The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.
         Taylor, John, 1666 or 7-1687.
      
       
         
           1667
        
      
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             The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.
             Taylor, John, 1666 or 7-1687.
          
           [8], 144 p.
           
             printed for William Tompson, bookseller at Harborough in Leicestershire,
             London :
             1667.
          
           
             J.T. = John Taylor.
             In two books; Book II has caption title (the first word of which is in Greek characters); register and pagination are continuous.
             Pages stained with some loss of print.
             Reproduction of the original in the British Library.
          
        
      
    
     
       
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         eng
      
       
         
           Mathematics -- Early works to 1800.
           Navigation -- Early works to 1800.
           Dialing -- Early works to 1800.
        
      
    
     
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           THE
           SEMICIRCLE
           ON
           A
           SECTOR
           :
           In
           two
           Books
           .
           CONTAINING
           The
           description
           of
           a
           general
           and
           portable
           Instrument
           ;
           whereby
           most
           Problems
           (
           reducible
           to
           Instrumental
           Practice
           )
           in
           
             Astronomy
             ,
             Trigonometry
             ,
             Arithmetick
             ,
             Geometry
             ,
             Geography
             ,
             Topography
             ,
             Navigation
             ,
             Dyalling
             ,
          
           &c.
           are
           speedily
           and
           exactly
           resolved
           .
        
         
           By
           
             J.
             T.
          
           
        
         
           LONDON
           ,
           Printed
           for
           
             William
             Tompson
          
           ,
           Bookseller
           at
           Harborough
           in
           Leicestershire
           .
           1667.
           
        
      
       
         
         
         
           To
           the
           Reader
           .
        
         
           ALL
           that
           is
           intended
           in
           this
           Treatise
           ,
           is
           to
           acquaint
           thee
           with
           an
           Instrument
           ,
           that
           is
           both
           portable
           and
           general
           ,
           of
           no
           great
           price
           ,
           easie
           carriage
           ,
           yet
           of
           a
           speedy
           and
           Accurate
           dispatch
           in
           the
           most
           difficult
           Problems
           in
           Astronomy
           ,
           &c.
           
           The
           lines
           for
           the
           most
           part
           have
           been
           formerly
           published
           by
           Mr.
           Gunter
           ,
           the
           famous
           Mr.
           Foster
           ,
           Mr.
           
             White
             ,
             &c.
          
           The
           reduction
           of
           the
           28.
           
           Cases
           of
           Spherical
           Triangles
           unto
           II.
           Problems
           ,
           I
           first
           learned
           from
           the
           reverend
           Mr.
           Palmers
           Catholick
           Planisphere
           .
           Many
           of
           the
           proportions
           in
           the
           Treatise
           of
           Dyalling
           are
           taken
           from
           (
           though
           first
           compared
           with
           the
           Globe
           )
           my
           worthy
           Friend
           (
           to
           whom
           I
           am
           indebted
           in
           all
           the
           Obligations
           of
           Civility
           ,
           and
           without
           whose
           encouragement
           this
           had
           never
           adventured
           the
           publick
           Test
           )
           Mr.
           
             John
             Collins
          
           .
           The
           applying
           Mr.
           Fosters
           Line
           of
           Versed
           Sines
           unto
           the
           Sector
           was
           first
           published
           by
           Mr.
           
             John
             
             Brown
          
           ,
           Mathematical
           Instrument-Maker
           ,
           at
           the
           Sphere
           and
           Sun-Dial
           in
           the
           
             Minories
             ,
             London
             ,
             Anno
             1660.
          
