
             heer
             you
             see
             or
             else
             you
             may
             against
             12
             ,
             set
             down
             78.
             09
             ,
             and
             then
             take
             out
             3.
             45
             ,
             as
             often
             as
             you
             can
             ,
             till
             you
             come
             to
             the
             substile
             ,
             and
             then
             
             what
             the
             remainder
             wants
             of
             .
             3.
             45
             ,
             must
             be
             set
             on
             the
             other
             side
             of
             the
             substile
             ,
             and
             3.
             45
             added
             to
             that
             ,
             till
             you
             come
             to
             as
             late
             as
             the
             sun
             will
             shew
             on
             the
             Plain
             ,
             as
             here
             you
             see
             to
             1
             /
             4
             past
             8.
             
             
               
                 
                   45
                   36
                
                 
                   .
                
                 
                   sub
                   .
                
                 
                   48
                   09
                
                 
                   2
                
              
               
                 
                   41
                   51
                
                 
                   8
                
                 
                   03
                   095
                
                 
                   51
                   54
                
                 
                   .
                
              
               
                 
                   38
                   06
                
                 
                   .
                
                 
                   06
                   54.
                   
                
                 
                   55
                   39
                
                 
                   .
                
              
               
                 
                   34
                   21
                
                 
                   .
                
                 
                   10
                   39.
                   
                
                 
                   59
                   24
                
                 
                   .
                
              
               
                 
                   30
                   36
                
                 
                   .
                
                 
                   14
                   24.
                   
                
                 
                   63
                   09
                
                 
                   1
                
              
               
                 
                   26
                   51
                
                 
                   7
                
                 
                   18
                   094
                
                 
                   66
                   54
                
                 
                   .
                
              
               
                 
                   23
                   06
                
                 
                   .
                
                 
                   21
                   54.
                   
                
                 
                   70
                   39
                
                 
                   .
                
              
               
                 
                   19
                   21
                
                 
                   .
                
                 
                   25
                   39.
                   
                
                 
                   74
                   24
                
                 
                   .
                
              
               
                 
                   15
                   36
                
                 
                   .
                
                 
                   29
                   24.
                   
                
                 
                   78
                   09
                
                 
                   12
                
              
               
                 
                   11
                   51
                
                 
                   6
                
                 
                   33
                   093
                
                 
                    
                
                 
                    
                
              
               
                 
                   08
                   06
                
                 
                   .
                
                 
                   36
                   54.
                   
                
                 
                    
                
                 
                    
                
              
               
                 
                   04
                   21
                
                 
                   .
                
                 
                   40
                   39.
                   
                
                 
                    
                
                 
                    
                
              
               
                 
                   00
                   36
                
                 
                   .
                
                 
                   44
                   24.
                   
                
                 
                    
                
                 
                    
                
              
            
          
           
           
             The
             stile
             or
             gnomon
             is
             to
             be
             erected
             right
             over
             the
             substile
             ,
             to
             the
             angle
             of
             9.
             16
             ,
             and
             augmented
             as
             much
             as
             from
             H
             to
             ●
             ,
             and
             from
             I
             to
             ●
             ,
             at
             the
             nighest
             distance
             from
             the
             point
             H
             and
             I
             ,
             drawnin
             the
             substile
             line
             on
             the
             Plain
             ,
             this
             way
             is
             as
             easy
             ,
             speedy
             ,
             artificial
             and
             true
             ,
             as
             any
             extant
             ,
             (
             if
             your
             scales
             be
             true
             ,
             )
             and
             improveth
             the
             Plain
             at
             the
             first
             ,
             to
             the
             most
             possible
             best
             advantage
             that
             can
             be
             .
          
           
             Note
             also
             ,
             if
             in
             striving
             to
             put
             as
             many
             hours
             as
             you
             can
             on
             the
             Plain
             ,
             the
             sum
             of
             the
             two
             natural
             numbers
             added
             together
             ,
             comes
             to
             above
             40000
             :
             then
             you
             must
             reject
             something
             of
             one
             of
             them
             ,
             for
             they
             will
             not
             be
             comely
             nor
             convenient
             ,
             nor
             far
             enough
             asunder
             .
             I
             might
             have
             enlarged
             to
             declining
             ,
             reclining
             plaines
             ,
             out
             my
             intention
             is
             not
             to
             make
             a
             business
             of
             it
             ,
             but
             onely
             to
             give
             a
             taste
             
             of
             the
             usefulness
             ,
             and
             convenience
             of
             a
             Joynt-rule
             ,
             as
             now
             it
             is
             improved
             .
          
           
             
               An
               Advertisement
               relating
               to
               Dialling
               .
            
             
               These
               directions
               are
               sufficient
               for
               any
               Horizontal
               ,
               direct
               ,
               and
               declining
               vertical
               Dials
               ,
               but
               for
               all
               other
               ,
               as
               East
               or
               West
               inclining
               and
               reclining
               ,
               direct
               ,
               or
               reclining
               and
               inclining
               ,
               and
               declining
               polars
               ;
               and
               all
               declining
               and
               inclining
               :
               and
               reclinimg
               Plains
               ,
               the
               perfect
               knowledge
               of
               their
               affections
               and
               scituations
               is
               very
               hard
               to
               conceive
               ,
               but
               much
               more
               hard
               to
               remember
               by
               roat
               ,
               or
               knowingly
               .
            
             
             
               A
               representation
               of
               which
               instrument
               take
               in
               this
               place
               ,
               let
               the
               circle
               E
               S
               W
               N
               ,
               represent
               the
               Horizon
               S
               AE
               P
               N
               the
               Meridian
               ,
               AE
               W
               AE
               E
               the
               Equinoctial
               ,
               P
               C
               P
               the
               Pole
               ,
               being
               the
               points
               of
               an
               Axes
               made
               of
               th●ead
               ,
               and
               passing
               through
               the
               center
               C
               W
               C
               E
               being
               shaddowed
               ,
               represents
               the
               Plain
               ,
               set
               to
               a
               declination
               ,
               and
               to
               a
               reclination
               45
               degrees
               ,
               F
               the
               foot
               that
               doth
               support
               it
               ;
               moreover
               the
               Plain
               hat●
               an
               Arch
               fitted
               to
               it
               ,
               to
               get
               the
               stiles
               height
               ,
               and
               to
               set
               it
               
               to
               any
               angle
               ;
               also
               the
               Horizon
               and
               Plain
               is
               made
               to
               turn
               round
               ,
               which
               could
               not
               well
               be
               expressed
               in
               this
               figure
               ,
               or
               representation
               of
               the
               instrument
               :
               a
               more
               liveley
               and
               easie
               help
               cannot
               likely
               be
               invented
               ,
               and
               for
               cheapness
               the
               like
               may
               be
               in
               a
               half
               circle
               ,
               or
               made
               of
               Past-board
               .
            
             
               The
               use
               and
               application
               of
               which
               figure
               ,
               I
               shall
               not
               now
               speak
               of
               ,
               partly
               because
               of
               the
               facillity
               in
               the
               using
               thereof
               ,
               and
               partly
               because
               of
               the
               difficulty
               in
               description
               thereof
               ;
               and
               lastly
               ,
               wheresoever
               you
               shall
               buy
               the
               same
               ,
               at
               either
               of
               the
               three
               places
               ,
               the
               use
               will
               be
               taught
               you
               gratis
               .
            
             
               Also
               note
               ,
               that
               half
               of
               the
               Instrument
               may
               be
               made
               either
               in
               b●ass
               ,
               or
               pastboard
               ,
               and
               be
               made
               to
               fold
               down
               in
               a
               book
               ,
               and
               perform
               the
               uses
               thereof
               indifferently
               well
               ,
               for
               many
               purposes
               ,
               as
               to
               the
               Affections
               of
               Dials
               .
            
             
               Then
               having
               discovered
               the
               affections
               ,
               you
               may
               by
               the
               Canno●s
               in
               the
               seventh
               Chapter
               of
               Mr.
               
                 Windgates
                 R●le
                 of
                 proportion
              
               ,
               finde
               all
               the
               requisites
               ,
               
               and
               then
               to
               speed
               the
               laying
               down
               the
               hours
               ,
               you
               may
               do
               it
               by
               help
               of
               the
               tangent
               of
               45
               ,
               as
               you
               did
               in
               erect
               decliners
               ;
               for
               after
               you
               have
               drawn
               the
               hours
               of
               6
               ,
               and
               three
               ,
               or
               nine
               ,
               and
               made
               a
               parallelogram
               by
               two
               lines
               parallel
               to
               12
               ,
               and
               one
               parallel
               to
               six
               ,
               then
               making
               the
               distance
               from
               six
               to
               nine
               ,
               or
               three
               in
               the
               line
               parallel
               to
               12
               ,
               a
               tangent
               of
               45
               ,
               and
               the
               distance
               from
               12
               to
               9
               ,
               or
               3
               in
               the
               parallel
               to
               6
               ,
               likewise
               a
               tangent
               of
               45
               ,
               in
               the
               sector
               ,
               and
               laying
               off
               15
               ,
               30.
               45
               ,
               or
               the
               respective
               quarters
               from
               6
               ,
               and
               12
               ,
               in
               those
               two
               lines
               ,
               and
               they
               shall
               be
               the
               true
               points
               to
               draw
               the
               hour
               lines
               ,
               by
               laying
               a
               ruler
               to
               them
               ,
               and
               the
               center
               of
               the
               Dial
               ;
               but
               for
               those
               that
               have
               no
               centers
               ,
               the
               rule
               of
               augmenting
               the
               stile
               in
               far
               decliners
               ,
               serveth
               for
               these
               also
               .
            
          
        
         
           
           
             CHAP.
             XVI
          
           
             To
             finde
             a
             perpendicular
             altitude
             at
             one
             or
             two
             stations
             ,
             and
             observations
             by
             by
             the
             degrees
             on
             the
             rule
             .
          
           
             OPen
             the
             rule
             to
             his
             Angle
             of
             60
             degrees
             ,
             then
             looking
             up
             to
             the
             top
             or
             point
             of
             altitude
             you
             would
             observe
             the
             height
             of
             ,
             as
             you
             do
             when
             you
             take
             the
             Suns
             altitude
             ,
             and
             note
             the
             degrees
             and
             parts
             the
             thread
             cuts
             ,
             and
             write
             it
             down
             in
             chaulk
             or
             ink
             ,
             that
             you
             forget
             it
             not
             .
          
           
             Then
             measure
             from
             the
             place
             you
             stand
             to
             the
             foot
             or
             base
             of
             the
             object
             ,
             (
             being
             right
             under
             the
             top
             of
             the
             object
             ,
             whose
             height
             you
             would
             measure
             )
             in
             feet
             ,
             yards
             ,
             or
             any
             other
             parts
             .
          
           
             Thirdly
             ,
             consider
             this
             being
             a
             right
             angled
             Plain
             triangle
             ,
             if
             you
             have
             the
             angle
             at
             the
             top
             ,
             the
             angle
             at
             the
             base
             is
             always
             the
             complement
             thereof
             .
          
           
           
             These
             things
             being
             premised
             ,
             the
             proportion
             holds
             ,
             as
             the
             fine
             of
             the
             angle
             opposite
             to
             the
             measured
             side
             ,
             (
             or
             base
             )
             being
             the
             complement
             of
             the
             angle
             found
             ,
             is
             to
             the
             base
             or
             measured
             side
             :
             so
             is
             the
             angle
             found
             ,
             to
             the
             height
             required
             .
          
           
             Always
             remember
             to
             adde
             the
             height
             of
             your
             eye
             from
             the
             ground
             ,
             (
             at
             the
             time
             of
             taking
             the
             angle
             )
             to
             the
             altitude
             found
             .
          
           
             For
             the
             operation
             of
             this
             ,
             extend
             the
             compasses
             from
             the
             sine
             of
             the
             complement
             of
             the
             Angle
             found
             to
             the
             number
             of
             the
             measured
             side
             ,
             on
             the
             line
             of
             numbers
             that
             distance
             applied
             the
             same
             way
             ,
             from
             the
             sine
             of
             the
             Angle
             found
             ,
             shall
             reach
             on
             the
             line
             of
             numbers
             to
             altitude
             required
             .
          
           
             
               Example
               at
               one
               station
               .
            
             
               I
               open
               my
               rule
               ,
               and
               hang
               on
               the
               thread
               and
               plummet
               on
               the
               center
               ,
               and
               observing
               the
               Angle
               at
               C
               ,
               I
               finde
               it
               to
               be
               41.
               45
               ,
               and
               the
               Angle
               at
               B
               the
               complement
               of
               it
               48.
               15
               ,
               and
               the
               
               measure
               from
               C
               to
               A
               271
               feet
               :
               then
               the
               work
               being
               so
               prepared
               ,
               is
               thus
               :
            
             
               As
               the
               Sine
               of
               48.
               15
               ,
               
               Is
               to
               271
               the
               measure
               of
               the
               side
               opposite
               to
               it
               :
               
               So
               is
               the
               Sine
               of
               the
               Angle
               41.
               45
               ,
               
               To
               242
               the
               measure
               of
               the
               side
               A
               B
               ,
               opposite
               to
               the
               Angle
               at
               C
               ,
               the
               height
               required
               .
            
             
               Again
               ,
               at
               the
               station
               D
               ,
               160
               foot
               from
               A
               ,
               I
               observe
               and
               finde
               the
               Angle
               D
               to
               be
               56.
               30
               ,
               the
               Angle
               at
               A
               is
               the
               complement
               thereof
               ,
               viz.
               33.
               30.
               
               This
               being
               prepared
               ,
               I
               extend
               my
               Compasses
               from
               the
               sine
               of
               33.
               30
               to
               160
               ,
               on
               the
               line
               of
               numbers
               the
               same
               extent
               will
               reach
               from
               the
               sine
               of
               56.
               30
               ,
               to
               242
               on
               the
               line
               of
               numbers
               ,
               lacking
               a
               small
               fraction
               ,
               with
               which
               I
               shall
               not
               trouble
               you
               .
            
          
           
             
               An
               example
               at
               two
               stations
               .
            
             
             
               As
               the
               Sine
               of
               the
               difference
               ,
               which
               is
               the
               Angle
               
                 C
                 B
                 D
              
               14.
               45
               ,
               
               Is
               to
               the
               side
               measured
               ,
               
                 viz.
                 D
                 C
              
               111
               feet
               on
               the
               numbers
               :
               
               So
               is
               the
               sine
               of
               the
               Angle
               at
               C
               41.
               45
               ,
               To
               the
               measure
               of
               the
               side
               
                 B
                 C
              
               ,
               the
               hypothenusa
               ,
               or
               measure
               from
               your
               eye
               to
               the
               top
               of
               the
               object
               ,
               viz.
               290
               feet
               .
            
             
               Again
               for
               the
               second
               Operation
               .
            
             
               As
               the
               sine
               of
               90
               the
               right
               angle
               at
               A
               ,
               
               Is
               to
               290
               the
               hypothenusa
               
                 B
                 D
              
               :
               
               
               So
               is
               the
               sine
               of
               56.
               30
               ,
               the
               Angle
               at
               the
               first
               station
               D
               ,
               
               To
               242
               ,
               the
               Altitude
               
                 B
                 A
              
               ,
               the
               thing
               required
               :
               :
               
               So
               also
               is
               the
               sine
               of
               the
               Angle
               at
               B
               33.
               70
               ,
               the
               complement
               of
               56
               ,
               30.
               
               
               To
               160
               the
               distance
               from
               D
               to
               A.
               
            
          
           
             
               To
               perform
               the
               same
               by
               the
               line
               of
               sines
               ,
               drawn
               from
               the
               center
               on
               the
               flat-side
               ,
               and
               the
               line
               of
               lines
               ,
               or
               equal
               parts
               or
               inches
               in
               ten
               parts
               .
            
             
               To
               work
               these
               or
               any
               other
               questions
               by
               the
               line
               of
               natural
               Sines
               and
               Tangents
               ,
               on
               the
               flat-side
               ,
               drawn
               from
               the
               center
               ,
               it
               is
               but
               changing
               the
               terms
               ,
               thus
               :
               As
               the
               measured
               distance
               taken
               out
               of
               the
               line
               of
               lines
               ,
               or
               any
               scale
               of
               equal
               parts
               ,
               is
               to
               the
               sine
               of
               the
               angle
               ,
               opposite
               to
               that
               measured
               side
               ,
               fitted
               across
               from
               one
               leg
               to
               the
               other
               ,
               the
               Sector
               so
               standing
               ,
               take
               out
               the
               parallel
               sine
               of
               the
               angle
               opposite
               to
               the
               enquired
               side
               ,
               and
               that
               measure
               shall
               reach
               on
               the
               line
               of
               equal
               parts
               ,
               to
               the
               measure
               of
               the
               Altitude
               requi●●●
               ▪
            
             
               
               
                 Example
                 as
                 before
                 .
              
               
                 Take
                 out
                 of
                 the
                 lines
                 or
                 inches
                 2.
                 71
                 ,
                 and
                 fit
                 it
                 in
                 the
                 sine
                 of
                 48.
                 15
                 ,
                 across
                 from
                 one
                 legge
                 to
                 the
                 other
                 ,
                 which
                 I
                 call
                 
                   A
                   parral
                   sine
                
                 ,
                 (
                 but
                 when
                 you
                 measure
                 from
                 the
                 center
                 onwards
                 the
                 end
                 ,
                 I
                 call
                 it
                 
                   A
                   latteral
                   sine
                
                 ,
                 )
                 then
                 take
                 out
                 the
                 parallel
                 sine
                 of
                 41.
                 45
                 ,
                 and
                 measure
                 it
                 on
                 the
                 line
                 of
                 inches
                 ,
                 or
                 equal
                 parts
                 ,
                 and
                 it
                 shall
                 reach
                 to
                 2
                 inches
                 42
                 parts
                 ,
                 or
                 242
                 ,
                 the
                 Altitude
                 required
                 .
              
               
                 After
                 the
                 same
                 manner
                 may
                 questions
                 be
                 wrought
                 on
                 the
                 line
                 of
                 lines
                 ,
                 sines
                 ,
                 or
                 tangents
                 alone
                 ,
                 or
                 any
                 one
                 with
                 the
                 other
                 ,
                 by
                 changing
                 the
                 Logarithmetical
                 Canon
                 from
                 the
                 first
                 to
                 the
                 second
                 ,
                 or
                 third
                 ,
                 and
                 the
                 second
                 or
                 third
                 to
                 the
                 first
                 to
                 second
                 ;
                 as
                 the
                 case
                 shall
                 require
                 ,
                 from
                 a
                 greater
                 to
                 a
                 less
                 ,
                 and
                 the
                 contrary
                 ;
                 for
                 the
                 fourth
                 is
                 always
                 the
                 same
                 ;
                 of
                 which
                 in
                 the
                 use
                 of
                 the
                 Sector
                 ,
                 by
                 
                   Edmund
                   Gunter
                
                 ,
                 you
                 may
                 finde
                 many
                 examples
                 ,
                 to
                 which
                 I
                 refer
                 you
                 .
              
               
                 Also
                 without
                 the
                 lines
                 of
                 sines
                 either
                 
                 natural
                 or
                 artificial
                 ,
                 you
                 may
                 find
                 altitudes
                 ,
                 by
                 putting
                 the
                 line
                 of
                 quadrat
                 ,
                 or
                 shadows
                 ,
                 on
                 the
                 Rule
                 as
                 in
                 a
                 quadrant
                 ,
                 then
                 the
                 directions
                 in
                 the
                 use
                 of
                 the
                 quadrat
                 ,
                 page
                 ▪
                 146
                 of
                 the
                 
                   Carpenters
                   Rule
                
                 ,
                 will
                 serve
                 your
                 turn
                 ,
                 which
                 runs
                 thus
                 :
              
               
                 As
                 100
                 (
                 or
                 50
                 according
                 as
                 it
                 is
                 divided
                 )
                 to
                 the
                 parts
                 cut
                 by
                 the
                 thread
                 ,
                 so
                 is
                 the
                 distance
                 measured
                 ,
                 to
                 the
                 height
                 required
                 ;
                 which
                 work
                 is
                 performed
                 by
                 the
                 line
                 of
                 numbers
                 onely
                 .
                 Or
                 again
                 ,
                 As
                 the
                 parts
                 cut
                 ,
                 to
                 100
                 or
                 50
                 :
                 so
                 is
                 the
                 height
                 to
                 the
                 distance
                 required
                 .
                 But
                 when
                 the
                 thread
                 falls
                 on
                 the
                 contrary
                 shadow
                 ,
                 that
                 is
                 ,
                 maketh
                 an
                 Angle
                 above
                 45
                 ,
                 then
                 the
                 work
                 is
                 just
                 the
                 contrary
                 to
                 the
                 former
                 .
              
               
                 What
                 is
                 spoken
                 here
                 of
                 taking
                 of
                 Altitudes
                 ,
                 may
                 be
                 applied
                 to
                 the
                 taking
                 of
                 distances
                 ;
                 for
                 if
                 the
                 Sector
                 be
                 fitted
                 with
                 a
                 staff
                 ,
                 and
                 a
                 ball-socket
                 ,
                 you
                 may
                 turn
                 it
                 either
                 horizontal
                 ,
                 or
                 perpendicular
                 ,
                 and
                 so
                 take
                 any
                 Angle
                 with
                 it
                 ,
                 very
                 conveniently
                 and
                 
                 readily
                 by
                 the
                 same
                 rules
                 and
                 directions
                 as
                 were
                 given
                 for
                 the
                 finding
                 of
                 Altitudes
                 .
              
            
          
        
         
           
             CHAP.
             XVII
             .
          
           
             
               The
               use
               of
               certain
               lines
               for
               the
               mensuration
               of
               superficial
               and
               solid
               bodies
               ,
               usually
               inserted
               on
               Ioynt-Rules
               for
               the
               use
               of
               Work-men
               ,
               of
               several
               sorts
               and
               kindes
               .
            
             
               FIrst
               the
               most
               general
               and
               received
               line
               for
               mensuration
               of
               Magnitudes
               ,
               is
               a
               foot
               divided
               into
               12
               inches
               ,
               and
               those
               inches
               into
               8
               ,
               10
               ,
               12
               ,
               or
               more
               parts
               ;
               but
               this
               being
               not
               so
               apt
               for
               application
               to
               the
               numbers
               ,
               I
               shall
               not
               insist
               of
               it
               here
               ,
               but
               rather
               refer
               you
               to
               the
               
                 Carpenters
                 Rule
              
               ;
               yet
               nevertheless
               those
               inches
               ,
               laid
               by
               a
               line
               of
               foot
               measure
               ,
               doth
               by
               occular
               inspection
               onely
               ,
               serve
               to
               reduce
               foot
               measure
               to
               inches
               ,
               and
               inches
               likewise
               to
               foot
               measure
               ,
               and
               some
               other
               conclusions
               also
               .
            
             
             
               1.
               
               As
               first
               ,
               The
               price
               of
               any
               commodity
               at
               five
               score
               to
               the
               hundred
               ,
               either
               tale
               or
               weight
               ,
               being
               given
               ,
               to
               finde
               the
               price
               of
               one
               in
               number
               ,
               or
               one
               pound
               in
               weight
               .
            
             
               As
               suppose
               at
               two
               pence
               half-peny
               a
               pound
               ,
               (
               or
               one
               )
               I
               demand
               to
               what
               cometh
               the
               hundred
               weight
               ,
               (
               or
               five-score
               ,
               )
               counting
               so
               many
               pound
               to
               the
               hundred
               weight
               ?
            
             
               If
               you
               look
               for
               two
               inches
               and
               a
               half
               ,
               representing
               two
               pence
               two
               farthings
               ,
               right
               against
               it
               on
               the
               foot
               measure
               ,
               you
               have
               21
               very
               near
               ;
               for
               if
               you
               conceive
               the
               space
               between
               20
               and
               21
               ,
               to
               be
               parted
               into
               12
               parts
               ,
               this
               will
               be
               found
               to
               contain
               ten
               of
               them
               ,
               for
               the
               odde
               ten
               pence
               .
               But
               for
               the
               more
               certain
               computation
               of
               the
               odde
               pence
               ,
               look
               how
               many
               farthings
               there
               is
               in
               the
               price
               of
               one
               pound
               ,
               twice
               so
               many
               shillings
               ,
               and
               once
               so
               many
               pence
               is
               the
               remainder
               ,
               which
               if
               it
               be
               above
               12
               ,
               the
               12
               or
               12s
               ,
               being
               substracted
               ,
               the
               remainder
               is
               the
               precise
               number
               of
               pence
               ,
               
               above
               the
               shillings
               there
               expressed
               ;
               and
               on
               the
               contrary
               ,
               at
               any
               price
               the
               C
               hundred
               ,
               or
               5
               score
               ,
               to
               finde
               the
               price
               of
               one
               ,
               or
               1l
               .
               As
               suppose
               at
               40s
               .
               the
               C.
               or
               5
               score
               ,
               look
               for
               40
               in
               the
               foot
               measure
               ,
               and
               right
               against
               it
               in
               the
               inches
               ,
               you
               have
               4
               inches
               ,
               3
               quarters
               ,
               and
               1
               /
               4
               of
               a
               quarter
               ,
               which
               in
               this
               way
               of
               account
               is
               4
               pence
               3
               farthings
               ,
               and
               about
               a
               quarter
               of
               one
               farthing
               .
               Thus
               by
               the
               lines
               ,
               as
               they
               are
               divided
               ,
               it
               proceeds
               to
               12
               pence
               a
               pound
               ;
               but
               if
               you
               conceive
               the
               inches
               to
               be
               doubled
               ,
               and
               the
               foot
               measure
               also
               ,
               you
               shall
               have
               it
               to
               24
               d.
               or
               48
               d.
               the
               pound
               ,
               or
               one
               in
               tale
               ,
               of
               any
               commodity
               .
               As
               at
               18
               d.
               a
               piece
               ,
               or
               pound
               ,
               the
               price
               comes
               to
               7l
               .
               10s
               .
               the
               C.
               for
               then
               every
               ten
               strokes
               is
               20s
               in
               the
               foot
               measure
               ,
               and
               every
               inch
               is
               2
               pence
               ,
               and
               every
               eighth
               one
               farthing
               .
            
             
               2.
               
               Secondly
               ,
               for
               the
               buying
               of
               Timber
               at
               50
               foot
               to
               a
               Load
               ,
               at
               any
               price
               the
               load
               ,
               how
               much
               a
               foot
               .
            
             
               Here
               in
               resolving
               this
               ,
               the
               inches
               
               are
               to
               be
               doubled
               ,
               and
               the
               foot
               measure
               taken
               as
               it
               is
               :
               As
               at
               40
               shillings
               the
               Load
               ,
               40
               in
               the
               foot
               measure
               stands
               right
               against
               4
               inches
               3
               quarters
               and
               better
               ,
               which
               being
               doubled
               ,
               is
               9d
               .
               2
               far
               .
               1
               /
               2
               far
               .
               near
               ,
               for
               the
               price
               of
               one
               foot
               ;
               and
               on
               the
               contrary
               ,
               at
               5d
               .
               a
               foot
               ,
               is
               41s
               .
               8
               d.
               a
               load
               ,
               &c.
               
            
             
               3.
               
               For
               the
               great
               Hundred
               of
               112l
               .
               to
               the
               Hundred
               ,
               let
               the
               space
               of
               12
               inches
               be
               divided
               into
               112
               parts
               ,
               then
               the
               like
               rule
               holds
               for
               that
               also
               .
            
             
               For
               the
               inches
               being
               divided
               into
               quarters
               ,
               every
               quarter
               is
               a
               farthing
               ,
               and
               every
               eighth
               half
               a
               farthing
               ,
               and
               every
               division
               of
               the
               112
               is
               a
               shilling
               ,
               and
               every
               alteration
               of
               a
               farthing
               in
               the
               price
               of
               a
               pound
               ,
               makes
               a
               groat
               in
               the
               Hundred
               ,
               as
               thus
               :
               At
               3
               pence
               a
               pound
               is
               28s
               .
               the
               C.
               At
               3d.
               1
               q.
               a
               pound
               ,
               30s
               .
               and
               4d
               .
               the
               C.
               At
               3d.
               1
               /
               2
               the
               pound
               ,
               32s
               .
               8
               d.
               the
               C.
               At
               3
               d.
               3
               farthings
               ,
               35s
               .
               the
               C.
               Thus
               you
               see
               that
               every
               fraction
               at
               a
               farthing
               advance
               ,
               is
               4
               pence
               in
               the
               Hundred
               ;
               but
               for
               any
               other
               account
               ,
               as
               3
               pence
               
               farthing
               half
               farthing
               ,
               then
               count
               the
               fraction
               ,
               as
               1
               ,
               12th
               part
               of
               a
               shilling
               ,
               and
               nearer
               you
               cannot
               come
               by
               a
               bare
               occular
               inspection
               ;
               but
               the
               price
               of
               the
               Hundred
               being
               given
               ,
               the
               price
               of
               the
               pound
               you
               have
               as
               near
               by
               this
               occular
               inspection
               ,
               as
               any
               usual
               Coin
               is
               reducable
               ,
               viz.
               to
               the
               32
               part
               of
               a
               peny
               ,
               or
               nigher
               if
               you
               please
               .
            
             
               Again
               note
               here
               also
               ,
               you
               may
               double
               ,
               or
               quadruple
               the
               price
               :
               as
               to
               24d
               .
               or
               48d
               .
               the
               pound
               ,
               or
               any
               price
               between
               .
               As
               for
               example
               :
               At
               13d
               .
               a
               pound
               ,
               is
               6l
               .
               1s
               .
               4d
               .
               the
               C.
               At
               32d
               .
               or
               2s
               .
               8d
               .
               the
               pound
               ,
               is
               14l
               .
               19s
               .
               the
               C.
               and
               the
               like
               by
               dupling
               and
               quadrupling
               the
               inches
               ,
               and
               the
               112
               parts
               ,
               that
               layeth
               by
               it
               .
            
             
               4.
               
               These
               lines
               of
               equal
               parts
               serve
               as
               Scales
               ,
               for
               the
               protracting
               of
               any
               Draught
               of
               house
               or
               field
               ,
               or
               the
               like
               ;
               also
               for
               addition
               or
               substraction
               of
               any
               small
               number
               .
            
             
               5.
               
               Note
               that
               the
               line
               of
               foot
               measure
               ,
               may
               be
               applied
               for
               the
               reducing
               of
               any
               odde
               fraction
               to
               a
               decimal
               fraction
               ,
               
               as
               you
               may
               fee
               it
               in
               page
               64.
               of
               Mr.
               
                 Windgats
                 Arithmetick
              
               made
               easie
               .
            
          
           
             
               2.
               
               The
               use
               of
               the
               lines
               of
               decimal
               Timber
               and
               Board
               measure
               .
            
             
               The
               lines
               of
               decimal
               Timber
               and
               Board
               measure
               ,
               are
               fitted
               to
               agree
               with
               the
               tenths
               ,
               or
               foot
               measure
               ,
               as
               those
               lines
               in
               the
               first
               chapter
               of
               the
               
                 Carpenters
                 Rule
              
               ,
               are
               fitted
               to
               the
               inches
               ,
               and
               the
               use
               of
               them
               is
               thus
               :
               And
               first
               for
               the
               decimal
               Board
               measure
               .
               Suppose
               a
               Board
               is
               1
               foot
               50
               broad
               ,
               I
               look
               for
               150
               on
               that
               line
               ,
               and
               from
               that
               place
               to
               the
               end
               of
               the
               Rule
               forwards
               ,
               toward
               100
               ,
               so
               much
               in
               length
               must
               you
               have
               to
               make
               a
               foot
               of
               superficial
               or
               board
               measure
               .
            
             
               2.
               
               Or
               else
               thus
               :
               If
               you
               apply
               the
               end
               of
               the
               Rule
               next
               100
               ,
               to
               one
               edge
               of
               the
               breadth
               of
               a
               board
               ,
               or
               glass
               ,
               then
               right
               against
               the
               other
               edge
               of
               the
               board
               ,
               on
               that
               line
               of
               decimal
               board
               measure
               ,
               you
               shall
               finde
               the
               10ths
               and
               100s
               ,
               (
               or
               feet
               10ths
               &
               100
               )
               
               parts
               of
               a
               foot
               ,
               that
               you
               must
               have
               in
               length
               to
               make
               a
               foot
               superficial
               at
               that
               breadth
               .
            
             
               
                 Example
                 .
              
               
                 I
                 come
                 to
                 a
                 board
                 and
                 applying
                 the
                 upper
                 end
                 next
                 100
                 ,
                 even
                 to
                 one
                 edge
                 of
                 the
                 board
                 ,
                 the
                 other
                 reacheth
                 to
                 0.
                 8
                 tenths
                 ,
                 then
                 I
                 say
                 that
                 8
                 tenths
                 of
                 a
                 foot
                 length
                 at
                 that
                 breadth
                 ,
                 makes
                 a
                 foot
                 .
              
            
          
           
             
               3.
               
               The
               use
               of
               decimal
               Timber
               measure
               .
            
             
               The
               use
               of
               this
               is
               much
               like
               the
               Board
               measure
               ,
               onely
               here
               you
               must
               have
               a
               respect
               to
               the
               squareness
               of
               the
               piece
               ,
               and
               not
               to
               the
               breadth
               onely
               ;
               for
               after
               you
               know
               how
               much
               the
               piece
               of
               timber
               or
               stone
               is
               square
               ,
               in
               feet
               and
               100
               parts
               ,
               then
               look
               that
               number
               on
               the
               line
               of
               decimal
               Timber
               measure
               ,
               and
               from
               thence
               to
               the
               end
               of
               the
               Rule
               ,
               is
               the
               length
               that
               goes
               to
               make
               a
               foot
               of
               timber
               .
            
             
               
               
                 Example
                 .
              
               
                 At
                 14
                 ,
                 or
                 1.
                 40.
                 parts
                 of
                 a
                 foot
                 square
                 ,
                 look
                 the
                 same
                 on
                 the
                 rule
                 ,
                 and
                 from
                 thence
                 to
                 the
                 end
                 where
                 40
                 is
                 ,
                 is
                 the
                 length
                 of
                 a
                 foot
                 of
                 Timber
                 ,
                 at
                 that
                 squareness
                 ,
                 being
                 about
                 51
                 parts
                 of
                 a
                 foot
                 divided
                 into
                 a
                 100
                 parts
                 .
              
            
          
           
             
               5.
               
               The
               use
               of
               the
               line
               of
               decimal
               yard
               measure
               ,
               also
               running
               yard
               measure
               ,
               according
               to
               the
               inches
               or
               decimal
               parts
               of
               a
               foot
               .
            
             
               The
               decimal
               yard
               measure
               ,
               is
               nothing
               else
               but
               a
               yard
               or
               3
               foot
               divided
               into
               a
               100
               parts
               ,
               and
               used
               in
               the
               same
               manner
               as
               the
               foot
               measure
               is
               ,
               for
               if
               you
               take
               the
               length
               ,
               and
               the
               breadth
               in
               that
               measure
               ,
               and
               multiply
               it
               together
               ,
               you
               shall
               have
               the
               content
               ,
               in
               yards
               and
               100
               parts
               of
               a
               yard
               .
            
             
               
                 Example
                 .
              
               
                 Suppose
                 a
                 peece
                 of
                 plastering
                 is
                 4
                 yards
                 78
                 parts
                 one
                 way
                 ;
                 and
                 7.
                 35
                 parts
                 another
                 way
                 :
                 being
                 multiplied
                 together
                 makes
                 35
                 yards
                 ,
                 and
                 9954.
                 of
                 10000
                 ,
                 which
                 is
                 very
                 neer
                 36
                 yards
                 .
              
               
               
                 5.
                 
                 But
                 the
                 decimal
                 running
                 yard
                 measure
                 is
                 fitted
                 to
                 the
                 foot
                 measure
                 ,
                 and
                 the
                 use
                 is
                 thus
                 :
                 Suppose
                 a
                 room
                 is
                 to
                 be
                 measured
                 that
                 is
                 7
                 foot
                 8
                 tenths
                 high
                 ,
                 and
                 I
                 would
                 know
                 how
                 much
                 makes
                 a
                 yard
                 ,
                 at
                 that
                 breadth
                 or
                 height
                 ;
                 look
                 foor
                 7f
                 .
                 8
                 10ths
                 on
                 the
                 line
                 of
                 decimal
                 running
                 yard
                 measure
                 ,
                 and
                 the
                 space
                 on
                 the
                 rule
                 ,
                 from
                 thence
                 to
                 the
                 end
                 next
                 100
                 is
                 the
                 true
                 length
                 ,
                 that
                 goeth
                 to
                 make
                 up
                 a
                 yard
                 of
                 superficial
                 measure
                 ,
                 at
                 that
                 breadth
                 or
                 height
                 .
              
               
                 But
                 if
                 the
                 peece
                 be
                 between
                 4
                 foot
                 5
                 10
                 broad
                 ,
                 and
                 2
                 foot
                 ,
                 then
                 the
                 table
                 at
                 the
                 end
                 of
                 the
                 line
                 ,
                 will
                 supply
                 the
                 defect
                 :
                 or
                 you
                 may
                 change
                 the
                 terms
                 ,
                 and
                 call
                 the
                 length
                 the
                 breadth
                 ,
                 and
                 the
                 contrary
                 .
                 But
                 if
                 it
                 be
                 under
                 2
                 foot
                 broad
                 ,
                 then
                 if
                 you
                 do
                 as
                 you
                 did
                 with
                 the
                 board
                 measure
                 ,
                 you
                 shall
                 have
                 your
                 desire
                 .
              
            
             
               
                 Example
                 .
              
               
                 At
                 1
                 foot
                 3
                 10th
                 broad
                 ,
                 6
                 foot
                 9
                 10ths
                 make
                 a
                 yard
                 .
              
               
                 6.
                 
                 But
                 if
                 the
                 running
                 yard
                 measure
                 
                 be
                 made
                 to
                 agree
                 with
                 the
                 inches
                 ,
                 then
                 measure
                 the
                 height
                 of
                 the
                 room
                 in
                 feet
                 and
                 inches
                 ,
                 and
                 if
                 you
                 take
                 a
                 pair
                 of
                 compasses
                 ,
                 and
                 measure
                 from
                 that
                 place
                 ,
                 to
                 the
                 end
                 of
                 the
                 rule
                 ,
                 then
                 turn
                 the
                 compasses
                 set
                 at
                 that
                 distance
                 as
                 many
                 times
                 as
                 you
                 can
                 about
                 the
                 room
                 ,
                 so
                 many
                 yards
                 is
                 there
                 in
                 the
                 room
                 .
              
            
          
           
             
               7.
               
               The
               use
               of
               the
               line
               of
               decimal
               round
               measure
               ,
               commonly
               called
               Girt-measure
               ,
               which
               is
               when
               the
               circumference
               of
               a
               round
               Cillender
               ,
               or
               piller
               given
               in
               inches
               or
               ten
               parts
               of
               a
               foot
               .
            
             
               First
               (
               for
               Girt-measure
               according
               to
               inches
               ,
               being
               the
               most
               usual
               measure
               )
               now
               much
               the
               pillar
               is
               about
               ,
               then
               look
               for
               the
               same
               number
               on
               the
               line
               of
               Girt-measure
               ,
               and
               from
               thence
               to
               the
               end
               of
               the
               rule
               ,
               is
               the
               length
               that
               goeth
               to
               make
               a
               foot
               of
               Timber
               .
            
             
               But
               if
               it
               be
               under
               30
               inches
               about
               ,
               then
               you
               must
               have
               above
               two
               foot
               in
               length
               ,
               and
               then
               a
               table
               at
               the
               end
               of
               the
               line
               ,
               or
               a
               repetition
               in
               another
               
               line
               ,
               will
               supply
               the
               defect
               .
               But
               if
               the
               line
               of
               Girt-measure
               be
               divided
               according
               to
               foot
               measure
               ,
               then
               use
               it
               as
               before
               ,
               seeking
               the
               decimal
               part
               on
               the
               line
               ,
               and
               from
               thence
               to
               the
               end
               is
               a
               foot
               .
            
          
           
             
               8.
               
               The
               use
               of
               a
               line
               of
               solid
               measure
               ,
               by
               having
               the
               Diameter
               of
               a
               round
               piece
               given
               in
               inches
               ,
               or
               foot
               measure
               .
            
             
               Take
               the
               diameter
               with
               a
               rule
               ,
               or
               a
               pair
               of
               Callipers
               ,
               and
               learn
               the
               measure
               either
               in
               inches
               ,
               or
               foot
               measure
               according
               as
               your
               line
               of
               Diameter
               is
               divided
               .
               Then
               look
               for
               the
               same
               number
               on
               the
               line
               of
               Diameter
               ,
               and
               from
               thence
               to
               the
               end
               of
               the
               rule
               forward
               ,
               is
               the
               length
               that
               makes
               a
               foot
               of
               timber
               at
               that
               diameter
               (
               or
               measure
               cross
               the
               end
               of
               the
               round
               piece
               of
               Timber
               or
               stone
               .
               )
            
             
               The
               Tables
               of
               all
               the
               under
               measure
               for
               all
               these
               lines
               follow
               .
            
             
             
               
                 
                   Decimal
                   Superficiall
                   under
                   M.
                   
                
                 
                   
                      
                  
                   
                     10th
                     .
                  
                   
                     F.
                     1000
                  
                   
                      
                  
                   
                     10
                  
                   
                     F.
                     1000
                     p.
                     
                  
                
                 
                   
                      
                  
                   
                     1
                  
                   
                     100.
                     00
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     848
                  
                
                 
                   
                      
                  
                   
                     2
                  
                   
                     50.
                     000
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     706
                  
                
                 
                   
                      
                  
                   
                     3
                  
                   
                     33.
                     300
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     570
                  
                
                 
                   
                      
                  
                   
                     4
                  
                   
                     25.
                     000
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     450
                  
                
                 
                   
                      
                  
                   
                     5
                  
                   
                     20.
                     000
                  
                   
                     3
                  
                   
                     3
                  
                   
                     3.
                     332
                  
                
                 
                   
                      
                  
                   
                     6
                  
                   
                     16.
                     600
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     217
                  
                
                 
                   
                      
                  
                   
                     7
                  
                   
                     14.
                     300
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     115
                  
                
                 
                   
                      
                  
                   
                     8
                  
                   
                     12.
                     500
                  
                   
                      
                  
                   
                      
                  
                   
                     3.
                     025
                  
                
                 
                   
                      
                  
                   
                     9
                  
                   
                     11.
                     120
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     940
                  
                
                 
                   
                     1
                  
                   
                     1
                  
                   
                     10.
                     000
                  
                   
                      
                  
                   
                     5
                  
                   
                     2.
                     850
                  
                
                 
                   
                      
                  
                   
                     1
                  
                   
                     9.
                     100
                  
                   
                      
                  
                   
                     6
                  
                   
                     2.
                     780
                  
                
                 
                   
                      
                  
                   
                     2
                  
                   
                     8.
                     340
                  
                   
                      
                  
                   
                     7
                  
                   
                     2.
                     700
                  
                
                 
                   
                      
                  
                   
                     3
                  
                   
                     7.
                     720
                  
                   
                      
                  
                   
                     8
                  
                   
                     2.
                     628
                  
                
                 
                   
                      
                  
                   
                     4
                  
                   
                     7.
                     150
                  
                   
                      
                  
                   
                     9
                  
                   
                     2.
                     560
                  
                
                 
                   
                      
                  
                   
                     5
                  
                   
                     6.
                     670
                  
                   
                      
                  
                   
                     4
                  
                   
                     2.
                     500
                  
                
                 
                   
                      
                  
                   
                     6
                  
                   
                     6.
                     ●60
                  
                   
                      
                  
                   
                      
                  
                   
                     ●
                     .
                     440
                  
                
                 
                   
                      
                  
                   
                     7
                  
                   
                     5.
                     888
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     382
                  
                
                 
                   
                      
                  
                   
                     8
                  
                   
                     5.
                     5●5
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     336
                  
                
                 
                   
                      
                  
                   
                     9
                  
                   
                     5.
                     260
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     273
                  
                
                 
                   
                     2
                  
                   
                     2
                  
                   
                     5.
                     000
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     213
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     4.
                     760
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     173
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     4.
                     5●6
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     127
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     4.
                     350
                  
                   
                      
                  
                   
                      
                  
                   
                     ●
                     .
                     083
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     4.
                     170
                  
                   
                      
                  
                   
                      
                  
                   
                     2.
                     042
                  
                
                 
                   
                      
                  
                   
                     5
                  
                   
                     4
                     ▪
                     000
                  
                   
                      
                  
                   
                     5
                  
                   
                     2.
                     000
                  
                
              
               
                 
                 
                   Decimall
                   Superficiall
                   .
                   M.
                   