           who
           bath
           very
           much
           assisted
           me
           ,
           by
           making
           ,
           adding
           unto
           ,
           and
           giving
           me
           freely
           the
           perusal
           of
           many
           Instruments
           ,
           according
           to
           any
           directions
           for
           Improvement
           ,
           that
           was
           proposed
           to
           him
           .
           After
           this
           account
           ,
           what
           hath
           been
           my
           part
           in
           this
           Work
           ,
           I
           hazard
           to
           thy
           censure
           ;
           and
           when
           I
           see
           others
           publish
           a
           more
           convenient
           ,
           speedy
           ,
           accurate
           ,
           and
           general
           Instrument
           ,
           I
           assure
           them
           to
           have
           as
           low
           thoughts
           of
           this
           ,
           as
           themselves
           .
           But
           here
           is
           so
           large
           a
           Catalogue
           of
           Errata's
           as
           would
           stagger
           my
           confidence
           at
           thy
           pardoning
           ,
           had
           they
           not
           been
           irrevocably
           committed
           before
           I
           received
           the
           least
           notice
           of
           them
           .
           The
           Printer
           writing
           me
           word
           (
           after
           I
           had
           corrected
           so
           much
           as
           came
           to
           my
           sight
           )
           that
           he
           could
           alter
           no
           Mistakes
           until
           the
           whole
           Book
           was
           printed
           :
           By
           which
           means
           he
           enforced
           me
           to
           do
           pennance
           in
           his
           Sheets
           ,
           for
           his
           own
           Crimes
           :
           Did
           not
           one
           gross
           mistake
           of
           his
           become
           my
           purgation
           ,
           viz.
           in
           
             lib.
             2.
          
           
           (
           throughout
           Chap.
           3.
           )
           where
           instead
           of
           the
           note
           of
           equality
           (
           marked
           thus
           =
           )
           he
           hath
           inserted
           the
           Algebraick
           note
           of
           Subtraction
           ,
           or
           Minoration
           (
           marked
           thus
           -
           )
        
         
           Nor
           hath
           the
           Engraver
           come
           behinde
           the
           Composer
           ,
           who
           so
           miserably
           mangleth
           Fig.
           13.
           that
           (
           at
           first
           sight
           )
           it
           would
           endanger
           branding
           of
           a
           mans
           Brains
           to
           spell
           the
           meaning
           thereof
           ,
           either
           in
           it self
           ,
           or
           in
           reference
           to
           the
           Book
           .
           All
           that
           I
           can
           help
           thee
           herein
           ,
           is
           this
           ;
           Whereas
           the
           Book
           mentions
           that
           Figure
           for
           an
           East
           Dyal
           ,
           if
           you
           account
           it
           (
           as
           now
           cut
           )
           a
           West
           Dyal
           ,
           and
           alter
           the
           names
           of
           the
           hours
           ,
           by
           putting
           Figures
           for
           the
           afternoon
           ,
           in
           the
           place
           of
           those
           there
           for
           the
           morning
           ;
           you
           will
           then
           have
           a
           true
           West
           Dyal
           of
           that
           Figure
           .
        
         
           The
           correction
           of
           Punctations
           would
           be
           an
           endless
           task
           ;
           for
           I
           finde
           some
           to
           be
           resolved
           ,
           ever
           since
           Valentine
           ,
           to
           recreate
           themselves
           at
           Spurn-point
           .
        
         
           What
           other
           material
           mistakes
           are
           in
           the
           Book
           (
           which
           ought
           to
           be
           corrected
           before
           reading
           thereof
           )
           you
           will
           finde
           mentioned
           in
           the
           Errata
           .
           Farewel
           .
        
         
           
             
               March
               29.
               1667.
               
            
          
           
             J.
             T.
             
          
        
      
       
         
         
           Errata
           .
        
         
           
             PAge
             4.
             line
             12.
          
           signs
           ,
           r.
           sines
           .
           
             p.
             8.
             l.
             5.
          
           seconds
           ,
           r.
           secants
           .
           
             p.
             12.
             l.
             9.
          
           all
           ,
           r.
           allone
           .
           
             p.
             13.
             l.
             9
             ,
             and
             10
             .
          
           sec
           .
           r.
           min.
           
             p.
             14.
             l.
             17.
          
           sec.
           r.
           min.
           
             p.
             22.
             l.
             14.
          
           any
           ,
           r.
           what
           .
           
             p.
             23.
             l.
             3.
          
           exact
           ,
           r.
           erect
           .
           
             p.
             24.
             l.
          
           ult
           .
           adde
           lib.
           
             2.
             p.
             25.
             l.
             2.
          
           signs
           ,
           r.
           sines
           .
           
             l.
             5
          
           .
           sign
           .
           r.
           sine
           .
           
             p.
             35.
             l.
             3
          
           .
           a
           mark
           ,
           r.
           an
           ark
           .
           
             p.
             37.
             l.
             22.
             20.
             r.
             22.
             and
             42.
             r.
             20.
             p.
             44.
             l.
             5.
             