                
                 
                   
                      
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                   
                     01
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                
                 
                   
                      
                  
                   
                     1.
                     962
                  
                   
                      
                  
                   
                     1.
                     320
                  
                
                 
                   
                      
                  
                   
                     1.
                     923
                  
                   
                      
                  
                   
                     1.
                     304
                  
                
                 
                   
                      
                  
                   
                     1.
                     816
                  
                   
                      
                  
                   
                     1.
                     286
                  
                
                 
                   
                      
                  
                   
                     1.
                     850
                  
                   
                      
                  
                   
                     1.
                     268
                  
                
                 
                   
                      
                  
                   
                     1.
                     820
                  
                   
                     8
                  
                   
                     1.
                     250
                  
                
                 
                   
                      
                  
                   
                     1.
                     785
                  
                   
                      
                  
                   
                     1.
                     237
                  
                
                 
                   
                      
                  
                   
                     1.
                     756
                  
                   
                      
                  
                   
                     1.
                     220
                  
                
                 
                   
                      
                  
                   
                     1.
                     726
                  
                   
                      
                  
                   
                     1.
                     207
                  
                
                 
                   
                      
                  
                   
                     1.
                     697
                  
                   
                      
                  
                   
                     1.
                     192
                  
                
                 
                   
                     6
                  
                   
                     1.
                     669
                  
                   
                      
                  
                   
                     1.
                     178
                  
                
                 
                   
                      
                  
                   
                     1.
                     640
                  
                   
                      
                  
                   
                     1.
                     164
                  
                
                 
                   
                      
                  
                   
                     1.
                     615
                  
                   
                      
                  
                   
                     1.
                     151
                  
                
                 
                   
                      
                  
                   
                     1.
                     589
                  
                   
                      
                  
                   
                     1.
                     138
                  
                
                 
                   
                      
                  
                   
                     1.
                     563
                  
                   
                      
                  
                   
                     1.
                     125
                  
                
                 
                   
                      
                  
                   
                     1.
                     538
                  
                   
                     9
                  
                   
                     1.
                     112
                  
                
                 
                   
                      
                  
                   
                     1.
                     516
                  
                   
                      
                  
                   
                     1.
                     100
                  
                
                 
                   
                      
                  
                   
                     1.
                     493
                  
                   
                      
                  
                   
                     1.
                     087
                  
                
                 
                   
                      
                  
                   
                     1.
                     472
                  
                   
                      
                  
                   
                     1.
                     076
                  
                
                 
                   
                      
                  
                   
                     1.
                     450
                  
                   
                      
                  
                   
                     1.
                     063
                  
                
                 
                   
                     7
                  
                   
                     1.
                     430
                  
                   
                      
                  
                   
                     1.
                     052
                  
                
                 
                   
                      
                  
                   
                     1.
                     409
                  
                   
                      
                  
                   
                     1.
                     041
                  
                
                 
                   
                      
                  
                   
                     1.
                     391
                  
                   
                      
                  
                   
                     1.
                     030
                  
                
                 
                   
                      
                  
                   
                     1.
                     373
                  
                   
                      
                  
                   
                     1.
                     020
                  
                
                 
                   
                      
                  
                   
                     1.
                     353
                  
                   
                      
                  
                   
                     1.
                     011
                  
                
                 
                   
                      
                  
                   
                     1.
                     337
                  
                   
                     10
                  
                   
                     1.
                     000
                  
                
              
               
                 
                 
                   Decimal
                   Solid
                   under
                   Measure
                   .
                
                 
                   
                      
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                   
                     10
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                
                 
                   
                     1
                  
                   
                     10000.
                     000
                  
                   
                      
                  
                   
                     14.
                     805
                  
                
                 
                   
                     2
                  
                   
                     2500.
                     000
                  
                   
                      
                  
                   
                     13.
                     735
                  
                
                 
                   
                     3
                  
                   
                     1100.
                     000
                  
                   
                      
                  
                   
                     12.
                     780
                  
                
                 
                   
                     4
                  
                   
                     630.
                     000
                  
                   
                      
                  
                   
                     11.
                     916
                  
                
                 
                   
                     5
                  
                   
                     400.
                     000
                  
                   
                     3
                  
                   
                     11.
                     125
                  
                
                 
                   
                     6
                  
                   
                     277.
                     900
                  
                   
                      
                  
                   
                     10.
                     415
                  
                
                 
                   
                     7
                  
                   
                     200.
                     430
                  
                   
                      
                  
                   
                     9.
                     760
                  
                
                 
                   
                     8
                  
                   
                     150.
                     660
                  
                   
                      
                  
                   
                     9.
                     125
                  
                
                 
                   
                     9
                  
                   
                     120.
                     350
                  
                   
                      
                  
                   
                     8.
                     625
                  
                
                 
                   
                     1
                  
                   
                     100.
                     0000
                  
                   
                      
                  
                   
                     8.
                     150
                  
                
                 
                   
                      
                  
                   
                     82.
                     800
                  
                   
                      
                  
                   
                     7.
                     700
                  
                
                 
                   
                      
                  
                   
                     96.
                     500
                  
                   
                      
                  
                   
                     7.
                     310
                  
                
                 
                   
                      
                  
                   
                     59.
                     390
                  
                   
                      
                  
                   
                     6.
                     900
                  
                
                 
                   
                      
                  
                   
                     51.
                     100
                  
                   
                      
                  
                   
                     6.
                     565
                  
                
                 
                   
                      
                  
                   
                     44.
                     500
                  
                   
                     4
                  
                   
                     6.
                     250
                  
                
                 
                   
                      
                  
                   
                     39.
                     150
                  
                   
                      
                  
                   
                     5.
                     945
                  
                
                 
                   
                      
                  
                   
                     34.
                     650
                  
                   
                      
                  
                   
                     5.
                     664
                  
                
                 
                   
                      
                  
                   
                     30.
                     850
                  
                   
                      
                  
                   
                     5.
                     404
                  
                
                 
                   
                      
                  
                   
                     27.
                     750
                  
                   
                      
                  
                   
                     5.
                     465
                  
                
                 
                   
                     2
                  
                   
                     25.
                     000
                  
                   
                      
                  
                   
                     4.
                     938
                  
                
                 
                   
                      
                  
                   
                     22.
                     700
                  
                   
                      
                  
                   
                     4.
                     720
                  
                
                 
                   
                      
                  
                   
                     20.
                     675
                  
                   
                      
                  
                   
                     4.
                     530
                  
                
                 
                   
                      
                  
                   
                     18.
                     920
                  
                   
                      
                  
                   
                     4.
                     342
                  
                
                 
                   
                      
                  
                   
                     17.
                     400
                  
                   
                      
                  
                   
                     4.
                     162
                  
                
                 
                   
                      
                  
                   
                     16.
                     000
                  
                   
                     5
                  
                   
                     4.
                     000
                  
                
              
               
                 
                 
                   Decimall
                   Solid
                   under
                   measure
                   .
                
                 
                   
                      
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                   
                     01.
                     
                  
                   
                     F.
                     100.
                     p.
                     
                  
                
                 
                   
                      
                  
                   
                     3.
                     825
                  
                   
                      
                  
                   
                     1.
                     738
                  
                
                 
                   
                      
                  
                   
                     3.
                     7●0
                  
                   
                      
                  
                   
                     1.
                     694
                  
                
                 
                   
                      
                  
                   
                     3.
                     524
                  
                   
                      
                  
                   
                     1.
                     651
                  
                
                 
                   
                      
                  
                   
                     3.
                     430
                  
                   
                      
                  
                   
                     1.
                     608
                  
                
                 
                   
                      
                  
                   
                     3.
                     310
                  
                   
                     8
                  
                   
                     1.
                     568
                  
                
                 
                   
                      
                  
                   
                     3.
                     188
                  
                   
                      
                  
                   
                     1.
                     528
                  
                
                 
                   
                      
                  
                   
                     3.
                     078
                  
                   
                      
                  
                   
                     1.
                     493
                  
                
                 
                   
                      
                  
                   
                     2.
                     968
                  
                   
                      
                  
                   
                     1.
                     458
                  
                
                 
                   
                      
                  
                   
                     2.
                     873
                  
                   
                      
                  
                   
                     1.
                     420
                  
                
                 
                   
                     6
                  
                   
                     2.
                     780
                  
                   
                      
                  
                   
                     1.
                     390
                  
                
                 
                   
                      
                  
                   
                     2.
                     688
                  
                   
                      
                  
                   
                     1.
                     356
                  
                
                 
                   
                      
                  
                   
                     2.
                     602
                  
                   
                      
                  
                   
                     1.
                     323
                  
                
                 
                   
                      
                  
                   
                     2.
                     521
                  
                   
                      
                  
                   
                     1.
                     297
                  
                
                 
                   
                      
                  
                   
                     2.
                     442
                  
                   
                      
                  
                   
                     1.
                     266
                  
                
                 
                   
                      
                  
                   
                     2.
                     366
                  
                   
                     9
                  
                   
                     1.
                     236
                  
                
                 
                   
                      
                  
                   
                     2.
                     294
                  
                   
                      
                  
                   
                     1.
                     208
                  
                
                 
                   
                      
                  
                   
                     2.
                     227
                  
                   
                      
                  
                   
                     1.
                     185
                  
                
                 
                   
                      
                  
                   
                     2.
                     160
                  
                   
                      
                  
                   
                     1.
                     160
                  
                
                 
                   
                      
                  
                   
                     2.
                     100
                  
                   
                      
                  
                   
                     1.
                     131
                  
                
                 
                   
                      
                  
                   
                     2.
                     043
                  
                   
                      
                  
                   
                     1.
                     109
                  
                
                 
                   
                     7
                  
                   
                      
                  
                   
                      
                  
                   
                      
                  
                
                 
                   
                      
                  
                   
                     1.
                     985
                  
                   
                      
                  
                   
                     1.
                     084
                  
                
                 
                   
                      
                  
                   
                     1.
                     93●
                  
                   
                      
                  
                   
                     1
                     ,
                     061
                  
                
                 
                   
                      
                  
                   
                     1.
                     878
                  
                   
                      
                  
                   
                     1.
                     041
                  
                
                 
                   
                      
                  
                   
                     1.
                     830
                  
                   
                      
                  
                   
                     1.
                     021
                  
                
                 
                   
                      
                  
                   
                     1.
                     781
                  
                   
                     10.
                     
                  
                   
                     1.
                     000
                  
                
              
               
                 
                 
                   Vnder
                   Yard-measure
                   for
                   feet
                   and
                   inches
                   from
                   one
                   inch'to
                   four
                   feet
                   six
                   inches
                
                 
                   
                     F.
                     
                  
                   
                     F.
                     1000.
                     
                  
                   
                     F.
                     
                  
                   
                     F.
                     1000.
                     
                  
                
                 
                   
                     In.
                     
                  
                   
                      
                  
                   
                     In.
                     
                  
                   
                      
                  
                
                 
                   
                     1
                  
                   
                     108.000
                  
                   
                      
                  
                   
                     3.
                     850
                  
                
                 
                   
                     2
                  
                   
                     54.
                     000
                  
                   
                      
                  
                   
                     3.
                     720
                  
                
                 
                   
                     3
                  
                   
                     36.
                     000
                  
                   
                      
                  
                   
                     3.
                     600
                  
                
                 
                   
                     4
                  
                   
                     27.
                     000
                  
                   
                     6
                  
                   
                     3.
                     482
                  
                
                 
                   
                     5
                  
                   
                     21.
                     600
                  
                   
                      
                  
                   
                     3.
                     373
                  
                
                 
                   
                     6
                  
                   
                     18.
                     000
                  
                   
                      
                  
                   
                     3.
                     271
                  
                
                 
                   
                     7
                  
                   
                     15.
                     420
                  
                   
                      
                  
                   
                     3.
                     175
                  
                
                 
                   
                     8
                  
                   
                     13.
                     520
                  
                   
                      
                  
                   
                     3.
                     085
                  
                
                 
                   
                     9
                  
                   
                     12.
                     000
                  
                   
                     3
                  
                   
                     3.
                     000
                  
                
                 
                   
                     10
                  
                   
                     10.
                     300
                  
                   
                      
                  
                   
                     2.
                     922
                  
                
                 
                   
                     11
                  
                   
                     9.
                     820
                  
                   
                      
                  
                   
                     2.
                     842
                  
                
                 
                   
                     1.
                     
                  
                   
                     9.
                     000
                  
                   
                      
                  
                   
                     2.
                     769
                  
                
                 
                   
                      
                  
                   
                     8.
                     320
                  
                   
                      
                  
                   
                     2.
                     710
                  
                
                 
                   
                      
                  
                   
                     7.
                     740
                  
                   
                      
                  
                   
                     2.
                     633
                  
                
                 
                   
                      
                  
                   
                     7.
                     201
                  
                   
                     6
                  
                   
                     2.
                     572
                  
                
                 
                   
                      
                  
                   
                     6.
                     760
                  
                   
                      
                  
                   
                     2.
                     512
                  
                
                 
                   
                      
                  
                   
                     6.
                     350
                  
                   
                      
                  
                   
                     2.
                     455
                  
                
                 
                   
                     6
                  
                   
                     6.
                     000
                  
                   
                      
                  
                   
                     2.
                     400
                  
                
                 
                   
                      
                  
                   
                     5.
                     680
                  
                   
                      
                  
                   
                     2.
                     345
                  
                
                 
                   
                      
                  
                   
                     5.
                     400
                  
                   
                      
                  
                   
                     2.
                     298
                  
                
                 
                   
                      
                  
                   
                     5.
                     140
                  
                   
                     4
                  
                   
                     2.
                     250
                  
                
                 
                   
                      
                  
                   
                     4.
                     906
                  
                   
                     1
                  
                   
                     2.
                     203
                  
                
                 
                   
                      
                  
                   
                     4.
                     695
                  
                   
                     2
                  
                   
                     2.
                     160
                  
                
                 
                   
                     2
                  
                   
                     5.
                     500
                  
                   
                     3
                  
                   
                     2.
                     119
                  
                
                 
                   
                      
                  
                   
                     4.
                     320
                  
                   
                     4
                  
                   
                     2.
                     073
                  
                
                 
                   
                      
                  
                   
                     4.
                     160
                  
                   
                     5
                  
                   
                     2.
                     037
                  
                
                 
                   
                      
                  
                   
                     4.
                     000
                  
                   
                     9
                  
                   
                     2.
                     000
                  
                
              
               
                 
                 
                   Vnder
                   yard
                   measure
                   according
                   to
                   Decimal
                   or
                   Foot
                   measure
                
                 
                   
                     F.
                     10.
                     
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                   
                      
                  
                   
                     F.
                     1000.
                     p.
                     
                  
                
                 
                   
                     1
                  
                   
                     90.
                     000
                  
                   
                     4
                  
                   
                     3.
                     7●0
                  
                
                 
                   
                     2
                  
                   
                     45.
                     000
                  
                   
                     5
                  
                   
                     3.
                     600
                  
                
                 
                   
                     3
                  
                   
                     30.
                     000
                  
                   
                     6
                  
                   
                     3.
                     461
                  
                
                 
                   
                     4
                  
                   
                     22.
                     500
                  
                   
                     7
                  
                   
                     3.
                     332
                  
                
                 
                   
                     5
                  
                   
                     18.
                     000
                  
                   
                     1
                  
                   
                     3.
                     211
                  
                
                 
                   
                     6
                  
                   
                     15.
                     000
                  
                   
                     9
                  
                   
                     3.
                     104
                  
                
                 
                   
                     7
                  
                   
                     12.
                     880
                  
                   
                     3
                  
                   
                     3.
                     000
                  
                
                 
                   
                     8
                  
                   
                     11.
                     200
                  
                   
                     1
                  
                   
                     2.
                     903
                  
                
                 
                   
                     9
                  
                   
                     10.
                     000
                  
                   
                     2
                  
                   
                     2.
                     812
                  
                
                 
                   
                     1
                  
                   
                     9.
                     000
                  
                   
                     3
                  
                   
                     2.
                     728
                  
                
                 
                   
                     1
                  
                   
                     8.
                     190
                  
                   
                     4
                  
                   
                     2.
                     648
                  
                
                 
                   
                     2
                  
                   
                     7.
                     510
                  
                   
                     5
                  
                   
                     2.
                     572
                  
                
                 
                   
                     3
                  
                   
                     6.
                     930
                  
                   
                     6
                  
                   
                     2.
                     502
                  
                
                 
                   
                     4
                  
                   
                     6.
                     430
                  
                   
                     7
                  
                   
                     2.
                     435
                  
                
                 
                   
                     5
                  
                   
                     6.
                     000
                  
                   
                     8
                  
                   
                     2.
                     370
                  
                
                 
                   
                     6
                  
                   
                     5.
                     625
                  
                   
                     9
                  
                   
                     2.
                     310
                  
                
                 
                   
                     7
                  
                   
                     5.
                     290
                  
                   
                     4
                  
                   
                     2.
                     250
                  
                
                 
                   
                     8
                  
                   
                     5.
                     000
                  
                   
                     1
                  
                   
                     2.
                     195
                  
                
                 
                   
                     9
                  
                   
                     4.
                     735
                  
                   
                     2
                  
                   
                     2.
                     142
                  
                
                 
                   
                     1
                  
                   
                     4.
                     500
                  
                   
                     3
                  
                   
                     2.
                     093
                  
                
                 
                   
                     1
                  
                   
                     4.
                     285
                  
                   
                     4
                  
                   
                     2.
                     046
                  
                
                 
                   
                     2
                  
                   
                     4.
                     092
                  
                   
                     5
                  
                   
                     2.
                     000
                  
                
                 
                   
                     3
                  
                   
                     3912
                  
                   
                      
                  
                   
                      
                  
                
              
               
                 
                 
                   Vnder
                   Girt-measure
                   .
                
                 
                   
                     Inc.
                     about
                  
                   
                     F.
                     in
                     .
                     100.
                     
                  
                   
                      
                  
                   
                     F.
                     in
                     .
                     100
                  
                
                 
                   
                     1
                  
                   
                     1809.6.81
                  
                   
                     24
                  
                   
                     3.1.87
                  
                
                 
                   
                     2
                  
                   
                     452.
                     4.
                     74
                  
                   
                     25
                  
                   
                     2.
                     10.
                     74
                  
                
                 
                   
                     3
                  
                   
                     201.
                     0.
                     77
                  
                   
                     26
                  
                   
                     2.
                     8.
                     12
                  
                
                 
                   
                     4
                  
                   
                     113.
                     1.
                     18
                  
                   
                     27
                  
                   
                     2.
                     5.
                     87
                  
                
                 
                   
                     5
                  
                   
                     72.
                     4.
                     60
                  
                   
                     28
                  
                   
                     2.
                     3.
                     70
                  
                
                 
                   
                     6
                  
                   
                     50.
                     3.
                     19
                  
                   
                     29
                  
                   
                     2.
                     1.
                     83
                  
                
                 
                   
                     7
                  
                   
                     39.
                     3.
                     22
                  
                   
                     30
                  
                   
                     2.
                     0.
                     13
                  
                
                 
                   
                     8
                  
                   
                     28.
                     4.
                     00
                  
                   
                     31
                  
                   
                     1.
                     10.
                     60
                  
                
                 
                   
                     9
                  
                   
                     22.
                     4.
                     09
                  
                   
                     32
                  
                   
                     1.
                     9.
                     21
                  
                
                 
                   
                     10
                  
                   
                     18.
                     1.
                     15
                  
                   
                     33
                  
                   
                     1.
                     7
                     
                     :
                     94
                  
                
                 
                   
                     11
                  
                   
                     14.
                     11.
                     46
                  
                   
                     34
                  
                   
                     1.
                     6.
                     78
                  
                
                 
                   
                     12
                  
                   
                     12.
                     6.
                     80
                  
                   
                     35
                  
                   
                     1.
                     5.
                     72
                  
                
                 
                   
                     13
                  
                   
                     10.
                     8.
                     09
                  
                   
                     36
                  
                   
                     1,4,75
                  
                
                 
                   
                     14
                  
                   
                     9.
                     2.
                     79
                  
                   
                     37
                  
                   
                     1.
                     3
                     ,
                     86
                  
                
                 
                   
                     15
                  
                   
                     8.
                     0.
                     51
                  
                   
                     38
                  
                   
                     1
                     ,
                     3
                     ,
                     04
                  
                
                 
                   
                     16
                  
                   
                     7.
                     0.
                     82
                  
                   
                     39
                  
                   
                     1
                     ,
                     2
                     ,
                     28
                  
                
                 
                   
                     17
                  
                   
                     6.
                     3.
                     14
                  
                   
                     40
                  
                   
                     1
                     ,
                     1
                     ,
                     57
                  
                
                 
                   
                     18
                  
                   
                     5.
                     7.
                     30
                  
                   
                     41
                  
                   
                     1
                     ,
                     0
                     ,
                     92
                  
                
                 
                   
                     19
                  
                   
                     5.
                     0.
                     15
                  
                   
                     42
                  
                   
                     1
                     ,
                     0
                     ,
                     31
                  
                
                 
                   
                     20
                  
                   
                     4.
                     6.
                     28
                  
                   
                     43
                  
                   
                     0
                     ,
                     11
                     ,
                     74
                  
                
                 
                   
                     21
                  
                   
                     4.
                     1.
                     24
                  
                   
                     44
                  
                   
                     0
                     ,
                     11
                     ,
                     22
                  
                
                 
                   
                     22
                  
                   
                     3.
                     8.
                     87
                  
                   
                     45
                  
                   
                     0
                     ,
                     10
                     ,
                     72
                  
                
                 
                   
                     23
                  
                   
                     3.
                     5.
                     04
                  
                   
                      
                  
                   
                      
                  
                
              
               
                 
                 
                   Vnder
                   measure
                   for
                   the
                   Diameter
                   in
                   Inches
                   and
                   quarters
                   .
                
                 
                   
                     In.
                     over
                  
                   
                     F.
                     in
                     10
                  
                   
                      
                  
                   
                     F.
                     in
                     100
                  
                
                 
                   
                     ,
                  
                   
                     29
                     3
                     0
                     ,
                     0
                     ,
                     00
                  
                   
                     ,
                  
                   
                     4
                     ,
                     8
                     ,
                     31
                  
                
                 
                   
                     ,
                  
                   
                     73●
                     ,
                     1
                     ,
                     00
                  
                   
                     ,
                  
                   
                     4
                     ,
                     4
                     ,
                     06
                  
                
                 
                   
                     ,
                  
                   
                     326
                     ,
                     0
                     ,
                     10
                  
                   
                     ,
                  
                   
                     4
                     ,
                     0
                     ,
                     27
                  
                
                 
                   
                     1
                  
                   
                     183
                     ,
                     ●
                     ,
                     80
                  
                   
                     7
                  
                   
                     3
                     ,
                     8
                     ,
                     89
                  
                
                 
                   
                     ,
                  
                   
                     118
                     ,
                     2
                     ,
                     16
                  
                   
                     ,
                  
                   
                     3
                     ,
                     5
                     ,
                     85
                  
                
                 
                   
                     ,
                  
                   
                     81
                     ,
                     ●
                     ,
                     00
                  
                   
                     ,
                  
                   
                     3
                     ,
                     3
                     ,
                     10
                  
                
                 
                   
                     ,
                  
                   
                     59
                     ,
                     10
                     ,
                     24
                  
                   
                     ,
                  
                   
                     3
                     ,
                     0
                     ,
                     61
                  
                
                 
                   
                     2
                  
                   
                     45
                     ,
                     10
                     ,
                     90
                  
                   
                     8
                  
                   
                     2
                     ,
                     10
                     ,
                     36
                  
                
                 
                   
                     ,
                  
                   
                     36
                     ,
                     2
                     ,
                     41
                  
                   
                     ,
                  
                   
                     2
                     ,
                     8
                     ,
                     31
                  
                
                 
                   
                     ,
                  
                   
                     29
                     ,
                     3
                     ,
                     93
                  
                   
                     ,
                  
                   
                     2
                     ,
                     6
                     ,
                     44
                  
                
                 
                   
                     ,
                  
                   
                     24
                     ,
                     3
                     ,
                     86
                  
                   
                     ,
                  
                   
                     2
                     ,
                     4
                     ,
                     73
                  
                
                 
                   
                     3
                  
                   
                     20
                     ,
                     4
                     ,
                     40
                  
                   
                     9
                  
                   
                     2
                     ,
                     3
                     ,
                     15
                  
                
                 
                   
                     ,
                  
                   
                     17
                     ,
                     4
                     ,
                     25
                  
                   
                     ,
                  
                   
                     1
                     ,
                     1
                     ,
                     70
                  
                
                 
                   
                     ,
                  
                   
                     15
                     ,
                     0
                     ,
                     00
                  
                   
                     ,
                  
                   
                     1
                     ,
                     0
                     ,
                     37
                  
                
                 
                   
                     ,
                  
                   
                     13
                     ,
                     1
                     ,
                     86
                  
                   
                     ,
                  
                   
                     1
                     ,
                     11
                     ,
                     13
                  
                
                 
                   
                     4
                  
                   
                     11
                     ,
                     5
                     ,
                     14
                  
                   
                     10
                  
                   
                     1
                     ,
                     9
                     ,
                     99
                  
                
                 
                   
                     ,
                  
                   
                     10
                     ,
                     1
                     ,
                     78
                  
                   
                     ,
                  
                   
                     1
                     ,
                     8
                     ,
                     93
                  
                
                 
                   
                     ,
                  
                   
                     9
                     ,
                     0
                     ,
                     62
                  
                   
                     ,
                  
                   
                     1
                     ,
                     7
                     ,
                     95
                  
                
                 
                   
                     ,
                  
                   
                     8
                     ,
                     1
                     ,
                     80
                  
                   
                     ,
                  
                   
                     1
                     ,
                     7
                     ,
                     04
                  
                
                 
                   
                     5
                  
                   
                     7
                     ,
                     3
                     ,
                     98
                  
                   
                     11
                  
                   
                     1
                     ,
                     6
                     ,
                     18
                  
                
                 
                   
                     ,
                  
                   
                     6
                     ,
                     7
                     ,
                     80
                  
                   
                     ,
                  
                   
                     1
                     ,
                     5
                     ,
                     38
                  
                
                 
                   
                     ,
                  
                   
                     6
                     ,
                     0
                     ,
                     72
                  
                   
                     ,
                  
                   
                     1
                     ,
                     4
                     ,
                     63
                  
                
                 
                   
                     ,
                  
                   
                     5
                     ,
                     6
                     ,
                     53
                  
                   
                     ,
                  
                   
                     1
                     ,
                     3
                     ,
                     93
                  
                
                 
                   
                     6
                  
                   
                     5
                     ,
                     1
                     ,
                     10
                  
                   
                     12
                  
                   
                     1
                     ,
                     3
                     ,
                     27
                  
                
                 
                   
                      
                  
                   
                      
                  
                   
                     13
                  
                   
                     1
                     ,
                     1
                     ,
                     02
                  
                
              
               
                 
                 
                   A
                   Table
                   of
                   the
                   number
                   of
                   bricks
                   in
                   a
                   rodd
                   of
                   Walling
                   at
                   any
                   Feet
                   high
                   ,
                   from
                   1
                   to
                   20
                   for
                   1
                   and
                   1
                   1
                   /
                   2
                
                 
                   
                     Feet
                     high
                     .
                  
                   
                     at
                     1
                     brick
                     thick
                     .
                  
                   
                     at
                     1
                     brick
                     &
                     1
                     /
                     2
                     thick
                  
                
                 
                   
                     1
                  
                   
                     176
                  
                   
                     264
                  
                
                 
                   
                     2
                  
                   
                     352
                  
                   
                     528
                  
                
                 
                   
                     3
                  
                   
                     528
                  
                   
                     792
                  
                
                 
                   
                     4
                  
                   
                     704
                  
                   
                     1056
                  
                
                 
                   
                     5
                  
                   
                     880
                  
                   
                     1320
                  
                
                 
                   
                     6
                  
                   
                     1136
                  
                   
                     1704
                  
                
                 
                   
                     7
                  
                   
                     1232
                  
                   
                     1848
                  
                
                 
                   
                     8
                  
                   
                     1408
                  
                   
                     2112
                  
                
                 
                   
                     9
                  
                   
                     1584
                  
                   
                     2376
                  
                
                 
                   
                     10
                  
                   
                     1760
                  
                   
                     2640
                  
                
                 
                   
                     11
                  
                   
                     1936
                  
                   
                     2904
                  
                
                 
                   
                     12
                  
                   
                     2112
                  
                   
                     3168
                  
                
                 
                   
                     13
                  
                   
                     2288
                  
                   
                     3432
                  
                
                 
                   
                     14
                  
                   
                     2464
                  
                   
                     3696
                  
                
                 
                   
                     15
                  
                   
                     2640
                  
                   
                     3960
                  
                
                 
                   
                     16
                  
                   
                     2816
                  
                   
                     4224
                  
                
                 
                   
                     16
                     1
                     /
                     2
                  
                   
                     2904
                  
                   
                     4356
                  
                
                 
                   
                     17
                  
                   
                     -2992
                  
                   
                     4488
                  
                
                 
                   
                     18
                  
                   
                     -3168
                  
                   
                     4752
                  
                
                 
                   
                     19
                  
                   
                     -3344
                  
                   
                     5010
                  
                
                 
                   
                     20
                  
                   
                     -3520
                  
                   
                     5280
                  
                
              
            
             
             
               If
               you
               would
               have
               this
               Table
               for
               1
               /
               2
               a
               brick
               ,
               take
               the
               half
               of
               the
               table
               for
               one
               brick
               .
               If
               for
               two
               bricks
               then
               double
               it
               .
               If
               for
               two
               and
               a
               1
               /
               2
               then
               ad
               both
               these
               together
               ;
               if
               for
               three
               ,
               double
               that
               for
               one
               brick
               and
               1
               /
               2.
               
            
             
               If
               you
               have
               any
               number
               of
               feet
               of
               brick
               work
               ,
               at
               half
               a
               brick
               ,
               one
               brick
               ,
               or
               two
               bricks
               ,
               or
               more
               ,
               and
               you
               would
               reduce
               it
               to
               one
               brick
               and
               half
               ,
               then
               say
               by
               the
               line
               of
               numbers
               as
               1.
               2.
               4.
               5
               or
               6
               is
               to
               three
               ,
               so
               is
               the
               number
               of
               feet
               at
               1
               /
               2
               1.
               2.
               2
               1
               
               /
               2
               or
               three
               bricks
               to
               the
               number
               of
               feet
               at
               one
               and
               1
               /
               2.
               
            
          
           
             
               The
               use
               of
               four
               scales
               ,
               called
               Circumfence
               ,
               Diameter
               ,
               Square
               equal
               ,
               Square
               inscribed
               .
            
             
               Suppose
               you
               have
               a
               circle
               whose
               diameter
               is
               10
               inches
               ,
               or
               10
               feet
               :
               and
               to
               this
               circle
               you
               would
               finde
               the
               Circumference
               ,
               or
               the
               side
               of
               a
               square
               equal
               ,
               or
               inscribed
               ,
               or
               having
               any
               one
               of
               the
               three
               ,
               to
               finde
               the
               other
               three
               ,
               do
               thus
               :
            
             
               Take
               the
               measure
               of
               the
               Circumference
               ,
               
               Diameter
               ,
               or
               either
               of
               the
               squares
               ,
               which
               is
               first
               given
               ,
               and
               open
               the
               compasses
               to
               the
               number
               of
               the
               given
               measure
               ,
               in
               its
               respective
               scale
               :
               the
               compasses
               so
               set
               ,
               if
               you
               apply
               it
               in
               the
               scale
               whose
               number
               you
               would
               know
               ,
               you
               shall
               have
               your
               desire
               .
            
             
               
                 Example
                 .
              
               
                 Suppose
                 a
                 circle
                 whose
                 diameter
                 is
                 10
                 inches
                 ,
                 and
                 to
                 it
                 I
                 would
                 know
                 the
                 Circumference
                 ,
                 take
                 10
                 out
                 of
                 the
                 diameter
                 scale
                 ,
                 and
                 in
                 the
                 Circumference
                 scale
                 it
                 shall
                 reach
                 to
                 31
                 42
                 ,
                 and
                 on
                 the
                 line
                 of
                 square
                 equal
                 8
                 86
                 ,
                 and
                 on
                 square
                 inscribed
                 7.
                 07.
                 
                 For
                 illustration
                 sake
                 ,
                 note
                 the
                 figure
                 .
              
            
             
               
                 The
                 use
                 of
                 the
                 line
                 to
                 divide
                 a
                 circle
                 into
                 any
                 number
                 of
                 parts
                 .
              
               
                 Take
                 the
                 Semidiameter
                 ,
                 or
                 Radius
                 of
                 the
                 circle
                 between
                 your
                 compasses
                 ,
                 and
                 fit
                 it
                 over
                 in
                 six
                 and
                 six
                 of
                 the
                 line
                 of
                 circles
                 ,
                 then
                 what
                 number
                 of
                 parts
                 you
                 would
                 have
                 ,
                 take
                 off
                 from
                 that
                 point
                 by
                 the
                 figure
                 in
                 the
                 line
                 of
                 circles
                 ,
                 and
                 it
                 shall
                 divide
                 the
                 circle
                 
                 
                 
                 
                 into
                 so
                 many
                 parts
                 .
                 As
                 suppose
                 I
                 would
                 have
                 the
                 former
                 circle
                 divided
                 into
                 nine
                 parts
                 ,
                 take
                 the
                 measure
                 from
                 the
                 center
                 to
                 the
                 circle
                 as
                 exactly
                 as
                 you
                 can
                 ,
                 fit
                 that
                 over
                 in
                 6
                 and
                 6
                 ,
                 then
                 take
                 out
                 9
                 and
                 9
                 ,
                 and
                 that
                 shall
                 divide
                 it
                 into
                 so
                 many
                 parts
                 ;
                 but
                 if
                 you
                 would
                 divide
                 a
                 wheel
                 into
                 any
                 odde
                 parts
                 ,
                 as
                 55.
                 63.
                 or
                 49
                 parts
                 ,
                 you
                 shall
                 finde
                 it
                 an
                 almost
                 impossible
                 thing
                 ,
                 to
                 take
                 a
                 part
                 so
                 exact
                 that
                 in
                 turning
                 about
                 so
                 many
                 times
                 ,
                 shall
                 not
                 miss
                 at
                 last
                 :
                 to
                 help
                 which
                 the
                 parts
                 the
                 rule
                 giveth
                 shall
                 fit
                 you
                 exact
                 enough
                 for
                 all
                 the
                 odde
                 parts
                 ,
                 then
                 the
                 even
                 will
                 easie
                 be
                 had
                 by
                 dividing
                 ,
                 therefore
                 usually
                 the
                 rule
                 is
                 divided
                 but
                 to
                 30
                 or
                 40
                 parts
                 .
                 So
                 that
                 for
                 this
                 use
                 as
                 the
                 finding
                 the
                 side
                 of
                 an
                 8
                 or
                 10
                 square
                 piece
                 ,
                 as
                 the
                 mast
                 of
                 a
                 ship
                 ,
                 or
                 a
                 newel
                 ,
                 or
                 a
                 post
                 ,
                 this
                 will
                 very
                 readily
                 ,
                 and
                 exactly
                 help
                 you
                 .
              
            
          
        
         
           
           
             CHAP
             XVIII
             .
          
           
             
               The
               use
               of
               Mr.
               Whites
               rule
               ,
               for
               the
               measuring
               of
               Timber
               and
               Board
               ,
               either
               by
               inches
               or
               foot
               measure
               .
            
             
               1.
               
               ANd
               first
               for
               superficial
               or
               board
               measure
               ,
               by
               the
               inches
               ,
               the
               breadth
               and
               length
               being
               given
               in
               inches
               ,
               and
               feet
               and
               parts
               ,
               slide
               or
               set
               12
               on
               any
               one
               side
               ,
               to
               the
               breadth
               in
               inches
               or
               parts
               on
               the
               other
               side
               ,
               then
               just
               against
               the
               length
               found
               on
               the
               first
               side
               ,
               where
               12
               was
               on
               the
               second
               side
               you
               shall
               have
               the
               content
               in
               feet
               ,
               and
               10ths
               ,
               or
               100
               parts
               required
               .
               Which
               by
               the
               rules
               of
               reduction
               by
               the
               foot
               measure
               ,
               you
               may
               reduce
               to
               inches
               and
               8
               parts
               .
            
             
               
                 Example
                 .
              
               
                 At
                 three
                 inches
                 broad
                 ,
                 and
                 20
                 foot
                 long
                 ,
                 you
                 shall
                 finde
                 it
                 to
                 be
                 5
                 foot
                 just
                 :
                 but
                 at
                 7
                 inches
                 broad
                 ,
                 and
                 the
                 same
                 length
                 it
                 will
                 come
                 to
                 11
                 foot
                 7
                 10th
                 fere
                 ,
                 or
                 8
                 inches
                 .
              
            
          
           
             
             
               2.
               
               The
               breadth
               being
               given
               in
               inches
               ,
               to
               finde
               how
               many
               inches
               in
               length
               goes
               to
               make
               a
               foot
               of
               board
               or
               flat
               measure
               .
            
             
               Set
               12
               on
               the
               first
               side
               ,
               to
               the
               breadth
               in
               inches
               on
               the
               second
               side
               ,
               then
               look
               for
               12
               on
               the
               second
               side
               ,
               and
               right
               against
               it
               on
               the
               first
               side
               ,
               is
               the
               number
               of
               inches
               ,
               that
               goes
               to
               make
               a
               foot
               Superficial
               ,
               at
               that
               breadth
               .
            
             
               
                 Example
                 .
              
               
                 At
                 three
                 inches
                 broad
                 you
                 shall
                 finde
                 48
                 inches
                 to
                 make
                 a
                 foot
                 .
              
            
          
           
             
               3.
               
               To
               work
               multiplication
               on
               the
               s●iding
               ,
               or
               Whites
               rule
               .
            
             
               Set
               one
               on
               the
               first
               side
               ,
               to
               the
               multiplicator
               on
               the
               second
               side
               ,
               then
               seek
               the
               multiplicand
               on
               the
               first
               ,
               and
               right
               against
               it
               on
               the
               second
               ,
               you
               shall
               finde
               the
               product
               .
            
             
               
                 Example
                 .
              
               
                 If
                 9
                 be
                 multiplied
                 by
                 16
                 ,
                 you
                 shall
                 finde
                 it
                 to
                 be
                 144.
                 
              
            
          
           
             
             
               4.
               
               To
               work
               division
               on
               the
               same
               rule
               .
            
             
               Set
               the
               divisor
               on
               the
               first
               side
               to
               one
               on
               the
               second
               ,
               then
               the
               dividend
               on
               the
               first
               ,
               shall
               on
               the
               second
               shew
               the
               quotient
               required
               .
            
             
               
                 Example
                 .
              
               
                 If
                 144
                 be
                 to
                 be
                 divided
                 by
                 16
                 ,
                 you
                 shall
                 finde
                 the
                 quotient
                 to
                 be
                 9.
                 
              
            
          
           
             
               3.
               
               To
               work
               the
               rule
               of
               3
               direct
               .
            
             
               Set
               the
               first
               term
               of
               the
               question
               ,
               sought
               out
               on
               the
               first
               line
               ,
               to
               the
               second
               term
               on
               the
               other
               ,
               (
               or
               second
               line
               :
               )
               then
               the
               third
               term
               sought
               on
               the
               first
               line
               ,
               right
               against
               it
               on
               the
               second
               ,
               you
               shall
               finde
               the
               fourth
               proportional
               term
               required
               .
            
             
               
                 Example
                 .
              
               
                 If
                 15
                 yard
                 1
                 /
                 2
                 ,
                 cost
                 37s
                 .
                 6d
                 .
                 what
                 cost
                 17
                 3
                 /
                 4
                 ?
                 facit
                 42s
                 .
                 10d
                 .
                 3
                 q.
                 for
                 if
                 you
                 set
                 15
                 1
                 /
                 2
                 right
                 against
                 37
                 1
                 /
                 2
                 ,
                 then
                 look
                 for
                 17
                 3
                 /
                 4
                 on
                 the
                 first
                 line
                 ,
                 (
                 where
                 15
                 1
                 /
                 2
                 was
                 found
                 ,
                 and
                 right
                 against
                 it
                 on
                 the
                 second
                 line
                 ,
                 is
                 neer
                 42
                 the
                 fractions
                 
                 are
                 all
                 decimal
                 ,
                 and
                 you
                 must
                 reduce
                 them
                 to
                 proper
                 fractions
                 accordingly
              
            
          
           
             
               To
               work
               the
               rule
               of
               3
               reverse
               .
            
             
               4.
               
               Set
               the
               first
               term
               sought
               out
               on
               the
               first
               line
               ,
               to
               the
               second
               being
               of
               the
               same
               denomination
               or
               kind
               to
               the
               second
               line
               ,
               or
               side
               .
               Then
               seek
               the
               third
               term
               on
               the
               second
               side
               ,
               and
               on
               the
               first
               you
               shall
               have
               the
               answer
               required
               .
            
             
               
                 Example
                 .
              
               
                 5.
                 
                 If
                 300
                 masons
                 build
                 an
                 edifice
                 in
                 28
                 days
                 ,
                 how
                 many
                 men
                 must
                 I
                 have
                 to
                 perform
                 the
                 same
                 in
                 six
                 days
                 ,
                 the
                 answer
                 will
                 be
                 found
                 to
                 be
                 1400.
                 
              
            
          
           
             
               6.
               
               To
               work
               the
               double
               rule
               of
               3
               direct
               .
            
             
               This
               is
               done
               by
               two
               workings
               :
               As
               thus
               for
               Example
               .
            
             
               If
               112
               l.
               or
               1
               C.
               weight
               ,
               cost
               12
               pence
               the
               carraiage
               for
               20
               miles
               ,
               what
               shall
               6
               C
               ,
               cost
               ,
               100
               miles
               ?
               Say
               first
               by
               the
               third
               rule
               last
               mentioned
               ,
               as
               1
               C.
               weight
               to
               12
               ,
               so
               is
               6
               C.
               weight
               to
               72.
               pence
               ,
               secondly
               say
               if
               6
               C.
               cost
               72
               pence
               or
               rather
               
               6s
               ,
               for
               20
               miles
               ?
               what
               shall
               100
               miles
               require
               ?
               the
               answer
               is
               30
               s.
               for
               if
               you
               set
               20
               against
               6
               then
               right
               against
               100
               is
               30
               ,
               the
               answer
               required
               .
            
          
           
             
               The
               use
               of
               Mr.
               Whites
               rule
               in
               measuring
               Timber
               round
               ,
               or
               square
               ,
               the
               square
               or
               girt
               being
               given
               in
               inches
               ,
               and
               the
               length
               in
               feet
               and
               inches
               .
            
             
               1.
               
               The
               inches
               that
               a
               piece
               of
               Timber
               is
               square
               ,
               being
               given
               :
               to
               finde
               how
               much
               in
               length
               makes
               a
               foot
               of
               Timber
               ,
               look
               the
               number
               of
               inches
               square
               on
               that
               side
               of
               the
               Timber
               line
               ,
               which
               is
               numbred
               with
               single
               figures
               from
               1
               to
               12
               ,
               and
               set
               it
               just
               against
               100
               on
               the
               other
               or
               second
               side
               ,
               then
               right
               against
               12
               at
               the
               lower
               ,
               (
               or
               some
               times
               the
               upper
               )
               end
               ,
               on
               the
               first
               line
               ,
               in
               the
               second
               you
               have
               the
               number
               of
               feet
               and
               inches
               required
               .
            
             
               Example
               .
               At
               4
               1
               /
               2
               inches
               square
               ,
               you
               must
               have
               7
               foot
               1
               inch
               1
               /
               3
               to
               make
               a
               foot
               of
               Timber
               .
               But
               if
               it
               be
               above
               12
               inches
               square
               ,
               then
               use
               the
               sixth
               Problem
               of
               the
               5th
               chapter
               of
               the
               
                 Carpenters
                 
                 Rule
              
               ,
               with
               the
               double
               figured
               side
               and
               Compasses
               .
            
             
               2.
               
               But
               if
               it
               be
               a
               round
               smooth
               stick
               ,
               of
               above
               12
               inches
               about
               ,
               and
               to
               it
               you
               would
               know
               how
               much
               in
               length
               makes
               a
               true
               foot
               ,
               then
               do
               thus
               .
               Set
               the
               one
               at
               the
               beginning
               of
               the
               double
               figured
               side
               ,
               next
               your
               left
               hand
               ,
               to
               the
               feet
               and
               inches
               about
               ,
               counted
               in
               the
               other
               side
               ,
               numbred
               with
               single
               figures
               from
               1
               to
               12
               ,
               then
               right
               against
               three
               foot
               six
               1
               /
               2
               inches
               ,
               in
               the
               single
               figures
               side
               next
               the
               right
               hand
               ,
               you
               have
               in
               the
               first
               side
               the
               number
               of
               feet
               ,
               and
               inches
               required
               .
            
             
               Example
               .
               A
               piece
               of
               12
               inches
               about
               ,
               requires
               11
               f
               7
               in
               :
               fere
               to
               make
               a
               foot
               .
               Again
               a
               piece
               of
               15
               inches
               about
               ,
               must
               have
               8
               foot
               1
               /
               2
               an
               inch
               in
               length
               ,
               to
               make
               a
               foot
               of
               timber
               .
            
             
               3.
               
               But
               if
               you
               would
               have
               it
               to
               be
               equal
               to
               the
               square
               ,
               made
               by
               the
               4th
               part
               of
               a
               line
               girt
               about
               the
               piece
               ,
               then
               instead
               of
               three
               foot
               6
               1
               /
               2
               inches
               make
               use
               of
               four
               foot
               ,
               and
               you
               shall
               have
               your
               desire
               .
            
             
             
               4.
               
               The
               side
               of
               a
               square
               being
               given
               in
               inches
               ,
               and
               the
               length
               in
               feet
               ,
               to
               find
               the
               content
               of
               a
               piece
               of
               timber
               .
            
             
               If
               it
               be
               under
               12
               inches
               square
               then
               work
               thus
               :
               set
               12
               at
               the
               beginning
               or
               end
               of
               the
               right
               hand
               side
               ,
               to
               the
               length
               counted
               on
               the
               other
               side
               ,
               then
               right
               against
               the
               inches
               square
               on
               the
               right
               side
               is
               the
               content
               on
               the
               left
               side
            
             
               Example
               .
               At
               30
               foot
               long
               ,
               &
               9
               inches
               square
               ,
               you
               shall
               find
               16
               foot
               11
               inches
               for
               the
               working
               this
               question
               ,
               12
               at
               the
               end
               must
               be
               used
               .
            