          
           At
           ,
           r.
           At.
           
             p.
             49.
             l.
             1
          
           .
           divided
           by
           ,
           r.
           dividing
           .
           
             p.
             62.
             l.
             15.
             dele
          
           ,
           a.
           
             In
             lib.
             1.
             chap.
             9.
             the
             pages
             are
             false
             numbred
             .
             But
             in
             chap.
             9.
             p.
             62.
             l.
             11
             .
          
           next
           ,
           r.
           exact
           .
           
             l.
             15
          
           .
           gauger
           ,
           r.
           gauge
           .
           l.24.the
           ,
           r.
           what
           .
           
             p.
             73.l
             .
             5
          
           .
           whereas
           ,
           r.
           where
           I.
           
             p.
             80.
             l.
             13
          
           .
           wherein
           ,
           r.
           whereof
           .
           l.22.pont
           ,
           r.
           point
           .
           
             p.
             91
             .
             l.1
          
           .
           the
           co-tangent
           ,
           r.
           half
           the
           co-tangent
           .
           
             l.
             19
          
           .
           L.
           r.
           P.
           
             p.
             92.
             l.
             21
          
           .
           PZ
           .
           r.
           PS
           .
           
             p.
             93.
             l.
             12
             .
             NSP
             ,
             r.
          
           NSZ
           .
           6.angle
           ,
           r.
           ark
           .
           
             throughout
             page
             96.
          
           
           Fig.
           
             3.
             r.
          
           Fig.
           
             7.
             and
          
           Fig.
           
             4.
             r.
          
           Fig.
           
             8.
             p.
             98.
             l.
             13.
          
           serve
           ,
           r.
           scrue
           .
           
             p.
             99
             .
             l.12
          
           .
           places
           ,
           r.
           plates
           .
           
             l.
             13.
          
           proportion
           ,
           r.
           perforation
           .
           l.9.serve
           ,
           r.
           scrue
           .
           l.4.serve
           ,
           r.
           scrue
           .
           21.serve
           ,
           r
           :
           scrue
           .
           
             p.
             111.
             l.
             20.
          
           lay
           in
           ,
           r.
           laying
           .
           l.14.of
           ,
           ●●
           and.
           
             p.
             128.
             l.
             3.
          
           
           Fig.
           
             12.
             r.
          
           Fig.
           
             13.
             l.
             22
          
           .
           ED.
           r.
           EC
           .
        
         
           In
           Fig.
           12.
           
           C.
           r.
           A.
           at
           the
           end
           of
           the
           line
           G.
           
        
      
       
         
         
           To
           the
           Right
           Honorable
           ,
           The
           Lord
           SHERARD
           ,
           Baron
           of
           Letrim
           .
        
         
           
             My
             Lord
             ,
          
        
         
           SInce
           the
           trifling
           Treatise
           of
           an
           Almanack
           hath
           usurped
           a
           custom
           to
           pinnion
           some
           Honourable
           Name
           to
           the
           Patronage
           of
           the
           Authors
           Follies
           ;
           had
           we
           not
           certain
           evidence
           from
           the
           uncertainty
           of
           their
           Predictions
           ,
           that
           their
           Brains
           (
           like
           their
           great
           Oracles
           the
           Planets
           )
           are
           often
           wandring
           ;
           it
           might
           be
           deemed
           a
           Crime
           ,
           beyond
           the
           benefit
           of
           the
           Clergy
           ,
           to
           prefix
           before
           any
           Book
           ,
           a
           dedication
           to
           a
           Noble
           Person
           :
           Or
           when
           I
           read
           the
           unreasonableness
           of
           others
           in
           those
           Addresses
           ,
           imploring
           their
           Patrons
           to
           be
           their
           