             
               But
               if
               it
               be
               above
               12
               inches
               square
               ,
               then
               ser
               one
               at
               the
               beginning
               ,
               or
               10
               at
               the
               end
               of
               the
               right
               hand
               side
               ,
               to
               the
               length
               counted
               on
               the
               other
               side
               ,
               then
               the
               number
               of
               inches
               or
               rather
               feet
               and
               inches
               ,
               counted
               on
               the
               first
               side
               ,
               shall
               shew
               on
               the
               second
               the
               feet
               and
               parts
               required
               .
            
             
               Example
               .
               At
               1
               f.
               6
               inch
               .
               square
               ,
               and
               30
               foot
               long
               ,
               you
               shall
               finde
               67
               feet
               and
               about
               a
               1
               /
               2.
               
            
             
               5.
               
               To
               measure
               a
               round
               piece
               by
               having
               the
               length
               ,
               and
               the
               number
               of
               
               inches
               about
               ,
               being
               a
               smooth
               piece
               ,
               and
               to
               measure
               true
               ,
               and
               just
               measure
               ,
               then
               proceed
               thus
               :
            
             
               Set
               3
               f.
               6
               1
               /
               2
               inches
               on
               the
               right
               side
               ,
               to
               the
               length
               on
               the
               other
               side
               ,
               then
               the
               feet
               and
               inches
               about
               ,
               on
               the
               first
               side
               ,
               shall
               shew
               on
               the
               second
               or
               left
               ,
               the
               content
               required
               .
               As
               at
               20
               inches
               about
               ,
               and
               20
               foot
               long
               ,
               the
               content
               will
               be
               found
               to
               be
               about
               4
               foot
               5
               inches
               .
            
             
               But
               if
               you
               give
               the
               usual
               allowance
               ,
               that
               is
               made
               by
               dupling
               the
               string
               4
               times
               ,
               that
               girts
               the
               piece
               :
               then
               you
               must
               set
               4
               foot
               on
               the
               right
               side
               ,
               to
               the
               length
               on
               the
               other
               ,
               then
               at
               1
               foot
               8
               inches
               about
               ,
               the
               last
               example
               you
               shall
               finde
               but
               three
               foot
               6
               inches
               .
            
             
               6.
               ●
               astly
               ,
               if
               the
               rule
               be
               made
               fit
               for
               foot
               measure
               ,
               onely
               then
               the
               point
               of
               12
               is
               altogether
               neglected
               ,
               and
               one
               onely
               made
               use
               of
               as
               a
               standing
               number
               :
               and
               the
               point
               at
               three
               foot
               6
               1
               /
               2
               will
               be
               at
               three
               foot
               54
               parts
               ;
               and
               the
               four
               will
               be
               the
               same
               ,
               and
               the
               same
               
               directions
               in
               every
               respect
               ,
               serve
               the
               turn
               .
               And
               because
               I
               call
               it
               
                 Mr.
                 Whites
              
               rule
               ,
               being
               the
               contriver
               thereof
               ,
               according
               to
               feet
               and
               inches
               ,
               I
               have
               therefore
               fitted
               these
               directions
               accordingly
               ,
               and
               there
               are
               sufficient
               to
               the
               ingenious
               practitioner
               .
            
          
        
         
           
             CHAP.
             XIX
             .
          
           
             Certain
             Propositions
             to
             finde
             the
             hour
             ,
             and
             the
             Azimuth
             ,
             by
             the
             lines
             on
             the
             Sector
             .
          
           
             
               PROP.
               1.
               
            
             
               HAving
               the
               latitude
               ,
               and
               complements
               of
               the
               declination
               ,
               and
               Suns
               altitude
               ,
               and
               the
               hour
               from
               noon
               ,
               to
               finde
               the
               Suns
               Azimuth
               ,
               〈◊〉
               that
               time
               .
            
             
               Take
               the
               right
               sine
               of
               the
               complement
               of
               the
               Suns
               altitude
               ,
               and
               mak
               〈◊〉
               it
               a
               parallel
               sine
               in
               the
               sine
               of
               th
               〈◊〉
               hour
               from
               noon
               ,
               (
               counting
               15
               degree
               〈◊〉
               for
               an
               hour
               ,
               and
               1
               degree
               for
               for
               minutes
               )
               counted
               from
               the
               center
               .
            
             
             
               The
               Sector
               so
               set
               ,
               take
               the
               right
               sine
               of
               the
               complement
               of
               the
               declination
               ,
               and
               carry
               it
               parallel
               till
               the
               compasses
               stay
               in
               like
               sines
               ,
               and
               the
               sine
               wherein
               they
               stay
               shall
               be
               the
               sine
               of
               the
               Azimuth
               required
               .
            
             
               
                 Or
                 else
                 thus
                 :
              
               
                 Take
                 the
                 right
                 sine
                 of
                 the
                 declination
                 ,
                 make
                 it
                 a
                 parallel
                 in
                 the
                 cosine
                 of
                 the
                 Suns
                 altitude
                 ,
                 then
                 take
                 the
                 parallel
                 sine
                 of
                 the
                 hour
                 from
                 noon
                 ,
                 and
                 it
                 shall
                 be
                 the
                 latteral
                 or
                 right
                 sine
                 of
                 the
                 Azimuth
                 from
                 the
                 south
                 required
                 .
                 If
                 it
                 be
                 between
                 six
                 in
                 the
                 morning
                 ,
                 and
                 6
                 at
                 night
                 ;
                 or
                 from
                 the
                 north
                 ,
                 if
                 it
                 be
                 before
                 or
                 after
                 six
                 :
                 and
                 so
                 likewise
                 is
                 the
                 Azimuth
                 .
              
            
          
           
             
               PROP.
               2.
               
            
             
               Having
               the
               Azimuth
               from
               south
               or
               north
               ,
               the
               complement
               of
               the
               Suns
               altitude
               ,
               and
               declination
               ,
               to
               finde
               the
               hour
               .
            
             
               Take
               the
               latteral
               ,
               or
               right
               sine
               of
               the
               complement
               of
               the
               Suns
               altitude
               ,
               make
               it
               a
               ga●●llel
               in
               the
               cosine
               of
               the
               declination
               :
               the
               sector
               so
               sett
               ,
               ake
               out
               
               the
               parallel
               sine
               of
               the
               Azimuth
               ,
               and
               measure
               it
               from
               the
               center
               ,
               and
               it
               shall
               reach
               to
               the
               right
               sine
               of
               the
               hour
               from
               noon
               required
               .
               Or
               else
               as
               before
               .
            
             
               As
               the
               right
               sine
               of
               the
               complement
               of
               the
               Suns
               declination
               :
               is
               to
               the
               parallel
               sine
               of
               the
               Azimuth
               ,
               so
               is
               the
               right
               sine
               of
               coaltitude
               ,
               to
               the
               parallel
               sine
               of
               the
               hour
               from
               noon
               ,
               counting
               as
               before
               .
            
          
           
             
               PROP.
               3.
               
            
             
               Having
               the
               complements
               of
               the
               latitude
               ,
               Suns
               altitude
               ,
               and
               declination
               .
               to
               finde
               the
               Suns
               Azimuth
               from
               the
               north
               part
               of
               the
               horizon
               .
            
             
               1.
               
               First
               of
               the
               complement
               of
               the
               latitude
               ,
               and
               Suns
               present
               altitude
               finde
               the
               difference
               .
            
             
               2.
               
               And
               secondly
               count
               it
               on
               the
               line
               of
               sines
               from
               90
               toward
               the
               center
               .
            
             
               3.
               
               Take
               the
               distance
               from
               thence
               ,
               to
               the
               sine
               of
               the
               Suns
               declination
               ;
               but
               note
               when
               the
               latitude
               and
               declination
               differ
               ,
               as
               in
               winter
               you
               must
               counte
               the
               declination
               beyond
               the
               center
               ,
               
               and
               you
               must
               call
               it
               the
               Suns
               distance
               from
               the
               pole
               .
            
             
               4.
               
               Fourthly
               ,
               make
               that
               distance
               a
               parallel
               sine
               in
               the
               complement
               of
               the
               latitude
               .
            
             
               5.
               
               Fifthly
               ,
               then
               take
               out
               the
               parallel
               sine
               of
               90.
               
            
             
               6.
               
               And
               sixthly
               ,
               make
               it
               a
               parallel
               sine
               in
               the
               coaltitude
               .
            
             
               7.
               
               Seventhly
               ,
               then
               the
               sector
               so
               set
               ,
               take
               out
               parallel
               ,
               Radius
               ,
               or
               sine
               of
               90.
               
            
             
               And
               eighthly
               ,
               measure
               it
               on
               the
               line
               of
               sines
               from
               90
               towards
               ,
               (
               and
               if
               need
               be
               beyond
               )
               the
               center
               :
               and
               it
               shall
               reach
               to
               the
               versed
               sine
               of
               the
               Suns
               Azimuth
               from
               the
               north
               ,
               or
               if
               you
               count
               the
               other
               way
               from
               the
               south
               ,
               note
               that
               in
               working
               of
               these
               ,
               if
               the
               line
               of
               sines
               be
               too
               big
               ,
               then
               you
               have
               two
               or
               three
               smaller
               sines
               on
               the
               rule
               ,
               where
               on
               to
               begin
               and
               end
               the
               work
               .
            
             
               
               
                 Example
                 .
              
               
                 Latitude
                 51.
                 32
                 ,
                 Declination
                 18
                 ▪
                 30
                 ,
                 Altitude
                 48.
                 12
                 ,
                 you
                 shall
                 finde
                 the
                 Azimuth
                 to
                 be
                 130
                 from
                 the
                 North
                 ,
                 or
                 50
                 from
                 the
                 South
                 .
              
            
          
           
             
               PROP.
               4.
               
            
             
               Having
               the
               complements
               of
               the
               latitude
               and
               declination
               ,
               or
               Suns
               distance
               from
               the
               Pole
               ,
               and
               the
               Sun
               altitude
               given
               ,
               to
               finde
               the
               hour
               from
               East
               or
               West
               ,
               or
               else
               from
               noon
               .
            
             
               1.
               
               First
               of
               the
               complement
               of
               the
               latitude
               ,
               and
               Suns
               distance
               from
               the
               Pole
               ,
               finde
               the
               difference
               .
            
             
               2.
               
               Count
               this
               from
               the
               sine
               of
               9●
               toward
               the
               center
               .
            
             
               3.
               
               Take
               the
               distance
               from
               thence
               to
               the
               sine
               of
               the
               Suns
               altitude
               .
            
             
               4.
               
               Make
               that
               distance
               a
               parallel
               sine
               of
               the
               complement
               of
               the
               latitude
               .
            
             
               5.
               
               Take
               out
               the
               parallel
               sine
               of
               90
               degrees
               ,
               and
            
             
               6.
               
               Make
               that
               a
               parallel
               sine
               in
               the
               codeclination
               ,
               then
            
             
               7.
               
               Take
               out
               the
               parallel
               sine
               o●
               90
               again
               ,
               and
            
             
             
               8.
               
               Measure
               it
               from
               the
               sine
               of
               90
               toward
               the
               center
               ,
               and
               it
               shall
               shew
               the
               versed
               sine
               of
               the
               hour
               from
               the
               North
               ,
               or
               the
               sine
               of
               the
               hour
               from
               East
               or
               West
               ;
               or
               if
               you
               reckon
               from
               90
               ,
               the
               hour
               from
               noon
               required
               .
            
             
               
                 Example
                 .
              
               
                 Latitude
                 51.
                 32
                 ,
                 Declination
                 North
                 20.
                 14
                 ,
                 Altitude
                 50.
                 55
                 ,
                 you
                 shall
                 finde
                 the
                 hour
                 from
                 the
                 North
                 to
                 be
                 10
                 houres
                 ,
                 or
                 10
                 a
                 clock
                 in
                 the
                 forenoon
                 ,
                 or
                 4
                 hours
                 past
                 6
                 ,
                 or
                 two
                 short
                 of
                 noon
                 ,
                 according
                 to
                 each
                 proper
                 reckoning
                 .
              
            
          
           
             
               PROP.
               5.
               
            
             
               Having
               the
               latitude
               ,
               the
               complement
               of
               the
               Suns
               declination
               ,
               the
               Suns
               present
               altitude
               ,
               and
               Meridian
               altitude
               for
               that
               day
               ,
               to
               finde
               the
               hour
               .
            
             
               Make
               the
               lateral
               secant
               of
               the
               latitude
               ,
               a
               parallel
               sine
               in
               the
               codeclination
               ,
               then
               take
               the
               distance
               from
               the
               Suns
               Meridian
               altitude
               ,
               to
               his
               present
               altitude
               ,
               and
               lay
               it
               from
               the
               
               cer
               on
               both
               sides
               of
               the
               line
               of
               sines
               ,
               and
               take
               the
               parallel
               distance
               between
               those
               two
               points
               ,
               and
               measure
               it
               from
               90
               on
               a
               line
               of
               sines
               ,
               of
               the
               same
               Radius
               the
               secants
               be
               ,
               (
               as
               the
               small
               adjoyning
               sine
               is
               ,
               )
               and
               it
               shall
               shew
               the
               versed
               sine
               of
               the
               hour
               from
               noon
               ,
               or
               the
               right
               sine
               before
               or
               after
               6
               ,
               towards
               noon
               or
               night
               .
            
          
           
             
               PROP.
               6.
               
            
             
               Having
               the
               Latitude
               ,
               Declination
               ,
               and
               Suns
               Altitude
               ,
               to
               finde
               the
               Suns
               Azimuth
               .
            
             
               Take
               the
               latteral
               secant
               of
               the
               latitude
               ,
               and
               make
               it
               a
               parallel
               in
               the
               complement
               of
               the
               Altitude
               :
            
             
               Then
               take
               the
               distance
               ,
               between
               the
               sun●
               of
               the
               complement●
               of
               of
               Museum
               the
               latitude
               and
               altitude
               ,
               (
               if
               under
               90
               ,
               )
               and
               the
               sons
               declination
               ,
               and
               lay
               it
               from
               the
               center
               on
               the
               line
               of
               sines
               ,
               that
               parallel
               distance
               taken
               ,
               and
               measured
               on
               the
               sines
               (
               of
               the
               same
               Radius
               the
               secant
               was
               )
               from
               90
               ,
               shall
               shew
               the
               versed
               sine
               of
               the
               Azimuth
               from
               noon
               .
            
             
             
               But
               if
               the
               sum
               of
               the
               colatitude
               ,
               and
               coaltitude
               exceed
               90
               ,
               then
               take
               the
               excess
               above
               90
               ,
               out
               of
               the
               natural
               sine
               from
               the
               center
               toward
               90
               ,
               and
               add
               that
               to
               the
               sine
               of
               the
               Suns
               declination
               towards
               90
               ,
               and
               then
               the
               parallel
               distance
               between
               those
               two
               points
               ,
               shall
               be
               the
               Azimuth
               required
               ,
               from
               noon
               ,
            
             
               But
               when
               the
               latitude
               and
               declination
               are
               unlike
               ,
               as
               with
               us
               (
               in
               the
               northern
               parts
               )
               in
               winter
               ,
               then
               you
               must
               take
               the
               declination
               out
               of
               the
               excess
               ,
               or
               the
               lesser
               out
               of
               the
               greater
               ,
               and
               lay
               the
               rest
               from
               the
               center
               ,
               and
               the
               parallel
               distance
               ,
               shall
               be
               versed
               sine
               of
               the
               Azimuth
               from
               noon
               .
            
             
               
                 Example
                 .
              
               
                 At
                 18
                 15
                 altitude
                 ,
                 latitude
                 51
                 32
                 ,
                 declination
                 13
                 15
                 South
                 .
              
               
                 The
                 sum
                 of
                 the
                 colatitude
                 ,
                 and
                 the
                 coaltitude
                 is
                 109.
                 37
                 ,
                 then
                 count
                 the
                 center
                 for
                 90
                 ,
                 the
                 right
                 sine
                 of
                 10
                 for
                 a
                 100
                 and
                 19
                 for
                 109
                 ,
                 and
                 37
                 minutes
                 forwarder
                 ,
                 there
                 set
                 the
                 point
                 of
                 the
                 Compasses
                 ,
                 then
                 take
                 from
                 thence
                 to
                 
                 the
                 right
                 sine
                 of
                 the
                 declination
                 ,
                 and
                 lay
                 this
                 distance
                 from
                 the
                 center
                 on
                 the
                 line
                 of
                 sines
                 ,
                 and
                 the
                 parallel
                 space
                 between
                 ,
                 is
                 the
                 versed
                 sine
                 of
                 the
                 Azimuth
                 required
                 .
              
            
          
           
             
               PROP.
               7.
               
            
             
               Having
               the
               length
               of
               the
               shaddow
               of
               any
               object
               standing
               perpendicular
               ,
               and
               the
               length
               of
               that
               object
               ,
               to
               finde
               the
               altitude
               .
            
             
               Take
               the
               Tangent
               of
               45
               ,
               and
               make
               it
               a
               parallel
               in
               the
               length
               of
               the
               shaddow
               in
               the
               line
               of
               lines
               ,
               then
               the
               parallel
               distance
               between
               the
               length
               of
               the
               object
               that
               casts
               the
               shaddow
               ,
               taken
               from
               the
               line
               of
               lines
               ,
               and
               measured
               on
               the
               line
               of
               tangents
               from
               the
               center
               ,
               shall
               reach
               to
               the
               Suns
               altitude
               required
               .
            
             
               
                 Example
                 .
              
               
                 If
                 the
                 object
                 be
                 40
                 parts
                 long
                 ,
                 and
                 the
                 shaddow
                 80
                 parts
                 ,
                 the
                 altitude
                 will
                 be
                 found
                 to
                 be
                 26.
                 35.
                 
                 But
                 if
                 you
                 have
                 the
                 altitude
                 ,
                 and
                 shaddow
                 ,
                 and
                 would
                 know
                 the
                 height
                 of
                 the
                 object
                 ,
                 then
                 work
                 thus
                 :
              
               
               
                 Take
                 the
                 length
                 of
                 the
                 shaddow
                 out
                 of
                 the
                 line
                 of
                 lines
                 ,
                 or
                 any
                 other
                 equal
                 parts
                 ,
                 and
                 make
                 it
                 a
                 parallel
                 tangent
                 of
                 45
                 ,
                 then
                 take
                 out
                 the
                 parallel
                 tangent
                 of
                 the
                 Suns
                 altitude
                 ,
                 and
                 measure
                 it
                 on
                 the
                 line
                 of
                 lines
                 ,
                 (
                 or
                 the
                 same
                 equal
                 parts
                 )
                 and
                 it
                 shall
                 shew
                 the
                 length
                 of
                 the
                 object
                 that
                 caused
                 the
                 shaddow
                 :
                 the
                 same
                 rule
                 doth
                 serve
                 in
                 taking
                 of
                 altitudes
                 by
                 the
                 rule
                 ,
                 as
                 in
                 the
                 18
                 chapter
                 ,
                 accounting
                 the
                 measure
                 from
                 the
                 station
                 to
                 the
                 object
                 ,
                 the
                 length
                 of
                 the
                 shaddow
                 ,
                 and
                 the
                 Suns
                 altitude
                 ,
                 the
                 angle
                 at
                 the
                 base
                 .
              
            
          
           
             
               PROP.
               8.
               
            
             
               To
               finde
               the
               Suns
               rising
               and
               setting
               in
               any
               latitude
               .
            
             
               Take
               the
               latteral
               cotangent
               of
               the
               latitude
               ,
               make
               it
               a
               parallel
               in
               the
               sines
               of
               90
               and
               90
               ,
               then
               take
               the
               latteral
               tangent
               of
               the
               Suns
               declination
               ,
               and
               carry
               it
               parallel
               in
               the
               sines
               till
               it
               stay
               in
               like
               sines
               ,
               that
               sine
               shall
               be
               the
               asentional
               difference
               between
               six
               ,
               and
               the
               time
               of
               rising
               before
               or
               
               after
               6
               ,
               counting
               15
               degrees
               to
               an
               hour
               .
            
          
           
             
               PROP.
               9.
               
            
             
               To
               finde
               the
               Amplitude
               in
               any
               latitude
               .
            
             
               Take
               the
               latteral
               sine
               of
               90
               ,
               and
               make
               it
               a
               parallel
               in
               the
               cosine
               of
               the
               latitude
               ,
               then
               the
               parallel
               sine
               of
               the
               declination
               ,
               taken
               and
               measured
               in
               the
               line
               of
               sines
               from
               the
               center
               ,
               shall
               give
               the
               amplitude
               required
               .
            
          
           
             
               PROP.
               10.
               
            
             
               To
               finde
               the
               Suns
               height
               at
               six
               in
               any
               latitude
               .
            
             
               Take
               the
               lateral
               or
               right
               sine
               of
               the
               declination
               ,
               and
               make
               it
               a
               parallel
               in
               the
               sine
               of
               90
               ,
               then
               take
               out
               the
               parallel
               sine
               of
               the
               latitude
               ,
               and
               measure
               it
               in
               the
               line
               of
               sines
               from
               the
               center
               ,
               and
               it
               shall
               reach
               to
               the
               altitude
               required
               .
            
             
               Note
               in
               working
               of
               any
               of
               these
               Propositions
               ,
               if
               the
               sines
               drawn
               from
               the
               center
               ,
               prove
               too
               large
               for
               your
               Compasses
               ,
               or
               to
               make
               a
               parallel
               sine
               
               or
               Tangent
               to
               a
               small
               number
               of
               degrees
               ,
               then
               you
               may
               use
               the
               smaller
               sine
               or
               tangent
               adjoyning
               ,
               that
               is
               set
               on
               the
               Rule
               ,
               and
               it
               will
               answer
               your
               desire
               .
               And
               note
               also
               in
               these
               Propositions
               ,
               the
               word
               right
               ,
               or
               
                 latteral
                 sine
              
               or
               tangent
               ,
               is
               to
               be
               taken
               right
               on
               from
               the
               center
               or
               beginning
               of
               the
               lines
               of
               sines
               ,
               or
               tangents
               ;
               and
               the
               word
               parallel
               always
               across
               from
               one
               leg
               to
               the
               other
               .
            
          
           
             
               PROP.
               11.
               
            
             
               To
               finde
               the
               Suns
               height
               at
               any
               time
               ,
               in
               any
               latitude
               .
            
             
               As
               the
               right
               Sine
               of
               90
               ,
               
               Is
               to
               the
               parallel
               cotangent
               of
               the
               latitude
               :
               
               So
               is
               the
               latteral
               or
               right
               Sine
               of
               the
               hour
               from
               6
               ,
               
               To
               the
               parallel
               tangent
               of
               a
               fourth
               ark
               ;
               which
               you
               must
               substract
               from
               the
               suns
               distance
               from
               the
               Pole
               ,
               and
               note
               the
               difference
               .
               Then
               ,
               
               As
               the
               right
               of
               the
               latitude
               ,
               
               To
               the
               parallel
               cosine
               of
               the
               fourth
               ark
               :
               
               
               So
               is
               the
               parallel
               cosine
               of
               the
               remainder
               ,
               
               To
               the
               latteral
               sine
               of
               the
               Altitude
               required
               .
            