             Dii
             Tutelares
          
           ,
           and
           prostrate
           their
           
           reputes
           to
           the
           unmannerly
           mangling
           of
           every
           Censurist
           ,
           under
           the
           notion
           of
           protecting
           (
           that
           is
           adopting
           )
           the
           Authors
           Ignorance
           ,
           or
           negligence
           ;
           it
           s
           enough
           to
           tempt
           the
           whole
           world
           to
           turn
           Democritians
           ,
           and
           hazard
           their
           spleens
           in
           laughing
           at
           such
           mens
           madness
           .
           My
           present
           design
           is
           only
           to
           give
           your
           Lordship
           my
           observance
           of
           your
           Commands
           about
           the
           Description
           and
           Improvement
           of
           the
           Sector
           ;
           and
           wherein
           I
           have
           erred
           (
           through
           mistake
           ,
           or
           defect
           )
           I
           despair
           not
           ,
           but
           from
           your
           Honor
           I
           shall
           meet
           with
           a
           pardon
           of
           Course
           to
           be
           granted
           unto
        
         
           
             Your
             Lordships
             most
             humble
             Servant
             
               J.
               T.
            
             
          
        
      
    
     
       
         
         
           THE
           SEMICIRCLE
           ON
           A
           SECTOR
           .
        
         
           
             LIB
             .
             I.
             
          
           
             
               CHAP.
               I.
               
            
             
               
                 A
                 Description
                 of
                 the
                 Instrument
                 ,
                 with
                 the
                 several
                 Lines
                 inscribed
                 thereon
                 .
              
            
             
               THe
               Instrument
               consists
               of
               three
               Rulers
               ,
               or
               Pieces
               ;
               two
               whereof
               are
               joyned
               together
               by
               a
               River
               that
               may
               open
               and
               shut
               to
               any
               Angle
               ,
               in
               fashion
               of
               the
               Sector
               ;
               or
               to
               use
               a
               courser
               comparison
               ,
               after
               the
               manner
               of
               Compasses
               .
               The
               third
               piece
               is
               loose
               ,
               or
               separable
               from
               them
               ,
               
               to
               be
               put
               into
               the
               Tenons
               at
               the
               end
               of
               the
               inward
               ledge
               of
               the
               joyned
               pieces
               ,
               and
               thereby
               constituting
               an
               aequilateral
               triangle
               .
            
             
               On
               these
               Rulers
               (
               after
               this
               manner
               put
               together
               )
               we
               take
               notice
               (
               for
               distinction
               sake
               )
               of
               the
               sides
               ,
               ledges
               ,
               ends
               ,
               and
               pieces
               .
            
             
               The
               sides
               are
               thus
               differenced
               ,
               one
               we
               call
               the
               quadrantal
               ,
               the
               other
               the
               proportional
               ,
               or
               sector
               side
               .
            
             
               The
               ledges
               are
               distinguished
               by
               naming
               one
               the
               inward
               ,
               the
               other
               the
               outward
               ledge
               .
            
             
               The
               ends
               are
               known
               in
               terming
               one
               the
               head
               (
               viz.
               where
               the
               two
               pieces
               are
               riveted
               together
               )
               the
               other
               the
               end
               .
            
             
               The
               pieces
               are
               discovered
               by
               styling
               one
               the
               fixed
               piece
               ,
               viz.
               that
               which
               hath
               the
               rivet
               upon
               it
               ;
               the
               second
               the
               movable
               piece
               ,
               which
               turns
               upon
               that
               rivet
               ,
               and
               the
               last
               the
               loose
               piece
               ,
               to
               be
               put
               into
               the
               tenons
               ,
               as
               before
               expressed
               .
            
             
               The
               quadrantal
               side
               of
               the
               joyned
               pieces
               is
               easily
               discerned
               ,
               having
               the
               names
               of
               the
               Moneths
               stamped
               on
               the
               movable
               piece
               ,
               and
               Par.
               Scale
               on
               the
               fixed
               piece
               .
               The
               quadrantal
               side
               of
               the
               loose
               piece
               is
               known
               by
               the
               degrees
               on
               the
               inward
               and
               outward
               limb
               .
            
             
               These
               directions
               are
               sufficient
               to
               instruct
               you
               how
               to
               put
               the
               Instrument
               together
               .
            
             
             
               Imagining
               the
               Instrument
               thus
               put
               together
               ,
               the
               lines
               upon
               the
               quadrantal
               side
               are
               these
               .
            
             
               First
               on
               the
               fixed
               piece
               ,
               next
               the
               outward
               limb
               ,
               is
               a
               line
               of
               12
               equal
               parts
               ;
               and
               each
               of
               those
               parts
               divided
               into
               30
               degrees
               ,
               marked
               from
               the
               end
               towards
               the
               head
               with
               ♈
               ♉
               ♊
               ,
               &c.
               representing
               the
               12
               Signs
               of
               the
               Zodiack
               ;
               the
               use
               of
               this
               line
               ,
               with
               the
               help
               of
               those
               under
               it
               ,
               was
               intended
               to
               find
               the
               hour
               of
               the
               night
               by
               the
               Moon
               .
            
             
               The
               next
               line
               to
               this
               is
               a
               line
               of
               twice
               12
               ,
               or
               24
               equal
               parts
               ,
               each
               division
               whereof
               cuts
               every
               15th
               degree
               of
               the
               former
               line
               :
               and
               therefore
               if
               the
               figures
               were
               set
               to
               every
               15th
               degree
               on
               this
               side
               the
               former
               line
               ,
               this
               second
               line
               would
               be
               useless
               ,
               and
               the
               former
               perform
               its
               office
               more
               distinctly
               .
               This
               line
               was
               intended
               an
               assistant
               for
               finding
               the
               hour
               by
               the
               Moon
               ;
               but
               is
               very
               ready
               to
               find
               the
               hour
               by
               any
               of
               the
               fixed
               Stars
               .
            
             
               The
               third
               line
               is
               a
               line
               of
               29●
               equal
               parts
               ,
               serving
               for
               the
               dayes
               of
               the
               Moons
               age
               ;
               in
               order
               to
               find
               the
               hour
               of
               the
               night
               by
               the
               Moon
               .
               But
               the
               operation
               is
               so
               tedious
               ,
               and
               far
               from
               exactness
               ,
               that
               I
               have
               no
               kindness
               for
               it
               ;
               and
               should
               place
               some
               other
               lines
               in
               the
               room
               of
               this
               and
               the
               former
               ,
               did
               I
               not
               resolve
               to
               impose
               upon
               no
               mans
               phantasie
               .
            
             
             
               The
               fourth
               line
               is
               a
               line
               of
               altitudes
               for
               a
               particular
               latitude
               ,
               noted
               at
               the
               end
               ,
               Par.
               Scale
               ,
               &c.
               
               This
               helpeth
               to
               find
               the
               hour
               and
               azimuth
               of
               the
               Sun
               ,
               or
               any
               fixed
               Star
               very
               exactly
               .
            
             
               The
               fifth
               line
               is
               a
               line
               of
               natural
               sines
               ;
               at
               the
               beginning
               whereof
               there
               is
               a
               pin
               ,
               or
               else
               an
               hole
               to
               put
               a
               pin
               into
               ,
               whereon
               to
               hang
               the
               thread
               and
               plummet
               for
               taking
               of
               altitudes
               .
               To
               this
               line
               of
               sines
               may
               be
               joyned
               a
               line
               of
               tangents
               to
               45
               degr
               .
               The
               use
               of
               the
               sines
               alone
               ,
               is
               to
               work
               proportions
               in
               signs
               .
               The
               use
               of
               the
               sines
               with
               the
               tangent
               line
               may
               be
               for
               any
               proportion
               in
               trigonometry
               ;
               but
               that
               I
               leave
               to
               liberty
               .
            
             
               The
               sixth
               line
               ,
               and
               last
               on
               this
               side
               the
               fixed
               piece
               is
               a
               line
               of
               versed
               sines
               ,
               numbred
               from
               the
               centre
               at
               the
               head
               to
               180
               at
               the
               end
               .
            
             
               On
               the
               quadrantal
               side
               of
               the
               movable
               piece
               ,
               the
               first
               line
               next
               the
               inward
               edge
               ,
               is
               a
               line
               of
               versed
               sines
               answering
               to
               that
               on
               the
               fixed
               piece
               .
               The
               use
               of
               these
               versed
               sines
               is
               various
               at
               pleasure
               .
            
             
               The
               second
               line
               from
               the
               inward
               edge
               ,
               is
               a
               line
               of
               hours
               and
               Azimuths
               serving
               to
               find
               the
               hour
               by
               the
               Sun
               or
               Stars
               ,
               or
               the
               Azimuth
               of
               the
               Sun
               ,
               or
               any
               fixed
               Star
               from
               the
               South
               .
            