          
           
             
               PROP.
               12.
               
            
             
               To
               finde
               when
               the
               Sun
               shall
               come
               to
               due
               East
               ,
               or
               West
               .
            
             
               Take
               the
               tangent
               of
               the
               latitude
               from
               the
               smaller
               tangents
               ,
               make
               it
               a
               parallel
               in
               the
               Sine
               of
               90
               ,
               then
               take
               the
               latteral
               tangent
               of
               the
               declination
               from
               the
               smaller
               tangents
               ,
               and
               carry
               it
               parallel
               in
               the
               Sines
               ,
               till
               it
               stay
               in
               like
               Sines
               ,
               and
               that
               Sine
               shall
               be
               the
               Sine
               of
               the
               hour
               required
               from
               6.
               
            
          
           
             
               PROP.
               13.
               
            
             
               To
               finde
               the
               Suns
               Altitude
               at
               East
               or
               West
               (
               or
               Vertical
               Circle
               .
               )
            
             
               As
               the
               latteral
               sine
               of
               declination
               ,
               
               Is
               to
               the
               parallel
               sine
               of
               the
               latitude
               :
               
               So
               is
               the
               parallel
               sine
               of
               90
               ,
               
               To
               the
               latteral
               sine
               of
               the
               Altitude
               required
               .
            
          
           
             
             
               PROP.
               14.
               
            
             
               To
               finde
               the
               Stiles
               height
               in
               upright
               declining
               Dials
               .
            
             
               As
               the
               right
               Sine
               of
               the
               complement
               of
               the
               latitude
               ,
               
               To
               the
               parallel
               sine
               of
               90
               :
               
               So
               the
               parallel
               sine
               complement
               of
               the
               Plains
               declination
               ,
               
               To
               the
               right
               sine
               of
               the
               Stiles
               elevation
            
          
           
             
               PROP.
               15.
               
            
             
               To
               finde
               the
               Substiles
               distance
               from
               the
               Meridian
               .
            
             
               As
               the
               lateral
               tangent
               of
               the
               colatitude
               ,
               
               To
               the
               parallel
               sine
               of
               90
               :
               
               So
               the
               parallel
               sine
               of
               the
               declination
               ,
               
               To
               the
               latteral
               tangent
               of
               the
               Substile
               from
               the
               Meridian
               .
            
          
           
             
               PROP.
               16.
               
            
             
               To
               finde
               the
               Inclination
               of
               Meridians
               .
            
             
               As
               the
               latteral
               tangent
               of
               the
               declination
               ,
               
               To
               the
               parallel
               sine
               of
               90
               :
               
               
               So
               is
               the
               parallel
               sine
               of
               the
               latitude
               ,
               
               To
               the
               latteral
               cotangent
               in
               the
               inclination
               of
               Meridians
               .
            
          
           
             
               PROP.
               17.
               
            
             
               To
               finde
               the
               hours
               distance
               from
               the
               Substile
               in
               all
               Plains
               .
            
             
               As
               the
               latteral
               tangent
               of
               the
               hour
               from
               the
               proper
               Meridian
               ,
               
               To
               the
               parallel
               sine
               of
               90
               :
               
               So
               is
               the
               parallel
               sine
               of
               the
               Stiles
               elevation
               ,
               
               To
               the
               latteral
               tangent
               of
               the
               hour
               from
               the
               substile
               .
            
          
           
             
               PROP.
               18.
               
            
             
               To
               finde
               the
               Angle
               of
               6
               from
               12
               ,
               in
               erect
               Decliners
               .
            
             
               As
               the
               latteral
               tangent
               of
               the
               complement
               of
               the
               latitude
               ,
               
               To
               the
               parallel
               sine
               of
               the
               declination
               of
               the
               Plain
               :
               
               So
               is
               the
               parallel
               sine
               of
               90
               :
               
               To
               the
               latteral
               tangent
               of
               the
               Angle
               between
               12
               and
               6.
               
            
             
             
               Thus
               you
               see
               the
               natural
               Sines
               and
               Tangents
               on
               the
               Sector
               ,
               may
               be
               used
               to
               operate
               any
               of
               the
               Canons
               that
               is
               performed
               by
               Logarithms
               ,
               or
               the
               artificial
               Sines
               and
               Tangents
               ,
               by
               changing
               the
               terms
               from
               the
               first
               to
               the
               third
               ,
               and
               the
               second
               to
               the
               first
               ,
               and
               the
               third
               to
               the
               second
               ,
               and
               the
               fourth
               must
               always
               be
               the
               fourth
               ,
               in
               both
               workings
               being
               the
               term
               required
               .
            
          
        
         
           
             CHAP.
             XX.
             
          
           
             A
             brief
             description
             ,
             and
             a
             short-touch
             of
             the
             use
             of
             the
             Serpentine-line
             ,
             or
             Numbers
             ,
             Sines
             ,
             Tangents
             ,
             and
             versed
             sine
             contrived
             in
             five
             (
             or
             rather
             15
             )
             turn
             .
          
           
             1.
             
             FIrst
             next
             the
             center
             is
             two
             circles
             divided
             one
             into
             60
             ,
             the
             other
             into
             100
             parts
             ,
             for
             the
             reducing
             of
             minutes
             to
             100
             parts
             ,
             and
             the
             contrary
             .
          
           
             2.
             
             You
             have
             in
             seven
             turnes
             two
             in
             pricks
             ,
             and
             five
             in
             divisions
             ,
             the
             first
             
             Radius
             of
             the
             sines
             (
             or
             Tangents
             being
             neer
             the
             matter
             ,
             alike
             to
             the
             first
             three
             degrees
             ,
             )
             ending
             at
             five
             degrees
             and
             44
             minutes
             .
          
           
             3.
             
             Thirdly
             ,
             you
             have
             in
             5
             turns
             the
             lines
             of
             numbers
             ,
             sines
             ,
             Tangents
             ,
             in
             three
             margents
             in
             divisions
             ,
             and
             the
             line
             of
             versed
             sines
             in
             pricks
             ,
             under
             the
             line
             of
             Tangents
             ,
             according
             to
             Mr.
             Gunters
             cross
             staff
             :
             the
             sines
             and
             Tangents
             beginning
             at
             5
             degrees
             ,
             and
             44
             minutes
             where
             the
             other
             ended
             ,
             and
             proceeding
             to
             90
             in
             the
             sines
             ,
             and
             45
             in
             the
             Tangents
             .
             And
             the
             line
             of
             numbers
             beginning
             at
             10
             ,
             and
             proceeding
             to
             100
             ,
             being
             one
             entire
             Radius
             ,
             and
             graduated
             into
             as
             many
             divisions
             as
             the
             largeness
             of
             the
             instrument
             will
             admit
             ,
             being
             from
             10
             to
             50
             into
             50
             parts
             ,
             and
             from
             50
             to
             100
             into
             20
             parts
             in
             one
             unit
             of
             increase
             ,
             but
             the
             Tangents
             are
             divided
             into
             single
             minutes
             from
             the
             beginning
             to
             the
             end
             ,
             both
             in
             the
             first
             ,
             second
             ,
             and
             third
             Radiusses
             ,
             and
             the
             sines
             into
             minutes
             ;
             also
             from
             30
             
             minutes
             to
             40
             degrees
             ,
             and
             from
             40
             to
             60
             ,
             into
             every
             two
             minutes
             ,
             and
             from
             60
             to
             80
             in
             every
             5th
             minute
             ,
             and
             from
             80
             to
             85
             every
             1oth
             ,
             and
             the
             rest
             as
             many
             as
             can
             be
             well
             discovered
             .
          
           
             The
             versed
             sines
             are
             set
             after
             the
             manner
             of
             Mr.
             Gunters
             Cross-staff
             ,
             and
             divided
             into
             every
             10th
             minutes
             beginning
             at
             0
             ,
             and
             proceeding
             to
             156
             going
             backwards
             under
             the
             line
             of
             Tangents
             .
          
           
             4.
             
             Fourthly
             ,
             beyond
             the
             Tangent
             of
             45
             in
             one
             single
             line
             ,
             for
             one
             turn
             is
             the
             secants
             to
             51
             degrees
             ,
             being
             nothing
             else
             but
             the
             sines
             reitterated
             beyond
             90.
             
          
           
             5.
             
             Fifthly
             ,
             you
             have
             the
             line
             of
             Tangents
             beyond
             45
             ,
             in
             5
             turnes
             to
             85
             degrees
             ,
             whereby
             all
             trouble
             of
             backward
             working
             is
             avoided
             .
          
           
             6.
             
             Sixthly
             ,
             you
             have
             in
             one
             circle
             the
             180
             degrees
             of
             a
             Semicircle
             ,
             and
             also
             a
             line
             of
             natural
             sines
             ,
             for
             finding
             of
             differences
             in
             sines
             ,
             for
             finding
             hour
             and
             Azimuth
             .
          
           
           
             7.
             
             Seventhly
             ,
             next
             the
             verge
             or
             outermost
             edge
             is
             a
             line
             of
             equal
             parts
             to
             get
             the
             Logarithm
             of
             any
             number
             ,
             or
             the
             Logarithm
             sine
             and
             Tangent
             of
             any
             ark
             or
             angle
             to
             four
             figures
             besides
             the
             carracteristick
             .
          
           
             8.
             
             Eightly
             and
             lastly
             ,
             in
             the
             space
             place
             between
             the
             ending
             of
             the
             middle
             five
             turnes
             ,
             and
             one
             half
             of
             the
             circle
             are
             three
             prickt
             lines
             fitted
             for
             reduction
             .
             The
             uppermost
             being
             for
             shillings
             ,
             pence
             ,
             and
             farthings
             .
             The
             next
             for
             pounds
             ,
             and
             ounces
             ,
             and
             quarters
             of
             small
             Averdupoies
             weight
             .
             The
             last
             for
             pounds
             ,
             shillings
             ,
             and
             pence
             ,
             and
             to
             be
             used
             thus
             :
             If
             you
             would
             reduce
             16
             s.
             3
             d.
             2
             q.
             to
             a
             decimal
             fraction
             ,
             lay
             the
             hair
             or
             edge
             of
             one
             of
             the
             legs
             of
             the
             index
             on
             16.
             3
             1
             /
             2
             in
             the
             line
             of
             l.
             s.
             d.
             and
             the
             hair
             shall
             cut
             on
             the
             equal
             parts
             81
             16
             ;
             and
             the
             contrary
             ,
             if
             you
             have
             a
             decimal
             fraction
             ,
             and
             would
             reduce
             it
             to
             a
             proper
             fraction
             ,
             the
             like
             may
             you
             do
             for
             shillings
             ,
             and
             pence
             ,
             and
             pounds
             ,
             and
             ounces
             .
          
           
             
             
               The
               uses
               of
               the
               lines
               follow
               .
            
             
               As
               to
               the
               use
               of
               these
               lines
               ,
               I
               shall
               in
               this
               place
               say
               but
               little
               ,
               and
               that
               for
               two
               reasons
               .
               First
               ,
               because
               this
               instrument
               is
               so
               contrived
               ,
               that
               the
               use
               is
               sooner
               learned
               then
               any
               other
               ,
               I
               speak
               as
               to
               the
               manner
               ,
               and
               way
               of
               using
               it
               ,
               because
               by
               means
               of
               first
               ,
               second
               ,
               and
               third
               radiusses
               ,
               in
               sines
               and
               Tangents
               ,
               the
               work
               is
               always
               right
               on
               ,
               one
               way
               or
               other
               ,
               according
               to
               the
               Canon
               whatsoever
               it
               be
               ,
               in
               any
               book
               that
               treats
               of
               the
               Logarithms
               ,
               as
               
                 Gunter
                 ,
                 Wells
                 ,
                 Oughtred
                 ,
                 Norwood
                 ,
              
               or
               others
               ,
               as
               in
               Oughtred
               from
               page
               62
               to
               107.
               
            
             
               Secondly
               ,
               and
               more
               especially
               ,
               because
               the
               more
               accurate
               ,
               and
               large
               handling
               thèreof
               is
               more
               then
               promised
               ,
               if
               not
               already
               performed
               by
               more
               abler
               pens
               ,
               and
               a
               large
               manuscript
               thereof
               by
               my
               Sires
               meanes
               ,
               provided
               many
               years
               ago
               ,
               though
               to
               this
               day
               not
               extant
               in
               print
               ;
               so
               for
               his
               sake
               I
               claiming
               my
               intrest
               therein
               ,
               make
               bold
               to
               present
               you
               with
               these
               
               few
               lines
               ,
               in
               order
               to
               the
               use
               of
               them
               :
               And
               first
               no●e
               .
            
             
               1.
               
               Which
               soever
               of
               the
               two
               legs
               is
               set
               to
               the
               first
               term
               in
               the
               question
               ,
               that
               I
               call
               the
               first
               leg
               always
               ,
               and
               the
               other
               being
               set
               to
               the
               second
               term
               ,
               I
               call
               the
               second
               leg
               .
            
             
               2.
               
               Secondly
               ,
               if
               one
               be
               the
               first
               or
               second
               term
               ,
               then
               for
               the
               better
               setting
               the
               index
               exactly
               ,
               you
               may
               set
               it
               to
               100
               ,
               for
               the
               error
               is
               like
               to
               be
               the
               least
               neerest
               the
               circumference
               .
            
             
               3.
               
               Thirdly
               ,
               be
               sure
               you
               keep
               a
               true
               account
               of
               the
               number
               of
               turnes
               between
               the
               first
               and
               second
               term
               .
            
             
               4.
               
               Fourthly
               ,
               observe
               which
               way
               you
               move
               ,
               from
               the
               first
               to
               the
               second
               term
               .
               To
               keep
               the
               like
               from
               the
               third
               to
               the
               fourth
               ,
               exept
               in
               the
               back
               rule
               of
               three
               ,
               and
               in
               such
               cases
               as
               the
               Canon
               requires
               the
               contrary
               .
            
             
               5.
               
               Fifthly
               in
               multiplication
               ,
               one
               is
               always
               the
               first
               term
               ,
               and
               the
               multiplycator
               or
               multiplycand
               the
               second
               ,
               and
               the
               product
               always
               the
               fourth
               .
               Also
               note
               that
               in
               multiplycation
               the
               
               product
               of
               two
               numbers
               multiplyed
               ,
               shall
               be
               in
               as
               many
               places
               as
               both
               the
               multiplycator
               ,
               and
               multiplycand
               ,
               except
               the
               least
               of
               them
               ,
               be
               less
               then
               the
               two
               first
               figures
               of
               the
               product
               ;
               moreover
               ,
               for
               your
               more
               certain
               assinging
               of
               the
               two
               last
               figures
               of
               four
               or
               six
               ,
               which
               is
               as
               many
               as
               you
               can
               see
               on
               this
               instrument
               ,
               multiply
               the
               two
               last
               in
               your
               minde
               ,
               and
               the
               product
               shall
               be
               the
               figure
               ,
               as
               in
               page
               28
               of
               the
               
                 Carpenters
                 Rule
              
               .
            
             
               6.
               
               In
               division
               ,
               the
               multiplicator
               is
               always
               the
               first
               term
               ,
               and
               one
               the
               second
               ,
               the
               dividend
               the
               third
               ,
               and
               the
               quotient
               the
               fourth
               ;
               also
               the
               quotient
               shall
               have
               as
               many
               figures
               as
               the
               dividend
               hath
               more
               then
               the
               divisor
               ,
               except
               the
               first
               figures
               of
               the
               divisor
               be
               greater
               then
               the
               dividends
               ,
               then
               it
               shall
               have
               one
               less
               .
               Also
               note
               ,
               the
               fraction
               after
               division
               ,
               is
               a
               decimal
               fraction
               ,
               and
               to
               be
               reduced
               as
               before
               .
            
             
               7.
               
               Note
               carefully
               whether
               the
               fourth
               proportional
               ought
               to
               be
               a
               greater
               or
               a
               less
               ,
               and
               resolve
               accordingly
               ,
               
               and
               note
               if
               one
               cometh
               between
               the
               third
               and
               fourth
               term
               ,
               then
               must
               the
               fourth
               be
               raised
               a
               Radius
               or
               a
               figure
               more
               ,
               and
               be
               careful
               to
               set
               the
               hairs
               exactly
               over
               the
               part
               representing
               the
               number
               or
               minutes
               of
               any
               degree
               .
            
             
               8.
               
               Always
               in
               direct
               proportion
               ,
               and
               Astronomical
               calculation
               ,
               set
               the
               first
               leg
               to
               the
               first
               term
               ,
               and
               the
               second
               leg
               to
               the
               second
               term
               ,
               and
               note
               how
               many
               circles
               is
               between
               ,
               then
               set
               the
               first
               leg
               to
               the
               third
               term
               ,
               and
               right
               under
               the
               second
               leg
               ,
               the
               same
               way
               ,
               and
               so
               many
               turnes
               between
               the
               third
               and
               fourth
               ,
               is
               always
               the
               fourth
               term
               required
               .
            
             
               
                 Example
                 .
              
               
                 As
                 1
                 ,
                 to
                 47
                 ,
                 so
                 is
                 240
                 ,
                 to
                 11280.
                 
                 As
                 the
                 sine
                 of
                 90
                 ,
                 to
                 51
                 degrees
                 30
                 minutes
                 ,
                 so
                 is
                 the
                 sine
                 of
                 80
                 to
                 50.
                 26.
                 
                 And
                 so
                 of
                 all
                 other
                 questions
                 according
                 to
                 their
                 respective
                 Canons
                 by
                 the
                 Logarithms
                 in
                 other
                 books
                 as
                 in
                 
                   Mr.
                   Oughtreds
                   Circles
                   of
                   Porportions
                
                 ,
                 from
                 page
                 62
                 to
                 107
                 ,
                 and
                 others
                 .
              
               
               
                 Here
                 followeth
                 the
                 working
                 of
                 certain
                 Propositions
                 by
                 the
                 Serpentine-line
                 .
              
               
                 Those
                 that
                 I
                 shall
                 insert
                 ,
                 are
                 onely
                 to
                 shew
                 the
                 manner
                 of
                 working
                 ,
                 and
                 knowing
                 of
                 that
                 once
                 ,
                 all
                 the
                 Canons
                 for
                 all
                 manner
                 of
                 questions
                 ,
                 either
                 in
                 Arithmetick
                 ,
                 Geometry
                 ,
                 Navigation
                 ,
                 or
                 Astronomy
                 ,
                 by
                 any
                 other
                 Author
                 ,
                 as
                 Mr.
                 Gunter
                 ,
                 Mr.
                 Oughtred
                 ,
                 Mr.
                 Windgate
                 ,
                 Mr.
                 Norwood
                 ,
                 or
                 others
                 ,
                 may
                 be
                 speedily
                 resolved
                 ,
                 and
                 as
                 exactly
                 as
                 by
                 the
                 Tables
                 ,
                 if
                 the
                 instrument
                 be
                 well
                 and
                 truly
                 made
                 .
                 And
                 first
                 ●or
                 the
                 hour
                 ,
                 according
                 to
                 Gunter
                 .
              
            
          
           
             
               PROP.
               1.
               
            
             
               Having
               the
               Latitude
               ,
               Declination
               ,
               and
               the
               Suns
               Altitude
               ,
               to
               finde
               the
               hour
               .
            
             
               Add
               the
               complement
               of
               the
               altitude
               ,
               the
               complement
               of
               the
               Suns
               present
               altitude
               ,
               and
               the
               distance
               of
               the
               Sun
               ,
               from
               the
               elevated
               Pole
               together
               ,
               and
               nore
               their
               sum
               ,
               and
               half
               sum
               ,
               and
               find
               the
               difference
               between
               their
               half
               sum
               
               and
               the
               complement
               of
               the
               Suns
               present
               altitude
               ,
               then
               work
               thus
               :
               for
               36
               42
               degrees
               high
               ,
               at
               23.
               32
               declination
               ,
               lat
               .
               51.
               32.
               
            
             
               Lay
               the
               first
               leg
               ,
               viz.
               that
               next
               your
               right
               hand
               being
               here
               most
               convenient
               ,
               on
               the
               Sine
               of
               90
               ,
               keeping
               that
               fixed
               there
               ,
               lay
               the
               other
               leg
               to
               the
               cosine
               of
               the
               latitude
               ,
               viz.
               38.
               28.
               and
               note
               the
               turns
               between
               ,
               which
               here
               is
               none
               between
               ,
               but
               it
               is
               found
               in
               the
               next
               over
               it
               ,
               then
               set
               the
               first
               leg
               to
               the
               Sine
               of
               66
               ,
               28
               ,
               the
               suns
               distance
               from
               the
               Pole
               ,
               and
               in
               the
               circle
               just
               over
               it
               you
               shall
               have
               the
               Sine
               of
               34.
               47.
               for
               the
               fourth
               Ark
               or
               Sine
               .
            
             
               
                 Then
                 in
                 the
                 second
                 Operation
                 .
              
               
                 Set
                 the
                 first
                 or
                 left
                 leg
                 to
                 the
                 sine
                 of
                 34.
                 47.
                 of
                 the
                 fourth
                 Sine
                 last
                 found
                 ,
                 then
                 keeping
                 that
                 fixed
                 there
                 ,
                 set
                 the
                 other
                 leg
                 to
                 the
                 Sine
                 of
                 the
                 half
                 sum
                 ,
                 viz.
                 79.
                 37.
                 then
                 remove
                 the
                 first
                 leg
                 to
                 the
                 Sine
                 of
                 the
                 difference
                 between
                 the
                 half
                 sum
                 ,
                 and
                 the
                 coaltitude
                 ,
                 viz.
                 
                 25.
                 49.
                 and
                 then
                 in
                 the
                 next
                 circle
                 ,
                 the
                 other
                 leg
                 shall
                 shew
                 the
                 sine
                 of
                 48.
                 34.
                 whose
                 half
                 distance
                 toward
                 90
                 ,
                 being
                 found
                 by
                 the
                 Scale
                 of
                 Logarithms
                 on
                 the
                 outermost
                 circle
                 ,
                 will
                 discover
                 the
                 Sine
                 of
                 an
                 Ark
                 ,
                 whose
                 complement
                 being
                 doubled
                 ,
                 and
                 turned
                 into
                 time
                 (
                 by
                 counting
                 15
                 degrees
                 to
                 an
                 hour
                 )
                 will
                 give
                 the
                 hour
                 required
                 ;
                 but
                 by
                 help
                 of
                 the
                 versed
                 Sines
                 all
                 this
                 trouble
                 is
                 saved
                 ;
                 for
                 when
                 the
                 index
                 or
                 second
                 leg
                 cuts
                 the
                 Sine
                 of
                 48.
                 34.
                 at
                 the
                 same
                 instant
                 it
                 cuts
                 the
                 versed
                 Sine
                 of
                 60
                 ,
                 the
                 hour
                 from
                 noon
                 required
                 ,
                 being
                 8
                 in
                 the
                 morning
                 ,
                 or
                 4
                 in
                 the
                 afternoon
                 ,
                 at
                 23.
                 32.
                 of
                 declination
                 ,
                 in
                 the
                 latitude
                 of
                 51.
                 32.
                 
              
            
          
           
             
               PROP.
               2.
               
            
             
               To
               work
               the
               same
               another
               way
               ,
               according
               to
               Mr.
               Collins
               .
            
          
           
             
             
               The
               Latitude
               and
               Declination
               given
               ,
               to
               finde
               the
               Suns
               height
               at
               6
               a
               clock
               ,
               Dec.
               23.
               31.
               
               Lat.
               51.
               32.
               
            
             
               Lay
               the
               left
               leg
               on
               the
               Sine
               of
               90
               ,
               and
               the
               other
               to
               the
               Sine
               of
               23.
               32
               ,
               and
               you
               shall
               finde
               one
               turn
               between
               upwards
               ,
               then
               the
               first
               leg
               laid
               on
               the
               Sine
               of
               the
               latitude
               51
               ,
               32
               ,
               the
               other
               leg
               shall
               shew
               the
               Sine
               of
               18.
               3.
               
               (
               in
               the
               second
               circle
               above
               51
               ,
               32.
               )
               for
               the
               Suns
               height
               at
               6
               required
               ,
               and
               this
               is
               fixed
               for
               one
               day
               .
            
             
               Then
               in
               summer
               time
               ,
               or
               north
               declinations
               ,
               by
               help
               of
               the
               line
               of
               natural
               sines
               ,
               in
               the
               second
               line
               ,
               finde
               the
               difference
               between
               the
               Suns
               present
               altitude
               ,
               and
               the
               latitude
               at
               6
               ,
               but
               in
               Winter
               or
               Southern
               (
               signs
               or
               )
               declinations
               ,
               add
               the
               two
               altitudes
               together
               ,
               in
               this
               manner
               .
               Lay
               one
               leg
               of
               the
               index
               to
               the
               natural
               sine
               of
               the
               altitude
               at
               6
               ,
               and
               the
               other
               to
               the
               altitude
               proposed
               ,
               the
               two
               legs
               so
               set
               ,
               bring
               one
               of
               them
               (
               viz
               the
               right
               )
               to
               the
               beginning
               of
               the
               line
               of
               natural
               
               sines
               ,
               and
               the
               other
               shall
               stay
               at
               the
               difference
               required
               ,
               but
               in
               Winter
               set
               one
               leg
               to
               the
               beginning
               of
               the
               sines
               ,
               and
               open
               the
               other
               to
               the
               height
               at
               6
               ,
               or
               rather
               depression
               under
               the
               Horizon
               at
               6
               ,
               (
               which
               is
               all
               one
               at
               like
               declinations
               ,
               North
               and
               South
               )
               then
               set
               the
               first
               leg
               to
               the
               present
               altitude
               of
               the
               Sun
               ,
               and
               the
               other
               shall
               shew
               the
               Sine
               of
               the
               sum
               of
               both
               added
               together
               ;
               which
               sum
               or
               difference
               is
               thus
               to
               be
               used
               :
            
             
               Lay
               the
               left
               leg
               to
               the
               Cosine
               of
               the
               declination
               ,
               and
               the
               other
               to
               the
               secant
               of
               the
               latitude
               ,
               counted
               beyond
               90
               ,
               as
               far
               as
               the
               secant
               of
               9
               ,
               40
               ;
               or
               rather
               lay
               the
               left
               leg
               on
               the
               Co-sine
               of
               the
               latitude
               ,
               and
               the
               other
               to
               the
               secant
               of
               the
               suns
               declination
               ,
               then
               the
               first
               leg
               laid
               on
               the
               sine
               of
               the
               sum
               in
               winter
               ,
               or
               difference
               in
               summer
               ,
               shall
               cause
               the
               other
               leg
               to
               fall
               on
               the
               sine
               of
               the
               hour
               from
               6
               ,
               toward
               noon
               in
               winter
               and
               summer
               ,
               except
               the
               altitude
               in
               summer
               be
               less
               then
               the
               altitude
               at
               6
               ,
               then
               it
               is
               the
               
               hour
               from
               6
               ,
               toward
               mid-night
               .
            
          
           
             
               PROP.
               3.
               
            
             
               Having
               the
               latitude
               ,
               Suns
               altitude
               and
               declination
               ,
               to
               finde
               the
               Suns
               Azimuth
               from
               east
               or
               west
               .
               Lat.
               51.
               32.
               
               Declin
               .
               23.
               30.
               
               Alt.
               49
               56.
               
            
             
               First
               you
               must
               get
               the
               Suns
               altitude
               ,
               or
               depression
               in
               the
               vertical
               Circle
               by
               this
               Cannon
               .
               Lay
               the
               first
               leg
               to
               the
               sine
               of
               the
               latitude
               ,
               and
               the
               second
               to
               the
               sine
               of
               90
               ,
               and
               you
               shall
               finde
               them
               both
               to
               be
               on
               the
               same
               line
               ,
               then
               the
               first
               leg
               laid
               on
               the
               sine
               of
               the
               declination
               ,
               shall
               cause
               the
               second
               (
               being
               carried
               with
               the
               first
               ,
               without
               moving
               the
               Angle
               first
               set
               )
               to
               fall
               on
               the
               sine
               of
               30.
               39.
               the
               Suns
               altitude
               in
               the
               Vertical
               Circle
               ;
               with
               which
               you
               must
               do
               ,
               as
               you
               did
               before
               with
               the
               altitude
               at
               6
               ,
               and
               the
               present
               altitude
               ,
               to
               finde
               the
               sum
               and
               difference
               by
               help
               of
               the
               line
               of
               natural
               Tangents
               ,
               then
               this
               proportion
               holds
               .
            
             
               Lay
               the
               first
               leg
               to
               the
               cosine
               of
               the
               
               Altitude
               ,
               (
               by
               counting
               the
               Altitude
               from
               20
               )
               and
               the
               second
               leg
               to
               the
               tangent
               of
               the
               latitude
               ,
               and
               observe
               which
               way
               ,
               and
               the
               turns
               between
               ;
               then
               the
               first
               leg
               removed
               and
               laid
               to
               the
               sine
               of
               the
               sum
               (
               before
               found
               )
               in
               winter
               ,
               or
               the
               difference
               in
               summer
               ,
               shall
               cause
               the
               second
               leg
               to
               fall
               on
               the
               sine
               of
               the
               Azimuth
               of
               the
               Sun
               ,
               from
               east
               or
               west
               toward
               noon
               ,
               if
               winter
               ;
               and
               also
               in
               summer
               ,
               when
               the
               Suns
               altitude
               is
               more
               than
               his
               altitude
               at
               the
               Vertical
               Circle
               ;
               but
               if
               less
               from
               the
               east
               or
               west
               ,
               toward
               north
               or
               mid-night
               meridian
               :
               Thus
               in
               our
               Example
               ,
               it
               will
               be
               found
               to
               be
               the
               sine
               of
               30
               degrees
               ,
               or
               60
               from
               the
               south
               ,
               the
               sine
               of
               the
               difference
               being
               found
               to
               be
               14.
               49.
               
            
          
           
             
               PROP.
               4.
               
            
             
               To
               finde
               the
               Azimuth
               ,
               according
               to
               Mr.
               Gunter
               ,
               by
               having
               the
               Latitude
               ,
               Suns
               Altitude
               ,
               and
               Declination
               given
               .
            
             
               First
               by
               the
               Suns
               declination
               get
               his
               
               distance
               from
               the
               Pole
               ,
               which
               in
               summer
               or
               North
               declinations
               ,
               is
               always
               the
               complement
               of
               the
               declination
               ,
               (
               likewise
               in
               south
               latitudes
               ,
               and
               south
               declinations
               )
               but
               when
               the
               latitude
               and
               declination
               is
               unlike
               ,
               then
               you
               must
               adde
               90
               to
               the
               declination
               ,
               and
               the
               sum
               is
               the
               distance
               from
               the
               elevated
               Pole.
               Having
               found
               the
               distance
               from
               the
               Pole
               ,
               adde
               that
               and
               the
               complement
               of
               the
               latitude
               ,
               and
               Suns
               altitude
               together
               ,
               finde
               the
               difference
               between
               their
               half
               sum
               ,
               and
               the
               Suns
               distance
               from
               the
               Pole
               ,
               then
               the
               proportion
               will
               be
               thus
               ,
               as
               in
               this
               Example
               :
               13
               Declination
               ,
               41.
               53.
               
               Alt.
               Lat.
               51.
               32.
               
            
             
               Lay
               one
               leg
               on
               the
               sine
               of
               90
               ,
               and
               the
               other
               to
               the
               sine
               of
               38
               ,
               28
               ,
               being
               so
               set
               ,
               remove
               the
               first
               leg
               to
               the
               sine
               complement
               of
               the
               altitude
               48.
               07.
               and
               the
               second
               legg
               shall
               fall
               on
               a
               fourth
               sine
               ,
               which
               will
               be
               found
               to
               be
               27.
               36.
               then
               set
               the
               first
               or
               left
               leg
               to
               27.
               36.
               the
               fourth
               sine
               ,
               and
               the
               second
               to
               81.
               47.
               the
               sine
               of
               the
               
               half
               sum
               ,
               then
               removing
               the
               first
               leg
               to
               the
               sine
               of
               the
               difference
               ,
               shall
               cause
               the
               second
               to
               shew
               two
               circles
               lower
               ,
               the
               versed
               sine
               of
               130
               ,
               the
               Azimuth
               required
               ,
               being
               counted
               from
               the
               North
               part
               of
               the
               horizon
               ,
               whose
               complement
               to
               180
               from
               the
               South
               is
               50
               degrees
               .
            
          
           
             
               Two
               other
               Canons
               to
               finde
               the
               hour
               of
               the
               day
               ,
               and
               Azimuth
               of
               the
               Sun
               ,
               by
               one
               operation
               ,
               by
               help
               of
               the
               natural
               sines
               :
               and
               first
               for
               the
               hour
               .
            
             
               Having
               the
               latitude
               ,
               the
               Suns
               Meridian
               ,
               and
               present
               altitude
               ,
               and
               declination
               ,
               to
               finde
               the
               hour
               from
               noon
               .
            
             
               First
               lay
               one
               leg
               to
               the
               Meridian
               altitude
               ,
               in
               the
               line
               of
               natural
               sines
               ,
               and
               the
               other
               to
               the
               sine
               of
               the
               altitude
               in
               the
               same
               line
               ,
               then
               bring
               the
               right
               leg
               to
               the
               beginning
               of
               the
               line
               of
               sines
               ,
               and
               the
               other
               〈…〉
               difference
               ,
               which
               difference
               〈…〉
               keep
               .
            
             
               Then
               lay
               one
               leg
               on
               the
               〈◊〉
               the
               declination
               ,
               and
               the
               other
               〈◊〉
               
               secant
               of
               the
               latitude
               ,
               and
               note
               the
               turnes
               between
               ,
               or
               rather
               lay
               the
               first
               leg
               to
               the
               cosine
               of
               the
               latitude
               ,
               and
               the
               other
               to
               the
               secant
               of
               the
               declination
               ,
               then
               the
               legs
               being
               so
               set
               ,
               bring
               the
               first
               or
               left
               leg
               to
               the
               sine
               of
               the
               difference
               first
               found
               ,
               and
               the
               other
               leg
               shall
               shew
               the
               versed
               sine
               of
               the
               hour
               from
               noon
               ,
               if
               the
               versed
               sines
               had
               been
               set
               thus
               ,
               
                 i.
                 e.
              
               the
               versed
               sine
               of
               90
               ,
               against
               the
               sine
               of
               90
               ,
               as
               in
               some
               instruments
               it
               is
               :
               But
               to
               remedy
               this
               defect
               ,
               do
               thus
               :
               keep
               the
               right
               leg
               there
               ,
               and
               open
               the
               other
               to
               the
               versed
               sine
               of
               0.
               or
               sine
               of
               90
               ,
               and
               note
               the
               turnes
               between
               ,
               then
               lay
               the
               leg
               that
               was
               on
               the
               sine
               of
               90
               ,
               (
               or
               versed
               sine
               of
               0
               ,
               )
               to
               the
               versed
               sine
               of
               90
               ,
               and
               the
               other
               leg
               shall
               shew
               the
               versed
               sine
               of
               the
               hour
               from
               noon
               ,
               counting
               from
               90.
               
            
             
               
                 Example
                 .
              
               
                 At
                 45
                 13
                 degrees
                 altitude
                 ,
                 declination
                 North
                 23
                 32.
                 latitude
                 51
                 32
                 ,
                 the
                 Meridian
                 altitude
                 is
                 62
                 ,
                 (
                 being
                 found
                 by
                 adding
                 colatitude
                 ,
                 and
                 declination
                 
                 together
                 ,
                 and
                 in
                 southern
                 declinations
                 by
                 substraction
                 .
                 )
                 Then
                 the
                 natural
                 sine
                 of
                 45.
                 42
                 ,
                 taken
                 from
                 62
                 ,
                 shall
                 be
                 the
                 sine
                 of
                 9.
                 38.
                 then
                 as
                 the
                 cosine
                 of
                 the
                 latitude
                 38.
                 28
                 ,
                 is
                 to
                 the
                 secant
                 of
                 declination
                 23.
                 32
                 ,
                 so
                 is
                 the
                 sine
                 of
                 9.
                 38.
                 to
                 the
                 sine
                 of
                 17.
                 05
                 )
                 or
                 versed
                 sine
                 of
                 45
                 ▪
                 if
                 they
                 were
                 placed
                 and
                 numbred
                 ,
                 as
                 in
                 some
                 instruments
                 they
                 be
                 :
                 but
                 to
                 help
                 i●
                 in
                 this
                 ,
                 say
                 :
                 as
                 the
                 versed
                 sine
                 of
                 0
                 ,
                 is
                 to
                 the
                 verses
                 of
                 114.
                 26
                 ,
                 so
                 is
                 the
                 versed
                 sine
                 of
                 90
                 ,
                 to
                 the
                 versed
                 sine
                 of
                 135
                 ,
                 whose
                 complement
                 to
                 180
                 is
                 the
                 angle
                 or
                 hour
                 from
                 noon
                 required
                 .
              
               
                 Secondly
                 ,
                 for
                 the
                 Suns
                 Azimuth
                 .
              
            
          
           
             
               Having
               the
               latitude
               ,
               declination
               ,
               and
               Suns
               altitude
               ,
               to
               finde
               the
               Azimuth
               from
               South
               or
               North.
               
            
             
               First
               add
               the
               complements
               of
               the
               coaltitude
               and
               cola●itude
               together
               ,
               then
               if
               the
               sum
               be
               under
               90
               ,
               take
               the
               distance
               between
               the
               cosine
               of
               it
               ,
               and
               the
               sine
               of
               the
               declination
               ,
               in
               the
               line
               
               of
               natural
               sines
               ,
               and
               measure
               it
               in
               the
               line
               of
               sines
               from
               the
               beginning
               ,
               and
               it
               shall
               give
               the
               sine
               of
               the
               difference
               ;
               but
               if
               the
               sum
               exceed
               90
               ,
               then
               when
               the
               latitude
               and
               declination
               is
               alike
               ,
               add
               the
               excess
               to
               the
               declination
               ;
               but
               if
               contrary
               substract
               one
               out
               of
               other
               ,
               and
               measuring
               the
               sum
               or
               remainder
               from
               the
               beginning
               of
               the
               sines
               ,
               you
               have
               the
               difference
               which
               you
               must
               keep
               .
            
             
               Then
               lay
               the
               first
               or
               left
               leg
               to
               the
               cosine
               of
               the
               altitude
               ,
               and
               the
               second
               to
               the
               secant
               of
               the
               latitude
               ,
               or
               else
               lay
               the
               first
               to
               the
               cosine
               of
               the
               latitude
               ,
               and
               the
               other
               to
               the
               secant
               of
               the
               altitude
               ,
               and
               more
               the
               turnes
               between
               ,
               then
               lay
               the
               first
               leg
               to
               the
               sine
               of
               the
               difference
               before
               found
               ,
               and
               the
               other
               shall
               shew
               the
               versed
               sine
               of
               the
               Azimuth
               from
               noon
               required
               ,
               if
               the
               versed
               sines
               be
               set
               as
               before
               is
               expressed
               ,
               that
               is
               to
               say
               90
               of
               right
               sines
               ,
               and
               versed
               sines
               together
               ,
               and
               numbred
               forwards
               as
               the
               sines
               be
               :
               but
               in
               the
               use
               of
               this
               instrument
               ,
               
               the
               remedy
               aforesaid
               supplyeth
               the
               defect
               .
            
             
               
                 Example
                 .
              
               
                 At
                 10.
                 19.
                 altitude
                 ,
                 23.
                 32.
                 declination
                 ,
                 51.
                 32.
                 latitude
                 ,
                 to
                 finde
                 the
                 Azimuth
                 .
              
               
                 The
                 sum
                 of
                 the
                 coaltitude
                 and
                 co-latitude
                 is
                 118.09
                 ,
                 the
                 excess
                 above
                 90
                 ,
                 with
                 right
                 sine
                 of
                 declination
                 added
                 is
                 60.
                 30
                 ,
                 found
                 by
                 natural
                 sines
                 ,
                 then
                 say
                 ,
                 As
                 the
                 cosine
                 of
                 latitude
                 ,
                 to
                 the
                 secant
                 of
                 altitude
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 difference
                 60.
                 30
                 ,
                 to
                 the
                 versed
                 sine
                 of
                 the
                 Azimuth
                 ,
                 but
                 here
                 to
                 the
                 versed
                 sine
                 of
                 an
                 ark
                 beyond
                 Radius
                 unknown
                 ,
                 then
                 as
                 the
                 versed
                 sine
                 of
                 0
                 ,
                 to
                 that
                 ark
                 ,
                 so
                 is
                 the
                 versed
                 sine
                 of
                 90
                 ,
                 to
                 the
                 ver
                 sine
                 of
                 60
                 ,
                 the
                 azimuth
                 from
                 noon
                 ,
                 whose
                 complement
                 to
                 180
                 ,
                 is
                 125
                 the
                 Azimuth
                 from
                 Sonth
                 required
                 .
              
               
                 Having
                 been
                 so
                 large
                 in
                 these
                 ,
                 I
                 shall
                 in
                 the
                 rest
                 contract
                 my self
                 as
                 to
                 the
                 repetition
                 ,
                 and
                 onely
                 give
                 the
                 Canon
                 for
                 the
                 propositions
                 following
                 ,
                 the
                 way
                 of
                 working
                 being
                 the
                 same
                 in
                 
                 all
                 other
                 ,
                 as
                 in
                 these
                 before
                 rehearsed
                 ,
                 and
                 note
                 also
                 what
                 is
                 to
                 be
                 done
                 by
                 the
                 Serpentine-line
                 ,
                 is
                 to
                 be
                 done
                 by
                 the
                 three
                 same
                 lines
                 of
                 numbers
                 ,
                 sines
                 and
                 Tangents
                 on
                 the
                 edge
                 of
                 the
                 Sector
                 ,
                 by
                 altering
                 the
                 term
                 leg
                 ,
                 to
                 to
                 the
                 point
                 of
                 the
                 Compasses
                 .
                 The
                 Canons
                 follow
                 .
              
            
          
           
             
               PROP
               4.
               
            
             
               Having
               latitude
               ,
               declination
               ,
               and
               hour
               given
               ,
               to
               finde
               the
               Suns
               altitude
               at
               that
               hour
               or
               quarter
               .
            
             
               And
               first
               for
               the
               hour
               of
               6.
               
            
             
               As
               the
               sine
               of
               90
               ,
               
               To
               the
               sine
               of
               the
               latitude
               51.
               30
               
               :
               
               So
               is
               the
               sine
               of
               the
               declination
               23.
               30
               ,
               
               To
               the
               sine
               of
               the
               altitude
               at
               6.
               18.
               13.
               
            
             
               Secondly
               ,
               for
               all
               hours
               in
               the
               Equinoctial
               .
            
          
           
             
               As
               the
               Radius
               or
               sine
               of
               90
               ,
               to
               the
               cosine
               of
               the
               latitude
               51.
               32
               :
               so
               is
               the
               sine
               of
               the
               Suns
               distance
               from
               6
               (
               in
               hours
               and
               minutes
               ,
               being
               turned
               into
               degrees
               and
               minutes
               30
               ,
               for
               8.
               or
               4
               ,
               )
            
             
             
               To
               the
               sine
               of
               the
               altitude
               of
               the
               Sun
               at
               the
               time
               required
               18.
               07
               ,
               But
               for
               all
               other
               times
               say
               ,
            
             
               As
               the
               sine
               of
               90
               ,
               
               To
               the
               cotangent
               of
               the
               latitude
               38.
               28.
               
               
               So
               is
               the
               sine
               of
               the
               Suns
               distance
               from
               6
               ,
               30
               ,
               0
               ,
               
               To
               the
               tangent
               of
               the
               4
               arke
               21
               40.
               
            
             
               Which
               fourth
               arke
               must
               be
               taken
               out
               of
               the
               Suns
               distance
               from
               the
               Pole
               66
               ,
               21
               ,
               (
               in
               Cancer
               )
               leaveth
               a
               residue
               44
               48
               ,
               which
               is
               called
               a
               fift
               ark
               .
               But
               for
               the
               hours
               before
               and
               after
               6
               ,
               you
               must
               add
               the
               fourth
               arke
               to
               the
               Suns
               distance
               from
               the
               Pole
               ,
               and
               the
               Sum
               is
               the
               fifth
               ark
               .
               Then
               say
               ,
            
             
               As
               the
               cosine
               of
               the
               fourth
               ark
               78
               ,
               20
               ,
               
               Is
               to
               the
               sine
               of
               the
               latitude
               51
               ,
               32
               :
               
               So
               is
               the
               cosine
               of
               the
               residue
               45
               ,
               12
               ,
               
               To
               the
               sine
               of
               the
               Suns
               altitude
               at
               8
               ,
               36
               ,
               42
               ,
               at
               that
               declination
               .
            
          
           
             
             
               PROP.
               6.
               
            
             
               Having
               the
               latitude
               ,
               declination
               ,
               and
               Azimuth
               ,
               to
               finde
               the
               Suns
               altitude
               at
               that
               Azimuth
               .
            
             
               And
               first
               to
               finde
               the
               Suns
               altitude
               at
               any
               Azimuth
               in
               the
               aequator
               .
               Then
               ,
               As
               the
               sine
               of
               90
               ,
               to
               the
               cosine
               of
               the
               Suns
               Azimuth
               from
               the
               South
               50
               ,
               0
               ,
               So
               is
               the
               cotangent
               of
               the
               latitude
               38
               ,
               28
               ,
               to
               the
               tangent
               of
               27
               ,
               03
               ▪
               the
               Suns
               altitude
               ,
               at
               that
               Azimuth
               required
               .
            