             
             
               The
               third
               and
               fourth
               lines
               are
               lines
               of
               Moneths
               ,
               marked
               with
               the
               respective
               names
               ,
               and
               each
               Moneth
               divided
               into
               so
               many
               parts
               as
               it
               contains
               dayes
               .
            
             
               The
               fifth
               line
               is
               a
               line
               of
               signs
               marked
               ♈
               Taurus
               ;
               Gemini
               ;
               ,
               &c.
               each
               sign
               being
               divided
               into
               30
               degr
               .
               and
               proceeding
               from
               ♈
               ,
               or
               Aries
               (
               which
               answers
               to
               the
               tenth
               of
               March
               )
               in
               the
               same
               order
               as
               the
               Moneths
               .
               The
               use
               of
               this
               line
               ,
               with
               help
               of
               the
               Moneths
               ,
               is
               to
               find
               the
               Suns
               place
               in
               the
               Zodiack
               .
            
             
               The
               sixth
               line
               is
               a
               line
               of
               the
               Suns
               right
               ascension
               ,
               commonly
               noted
               by
               hours
               from
               00
               to
               24.
               but
               better
               if
               divided
               by
               degrees
               or
               sines
               from
               00
               to
               360
               ,
               and
               both
               wayes
               proceeding
               backward
               and
               forward
               as
               the
               signs
               of
               the
               Zodiack
               ,
               or
               dayes
               of
               the
               Moneth
               .
            
             
               Lastly
               the
               outward
               edge
               or
               limb
               of
               the
               movable
               and
               loose
               piece
               both
               ,
               is
               graduated
               unto
               180
               degrees
               ,
               or
               two
               quadrants
               ;
               whose
               centre
               is
               the
               pin
               ,
               or
               pin-hole
               before
               mentioned
               ,
               at
               the
               beginning
               of
               the
               sines
               on
               this
               side
               the
               fixed
               piece
               .
               The
               perpendicular
               is
               at
               0
               /
               60
               upon
               the
               loose
               piece
               ;
               from
               whence
               reckoning
               along
               the
               outward
               edge
               of
               the
               loose
               piece
               ,
               till
               it
               intersects
               the
               produced
               line
               of
               sines
               at
               the
               end
               of
               the
               fixed
               piece
               ,
               you
               have
               90
               degrees
               .
               Or
               counting
               from
               0
               /
               60
               on
               the
               loose
               piece
               ,
               and
               continuing
               
               it
               along
               the
               degrees
               of
               the
               outward
               limb
               of
               the
               movable
               piece
               ,
               until
               they
               intersect
               the
               produced
               line
               of
               sines
               on
               the
               fixed
               piece
               at
               the
               head
               ,
               you
               have
               again
               90
               degrees
               ,
               which
               compleat
               the
               Semicircle
               .
               The
               use
               of
               this
               line
               is
               for
               taking
               of
               altitudes
               ,
               counting
               upon
               the
               former
               90.
               degr
               .
               when
               you
               hold
               the
               head
               of
               the
               fixed
               piece
               toward
               the
               Sun
               :
               and
               numbring
               upon
               this
               latter
               ,
               when
               you
               hold
               (
               which
               is
               best
               because
               the
               degrees
               are
               largest
               )
               the
               end
               of
               the
               fixed
               piece
               toward
               the
               Sun.
               There
               are
               other
               wayes
               of
               numbring
               these
               degrees
               for
               finding
               of
               the
               Azimuth
               ,
               &c.
               which
               shall
               be
               mentioned
               in
               their
               proper
               places
               .
            
             
               On
               the
               quadrantal
               side
               of
               the
               loose
               piece
               ,
               the
               inward
               edge
               or
               limb
               is
               graduated
               unto
               60
               degrees
               (
               or
               twice
               30
               ,
               which
               you
               please
               )
               whose
               centre
               is
               a
               pin
               at
               the
               head
               .
               The
               use
               of
               this
               is
               to
               find
               the
               altitude
               of
               the
               Sun
               ,
               or
               any
               Star
               ,
               without
               thread
               and
               plummet
               ;
               or
               to
               perform
               some
               uses
               of
               the
               Cross-staff
               .
               This
               is
               for
               large
               rules
               or
               instruments
               ,
               and
               therefore
               not
               illustrated
               here
               .
            