             
               Secondly
               ,
               to
               find
               it
               at
               all
               other
               times
               ,
               do
               thus
               :
               As
               the
               sine
               of
               the
               latitude
               51
               ,
               32
               ,
               To
               the
               sine
               of
               the
               Suns
               declination
               ,
               23
               ,
               32
               :
            
             
               So
               is
               the
               cosine
               of
               the
               Suns
               altitude
               in
               the
               aequator
               ,
               at
               the
               same
               Azimuth
               from
               the
               vertical
               ,
               viz.
               30
               ,
               to
               the
               sine
               of
               a
               4th
               .
               ark
               28
               ,
               16.
               
            
             
               Which
               fourth
               ark
               must
               be
               added
               to
               the
               Suns
               altitude
               at
               the
               aequator
               in
               all
               Azimuths
               under
               90
               ,
               from
               the
               meridian
               ,
               where
               the
               latitude
               and
               declination
               are
               alike
               .
            
             
             
               But
               in
               Azimuths
               more
               then
               90
               from
               the
               meridian
               ,
               take
               the
               altitude
               in
               the
               aequator
               out
               of
               the
               fourth
               ark
               ,
               and
               the
               sum
               or
               remainder
               shall
               be
               the
               altitude
               required
               ,
               viz.
               42
               ,
               56.
               
               But
               when
               the
               latitude
               and
               declination
               are
               unlike
               ,
               as
               with
               us
               in
               winter
               time
               ,
               then
               take
               the
               fourth
               ark
               out
               of
               the
               altitude
               at
               the
               aequator
               ,
               and
               you
               shall
               have
               the
               altitude
               belonging
               to
               that
               Azimuth
               required
               .
            
          
           
             
               PROP.
               7.
               
            
             
               Having
               the
               hour
               from
               noon
               ,
               and
               the
               altitude
               to
               find
               the
               Suns
               Azimuth
               at
               that
               time
               .
            
             
               As
               the
               cosine
               of
               the
               altitude
               ,
               
               To
               the
               sine
               of
               the
               hour
               ,
               
               So
               is
               the
               cosine
               of
               the
               Suns
               declination
               ,
               
               To
               the
               sine
               of
               the
               Azimuth
               required
               .
            
          
           
             
               PROP.
               8.
               
            
             
               Having
               the
               Suns
               Azimuth
               ,
               Altitude
               ,
               and
               declination
               ,
               to
               find
               tho
               hour
               of
               the
               day
               .
            
             
               As
               the
               cosine
               of
               the
               declination
               ,
               
               To
               the
               sine
               of
               the
               Suns
               Azimuth
               :
               
               
               So
               is
               the
               cosine
               of
               the
               altitude
               ,
               
               To
               the
               sine
               of
               the
               hour
               .
            
          
           
             
               PROP.
               9
               ,
            
             
               Having
               the
               latitude
               and
               declination
               ,
               to
               find
               when
               the
               Sun
               shall
               be
               due
               East
               or
               West
               .
            
             
               As
               the
               tangent
               of
               the
               latitude
               ,
               
               To
               the
               tangent
               of
               the
               Suns
               declination
               ,
               
               So
               is
               the
               sine
               of
               90
               ,
               
               To
               the
               cosine
               of
               the
               hour
               from
               noon
               .
            
          
           
             
               PROP.
               10.
               
            
             
               Having
               the
               latitude
               and
               Suns
               declination
               ,
               to
               find
               the
               Amplitude
               .
            
             
               As
               the
               cosine
               of
               the
               latitude
               ,
               
               To
               the
               sine
               of
               the
               declination
               :
               
               So
               is
               the
               sine
               of
               90
               ,
               
               To
               the
               sine
               of
               the
               amplitude
               from
               the
               East
               or
               West
               ,
               toward
               North
               or
               South
               ,
               according
               to
               the
               time
               of
               the
               day
               and
               year
               .
            
          
           
             
               PROP.
               11.
               
            
             
               The
               latitude
               and
               declination
               given
               ,
               to
               find
               the
               time
               of
               the
               Suns
               rising
               before
               or
               after
               6.
               
            
             
             
               As
               the
               cotangent
               of
               the
               latitude
               ,
               
               To
               the
               sine
               of
               90
               :
               
               So
               is
               the
               tangent
               of
               the
               Suns
               declination
               ,
               to
               the
               sine
               of
               the
               Suns
               ascentional
               difference
               between
               the
               hour
               of
               6
               and
               the
               Suns
               rising
               .
            
          
           
             
               PROP.
               12.
               
            
             
               Having
               the
               Suns
               place
               ,
               to
               find
               his
               declination
               ,
               and
               the
               contrary
               .
            
             
               As
               the
               sine
               of
               90
               ,
               
               To
               the
               suns
               distance
               from
               the
               next
               equinoctial
               point
               ,
               
               So
               is
               the
               sine
               of
               the
               suns
               greatest
               declination
               ,
               
               To
               the
               sine
               of
               his
               present
               declination
               required
               .
            
          
           
             
               PROP.
               13.
               
            
             
               The
               greatest
               and
               present
               declination
               given
               ,
               to
               find
               the
               Suns
               right
               ascension
               .
            
             
               As
               the
               tangent
               of
               the
               greatest
               declination
               ,
               
               To
               the
               sine
               of
               90
               :
               
               So
               the
               tangent
               of
               the
               present
               declination
               ,
               
               To
               the
               right
               ascension
               required
               .
            
             
             
               Onely
               you
               must
               regard
               to
               give
               it
               a
               right
               account
               by
               considering
               the
               time
               of
               the
               year
               ,
               and
               how
               many
               90s
               .
               past
               .
            
          
           
             
               PROR
               .
               14.
               
            
             
               To
               find
               an
               altitude
               by
               the
               length
               ,
               and
               shadow
               of
               any
               perpendicular
               object
               .
            
             
               Lay
               the
               hair
               on
               one
               legg
               to
               the
               length
               of
               the
               shadow
               found
               on
               the
               line
               of
               numbers
               ,
               and
               the
               hair
               of
               the
               other
               leg
               to
               the
               length
               of
               the
               object
               that
               caused
               the
               shadow
               found
               on
               the
               same
               line
               of
               the
               numbers
               ;
               then
               observe
               the
               lines
               between
               ,
               and
               which
               way
               when
               the
               legs
               are
               so
               set
               ,
               bring
               the
               first
               of
               them
               to
               the
               tangent
               of
               45
               ,
               and
               the
               other
               leg
               shall
               ●hew
               on
               the
               line
               of
               tangents
               ,
               so
               many
               turns
               between
               ,
               and
               the
               same
               way
               the
               tangent
               of
               the
               altitude
               required
               .
            
             
               Thus
               may
               you
               apply
               all
               manner
               of
               quest
               .
               to
               the
               Serpentine-line
               &
               work
               them
               by
               the
               same
               Canons
               ,
               that
               you
               use
               for
               the
               Logarithms
               in
               all
               or
               most
               Authors
               .
            
          
           
             
               PROP.
               15.
               
            
             
               To
               square
               ,
               and
               cube
               a
               number
               ,
               and
               to
               findethe
               square
               root
               ,
               or
               cube
               roat
               of
               a
               number
               .
            
             
             
               The
               squaring
               of
               a
               number
               ,
               is
               nothing
               else
               but
               the
               multiplying
               of
               the
               number
               by
               it self
               ,
               as
               to
               square
               12
               is
               to
               multiply
               12
               by
               12
               ,
               and
               then
               the
               cubing
               of
               12
               ,
               is
               to
               multiply
               the
               square
               144
               by
               12
               ,
               &
               that
               makes
               1728
               ,
               and
               the
               way
               to
               work
               it
               ,
               is
               thus
               :
               Set
               the
               first
               leg
               to
               1
               ,
               and
               the
               other
               to
               12
               ,
               then
               set
               the
               first
               to
               12
               ,
               and
               then
               the
               second
               shall
               reach
               to
               144
               ,
               then
               set
               the
               first
               to
               144
               ,
               and
               the
               second
               shall
               reach
               to
               1728
               ,
               the
               cube
               of
               12
               required
               :
               but
               note
               ,
               the
               number
               of
               figures
               in
               a
               cube
               ,
               that
               hath
               but
               one
               figure
               is
               certainly
               found
               by
               the
               line
               ,
               by
               the
               rule
               aforegoing
               :
               but
               if
               there
               be
               more
               figures
               then
               one
               ,
               so
               many
               times
               3
               must
               be
               added
               to
               the
               cube
               ,
               and
               so
               many
               times
               two
               to
               the
               square
               .
            
             
               To
               find
               the
               square
               root
               of
               a
               number
               ,
               do
               thus
               :
               Put
               a
               prick
               under
               the
               first
               ,
               the
               third
               ,
               the
               5th
               ,
               the
               7th
               ,
               &
               the
               number
               of
               pricks
               doth
               shew
               the
               number
               of
               figures
               in
               the
               root
               ;
               and
               note
               if
               the
               figures
               be
               even
               ,
               count
               the
               100
               to
               be
               the
               unit
               ,
               if
               odde
               as
               3
               ,
               5
               ,
               7
               ,
               9
               ,
               &c.
               
               the
               10
               at
               the
               beginning
               must
               be
               th●
               unit
               ,
               as
               for
               144
               ,
               the
               root
               consists
               of
               two
               figures
               ,
               because
               there
               is
               two
               pricks
               under
               the
               number
               ,
               and
               if
               you
               lay
               the
               index
               to
               144
               in
               the
               numbers
               ,
               it
               cu●s
               on
               the
               line
               of
               Logarithms
               15870
               ,
               the
               half
               of
               which
               is
               7915
               whereunto
               if
               you
               lay
               the
               index
               ,
               it
               shall
               shew
               the
               12
               the
               root
               required
               ;
               but
               if
               you
               would
               have
               the
               root
               of
               14+44
               ,
               then
               divide
               the
               space
               between
               that
               number
               ,
               and
               100
               you
               shall
               finde
               it
               come
               to
               8
               ,
               4140
               that
               is
               four
               turnes
               ,
               and
               4140
               for
               which
               four
               turnes
               ,
               you
               must
               count
               80000
               ,
               the
               half
               of
               which
               8,4140
               ,
               is
               4,2070
               ,
               whereunto
               if
               you
               lay
               the
               index
               ,
               and
               count
               from
               1444
               ●r
               100
               ,
               at
               the
               end
               you
               shall
               have
               it
               cut
               at
               38
               lack
               four
               of
               a
               100.
               
            
             
               To
               extract
               the
               cubique
               root
               of
               a
               number
               ,
               set
               the
               number
               down
               ,
               and
               put
               a
               point
               under
               the
               1
               ,
               the
               4th
               ,
               the
               7th
               ,
               and
               10th
               ,
               and
               look
               how
               many
               pricks
               ,
               so
               many
               figures
               must
               be
               in
               the
               root
               ,
               but
               to
               finde
               the
               unity
               you
               must
               consider
               ,
               if
               the
               prick
               falls
               on
               the
               last
               
               figure
               ,
               then
               the
               10
               is
               the
               unit
               at
               the
               beginning
               of
               the
               line
               ,
               as
               it
               doth
               in
               1728
               ,
               for
               the
               index
               laid
               on
               1728
               ,
               in
               the
               Log●rithms
               ,
               sheweth
               2,3760
               ,
               whose
               third
               part
               0,7920
               counted
               from
               10
               ,
               falls
               on
               12
               the
               root
               ,
               but
               in
               17280
               ,
               then
               you
               must
               conceive
               five
               whole
               turnes
               ,
               or
               1000
               to
               be
               added
               ,
               to
               give
               the
               number
               that
               is
               to
               be
               divided
               by
               three
               ,
               which
               number
               on
               the
               outermost
               circle
               in
               this
               place
               ,
               is
               12
               ,
               +3750.
               by
               conceiving
               10000
               to
               be
               added
               ,
               whose
               third
               part
               counted
               from
               10
               ,
               viz.
               two
               turnes
               or
               4.125
               ,
               shall
               fall
               in
               the
               numbers
               to
               be
               near
               26.
               
               But
               if
               the
               prick
               falls
               of
               the
               last
               but
               2
               ,
               as
               in
               172800
               then
               100
               at
               the
               end
               of
               the
               line
               ,
               must
               be
               the
               unit
               ,
               and
               you
               must
               count
               thus
               :
               count
               all
               the
               turnes
               from
               172830
               to
               the
               end
               of
               the
               line
               ,
               and
               you
               shall
               finde
               them
               to
               amount
               to
               7,6250
               ,
               whose
               third
               part
               2
               ,
               5413
               counted
               backward
               from
               100
               ,
               will
               fall
               on
               55,70
               the
               cubique
               root
               required
               .
            
          
           
             
             
               PROP.
               16.
               
            
             
               To
               work
               questions
               of
               interest
               or
               progression
               ,
               you
               must
               use
               the
               help
               of
               equal
               parts
               ,
               as
               in
               the
               extraction
               of
               roots
               ,
               as
               in
               this
               question
               ,
               if
               100
               l.
               yield
               106
               in
               one
               year
               ,
               what
               shall
               253
               yield
               in
               7
               year
               ?
               Set
               the
               first
               leg
               to
               10
               at
               the
               beginning
               ,
               in
               this
               case
               representing
               a
               100
               ,
               and
               the
               other
               to
               106
               ,
               and
               you
               shall
               finde
               the
               legs
               to
               open
               to
               253
               of
               the
               small
               divisions
               ,
               on
               the
               Logarithms
               ,
               multiply
               253
               by
               7
               ,
               it
               comes
               to
               1771
               ,
               now
               if
               you
               lay
               the
               hair
               upon
               253
               ,
               and
               from
               the
               place
               where
               the
               index
               cuts
               the
               Logarithms
               count
               onwards
               1771
               ,
               it
               shall
               stay
               on
               380
               l.
               8
               s.
               or
               rather
               thus
               :
               set
               one
               leg
               to
               the
               beginning
               of
               the
               Logarithms
               ,
               and
               the
               other
               to
               1771
               either
               forward
               or
               backward
               ,
               and
               then
               set
               the
               same
               first
               leg
               to
               the
               sum
               253
               ,
               and
               the
               second
               shall
               fall
               on
               380.
               8
               s.
               according
               to
               estimation
               ;
               the
               contrary
               work
               is
               to
               finde
               what
               a
               sum
               of
               money
               due
               at
               a
               time
               to
               ●ome
               ,
               is
               worth
               in
               ready
               money
               :
               this
               being
               premised
               here
               ,
               is
               enough
               
               for
               the
               ingenious
               to
               apply
               it
               to
               any
               question
               of
               this
               nature
               ,
               by
               the
               rules
               in
               other
               Authors
               .
               However
               you
               may
               shortly
               expect
               a
               more
               ample
               treatise
               ,
               in
               the
               mean
               time
               take
               this
               for
               a
               taste
               and
               farewell
               .
            
          
           
             
               The
               Use
               of
               the
               Almanack
               .
            
             
               Having
               the
               year
               ,
               to
               finde
               the
               day
               of
               the
               week
               the
               first
               of
               March
               is
               on
               in
               that
               year
               ,
               and
               Dominical
               letter
               also
               .
            
             
               First
               if
               it
               be
               a
               Leap-year
               ,
               then
               look
               for
               it
               in
               the
               row
               of
               Leap-year
               ,
               and
               in
               the
               column
               of
               week-days
               ,
               right
               over
               it
               is
               the
               day
               required
               ,
               and
               in
               the
               row
               of
               dominical
               letters
               is
               the
               Sunday
               letters
               also
               :
               but
               note
               the
               Dominical
               letter
               changeth
               the
               first
               of
               Ianuary
               ,
               but
               the
               week
               day
               the
               first
               of
               March
               ,
               so
               also
               doth
               the
               Epact
               .
            
             
               Example
               .
               In
               the
               year
               1660
               ,
               right
               over
               60
               which
               stands
               for
               1660
               ,
               there
               is
               G
               for
               the
               Dominical
               or
               Sunday
               letter
               ,
               beginning
               at
               Ianuary
               ,
               and
               T
               for
               thursday
               the
               day
               of
               the
               week
               the
               first
               of
               March
               is
               on
               ,
               and
               28
               underneath
               for
               the
               epact
               that
               year
               ,
               but
               in
               the
               year
               
               1661.
               being
               the
               next
               after
               1660
               the
               Leap-year
               ,
               count
               onwards
               toward
               your
               right
               hand
               ,
               and
               when
               you
               come
               to
               the
               last
               column
               ,
               begin
               again
               at
               the
               right
               hand
               ,
               and
               so
               count
               forwards
               till
               you
               come
               to
               the
               next
               Leap-year
               ,
               according
               to
               this
               account
               for
               61
               ,
               T
               is
               the
               dominical
               letter
               ,
               and
               Friday
               is
               the
               first
               of
               March.
               
            
             
               But
               to
               finde
               the
               Epact
               ,
               count
               how
               many
               years
               it
               is
               since
               the
               last
               Leap-year
               ,
               which
               can
               be
               but
               three
               ,
               for
               every
               4th
               is
               a
               Leap-year
               ,
               and
               adde
               so
               many
               times
               11
               to
               the
               epact
               in
               the
               Leap-year
               last
               past
               ,
               and
               the
               sum
               ,
               if
               under
               30
               ,
               is
               the
               Epact
               ;
               if
               above
               30
               ,
               then
               the
               remainder
               30
               or
               60
               ,
               being
               substracted
               is
               the
               Epact
               for
               that
               year
               .
               Example
               for
               1661.28
               the
               epact
               for
               1660
               ,
               and
               11
               being
               added
               makes
               39
               from
               which
               take
               30
               ,
               and
               there
               remaineth
               9
               ,
               for
               the
               Epact
               for
               the
               year
               1661
               the
               thing
               required
               .
            
             
               Note
               that
               in
               orderly
               counting
               the
               years
               ,
               when
               you
               come
               to
               the
               Leap-year
               ,
               you
               must
               neglect
               or
               slip
               one
               ,
               the
               
               reason
               is
               ,
               because
               every
               Leap-year
               hath
               two
               dominical
               letters
               ,
               and
               there
               also
               doth
               the
               week
               day
               change
               in
               the
               first
               of
               March
               ,
               so
               that
               for
               the
               day
               of
               the
               month
               ,
               in
               finding
               that
               the
               trouble
               of
               remembring
               the
               Leap-year
               is
               avoided
               .
            
             
               
                 To
                 find
                 the
                 day
                 of
                 the
                 Month.
                 
              
               
                 Having
                 found
                 the
                 day
                 of
                 the
                 week
                 ,
                 the
                 first
                 of
                 March
                 is
                 on
                 the
                 respective
                 year
                 ;
                 then
                 look
                 for
                 the
                 month
                 in
                 the
                 column
                 ,
                 and
                 row
                 of
                 months
                 :
                 then
                 all
                 the
                 daies
                 right
                 under
                 the
                 month
                 are
                 the
                 same
                 day
                 of
                 the
                 week
                 the
                 first
                 of
                 March
                 was
                 on
                 ,
                 then
                 in
                 regard
                 the
                 days
                 go
                 round
                 ,
                 that
                 is
                 change
                 orderly
                 every
                 seven
                 days
                 ,
                 you
                 may
                 find
                 any
                 other
                 successive
                 day
                 sought
                 for
                 .
              
               
                 
                   Example
                   .
                   About
                   the
                   middle
                   of
                   March
                   1661
                   on
                   a
                   Friday
                   ,
                   what
                   day
                   of
                   the
                   month
                   is
                   it
                   ?
                
                 
                   First
                   the
                   week
                   day
                   for
                   1661
                   is
                   Friday
                   ,
                   as
                   the
                   letter
                   F
                   on
                   the
                   next
                   collumn
                   beyond
                   60
                   she●et●
                   ;
                   then
                   I
                   look
                   for
                   1
                   among
                   the
                   months
                   ,
                   and
                   all
                   the
                   days
                   right
                   under
                   ,
                   viz.
                   1
                   ,
                   8
                   ,
                   15
                   ,
                   22
                   ,
                   
                   29.
                   in
                   March
                   ,
                   and
                   November
                   61
                   ,
                   are
                   Friday
                   ,
                   therefore
                   my
                   day
                   being
                   Friday
                   ,
                   and
                   about
                   the
                   middle
                   of
                   the
                   month
                   ,
                   I
                   conclude
                   it
                   is
                   the
                   15th
                   day
                   required
                   .
                
                 
                   Again
                   in
                   May
                   1661.
                   on
                   a
                   Saturday
                   about
                   the
                   end
                   of
                   May
                   ,
                   what
                   day
                   of
                   the
                   month
                   ?
                   May
                   is
                   the
                   third
                   month
                   ,
                   by
                   the
                   last
                   rule
                   I
                   find
                   that
                   the
                   24
                   and
                   31
                   are
                   Fridays
                   ,
                   therefore
                   this
                   must
                   needs
                   be
                   the
                   25
                   day
                   ,
                   for
                   the
                   first
                   of
                   Iune
                   is
                   the
                   next
                   Saturday
                   .
                
              
            
          
        
         
           FINIS
           .
        
      
    
     
       
         
           ERRATA
           .
        
         
           PAge
           23.
           l.
           4.
           adde
           1660
           p.
           24.
           l.
           6.
           for
           5
           hours
           r.
           4.
           l.
           9.
           for
           3.
           29.
           r.
           4.
           39.
           1.
           12
           for
           5.
           52.
           r.
           4
           ,
           52.
           l.
           13.
           for
           3.
           39.
           r.
           4.
           39.
           l.
           17
           for
           5
           hours
           52.
           r.
           4.
           52.
           p.
           27.
           l.
           ult
           .
           dele
           or
           11.
           03.
           
           p
           31.
           l
           4.
           for
           sun
           r.
           sum
           .
           p.
           50.
           l.
           8.
           for
           B
           r.
           A.
           p.
           50
           d
           CHAP.
           XII
           .
           p.
           51.
           r.
           16.
           for
           6.
           10.
           1.
           6
           to
           10.
           p.
           71.
           l.
           6.
           for
           7
           /
           4
           r.
           1
           /
           4.
           l.
           penult
           .
           for
           2
           afternoon
           ,
           r.
           1.
           p.
           74
           l.
           ult
           .
           for
           1.
           r.
           1
           ,
           2.
           p.
           83.
           l.
           18.
           for
           BC
           r.
           BD.
           p.
           69
           l.
           17.
           add
           measure
           ,
           p.
           129
           l.
           24.
           for
           right
           of
           ,
           r.
           
             right
             sine
             of
          
           .
           p.
           114
           l.
           9
           for
           18
           3.
           r
           18
           13.
           p.
           147
           1.
           2
           for
           20
           ,
           r.
           90
           p.
           163.
           l.
           16.
           for
           of
           ,
           r.
           on
           .