             
               In
               the
               empty
               spaces
               upon
               the
               quadrantal
               side
               may
               be
               engraven
               the
               names
               of
               some
               fixed
               Stars
               ,
               with
               their
               right
               ascensions
               and
               declinations
               .
            
             
               On
               the
               proportional
               side
               the
               lines
               issuing
               from
               the
               centre
               are
               the
               same
               upon
               the
               fixed
               
               and
               movable
               piece
               ,
               but
               happily
               transplaced
               (
               thanks
               to
               the
               first
               contriver
               )
               after
               this
               manner
               :
               The
               line
               that
               lies
               next
               the
               inward
               edge
               on
               the
               fixed
               piece
               hath
               his
               fellow
               or
               correspondent
               line
               toward
               the
               outward
               edge
               on
               the
               movable
               piece
               ;
               by
               which
               means
               these
               lines
               all
               meeting
               at
               the
               centre
               ,
               stand
               all
               at
               the
               same
               angle
               ,
               and
               give
               you
               the
               freedom
               from
               a
               great
               deal
               of
               trouble
               ,
               in
               working
               proportions
               by
               sines
               and
               tangents
               ,
               or
               laying
               down
               any
               sine
               or
               tangent
               to
               any
               Radius
               given
               ,
               &c.
               
            
             
               The
               lines
               issuing
               from
               the
               centre
               toward
               the
               outward
               edge
               of
               the
               movable
               piece
               ,
               whose
               fellow
               is
               next
               the
               inward
               edge
               of
               the
               fixed
               piece
               ,
               is
               a
               line
               of
               natural
               sines
               on
               the
               outward
               side
               ,
               marked
               at
               the
               end
               S
               ,
               and
               on
               the
               inward
               side
               a
               line
               of
               lines
               or
               equal
               parts
               ,
               noted
               at
               the
               end
               L
               ;
               the
               middle
               line
               serving
               for
               both
               of
               them
               .
            
             
               The
               lines
               issuing
               from
               the
               centre
               next
               the
               inward
               edge
               of
               the
               movable
               piece
               ,
               whose
               fellows
               are
               toward
               the
               outward
               edge
               of
               the
               fixed
               piece
               ,
               are
               lines
               of
               natural
               tangents
               ,
               which
               on
               the
               outward
               ,
               side
               of
               the
               line
               is
               divided
               to
               45
               the
               Radius
               ;
               and
               on
               the
               inward
               side
               of
               the
               lines
               (
               the
               middle
               line
               serving
               for
               both
               )
               at
               a
               quarter
               of
               the
               former
               Radius
               from
               the
               centre
               is
               another
               Radius
               noted
               45
               at
               the
               beginning
               ,
               and
               continued
               to
               the
               tangent
               75.
               
               
               These
               lines
               are
               noted
               at
               the
               end
               T.
               The
               use
               of
               these
               you
               will
               find
               Chap.
               3,4,5
               .
            
             
               Betwixt
               the
               lines
               of
               sines
               and
               tangents
               ,
               both
               upon
               the
               fixed
               and
               movable
               pieces
               ,
               is
               placed
               a
               line
               of
               seconds
               ,
               continued
               unto
               60
               ,
               and
               marked
               at
               the
               end
               .
               Se.
               
            
             
               Next
               to
               the
               outward
               edge
               on
               the
               fixed
               and
               movable
               piece
               (
               which
               is
               best
               discerned
               when
               those
               pieces
               are
               opened
               to
               the
               full
               length
               )
               is
               a
               line
               of
               Meridians
               divided
               to
               85
               ;
               whose
               use
               is
               for
               Navigation
               ,
               in
               describing
               Maps
               or
               Charts
               ,
               &c.
               
            
             
               In
               the
               vacant
               spaces
               you
               may
               have
               a
               line
               of
               chords
               ,
               sines
               ,
               and
               tangents
               ,
               to
               any
               Radius
               the
               space
               will
               bear
               ;
               and
               what
               other
               any
               one
               thinks
               best
               of
               ,
               as
               a
               line
               of
               latitudes
               and
               hours
               ,
               &c.
               
            
             
               On
               the
               proportional
               side
               of
               the
               loose
               piece
               are
               lines
               for
               measuring
               all
               manner
               of
               solids
               ,
               as
               Timber
               ,
               Stone
               ,
               &c.
               likewise
               for
               gaging
               of
               Vessels
               either
               in
               Wine
               or
               Ale
               measure
               .
            
             
               On
               the
               outward
               ledge
               of
               the
               movable
               and
               fixed
               piece
               ,
               both
               (
               which
               in
               use
               must
               be
               stretched
               out
               to
               the
               full
               length
               )
               is
               a
               line
               of
               artificial
               numbers
               ,
               sines
               ,
               tangents
               ,
               and
               versed
               sines
               .
               The
               first
               marked
               N
               ,
               the
               second
               S
               ,
               the
               third
               T
               ,
               the
               fourth
               VS
               .
            
             
               On
               the
               inward
               ledge
               of
               the
               movable
               piece
               
               is
               a
               line
               of
               12
               inches
               divided
               into
               halfs
               ,
               quarters
               ,
               half-quarters
               .
               Next
               to
               that
               is
               a
               prick'd
               line
               ,
               whose
               use
               is
               for
               computing
               of
               weight
               and
               carriages
               .
               Lastly
               a
               line
               of
               foot
               measure
               ,
               or
               a
               foot
               divided
               into
               ten
               parts
               ;
               and
               each
               of
               those
               subdivided
               into
               ten
               or
               twenty
               more
               .
            
             
               On
               the
               inward
               ledge
               of
               the
               loose
               piece
               you
               may
               have
               a
               line
               of
               circumference
               ,
               diameter
               ,
               square
               equal
               ,
               and
               square
               inscribed
               .
            
             
               There
               will
               still
               be
               requisite
               sights
               ,
               a
               thread
               and
               plummet
               .
               And
               if
               any
               go
               to
               the
               price
               of
               a
               sliding
               Index
               to
               find
               the
               shadow
               from
               the
               plains
               perpendicular
               ,
               in
               order
               to
               taking
               a
               plains
               declination
               ,
               and
               have
               a
               staff
               and
               a
               ballsocket
               ,
               the
               Instrument
               is
               compleated
               with
               its
               furniture
               .
               Proceed
               we
               now
               to
               the
               uses
               .
            
             
               Onely
               note
               by
               the
               way
               that
               Mr.
               Brown
               hath
               (
               for
               conveniency
               of
               carrying
               a
               pair
               of
               Compasses
               ,
               Pen
               ,
               Ink
               ,
               and
               Pencil
               )
               contrived
               the
               fixed
               piece
               and
               movable
               both
               to
               be
               hollow
               ,
               and
               then
               the
               pieces
               that
               cover
               those
               hollows
               do
               ,
               one
               supply
               the
               place
               of
               the
               loose
               piece
               for
               taking
               altitudes
               ;
               the
               other
               (
               being
               a
               sliding
               rule
               )
               for
               measuring
               solids
               ,
               and
               gaging
               Vessels
               without
               Compasses
               .
            
          
           
             
             
               CHAP.
               II.
               
            
             
               
                 Some
                 uses
                 of
                 the
                 quadrantal
                 side
                 of
                 the
                 Instrument
                 .
              
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   To
                   find
                   the
                   altitude
                   of
                   the
                   Sun
                   or
                   any
                   Star.
                   
                
              
               
                 HAng
                 the
                 thread
                 and
                 plummet
                 upon
                 the
                 pin
                 at
                 the
                 beginning
                 of
                 the
                 line
                 of
                 sines
                 on
                 the
                 fixed
                 piece
                 ,
                 and
                 (
                 having
                 two
                 sights
                 in
                 two
                 holes
                 parallel
                 to
                 that
                 line
                 )
                 raise
                 the
                 end
                 of
                 the
                 fixed
                 piece
                 ,
                 toward
                 the
                 Sun
                 until
                 the
                 rayes
                 pass
                 through
                 the
                 sights
                 (
                 but
                 when
                 the
                 Sun
                 is
                 in
                 a
                 cloud
                 ,
                 or
                 you
                 take
                 the
                 altitude
                 of
                 any
                 Star
                 ,
                 look
                 along
                 the
                 outward
                 ledge
                 of
                 the
                 fixed
                 piece
                 ,
                 until
                 it
                 be
                 even
                 with
                 the
                 middle
                 of
                 the
                 Sun
                 or
                 Star
                 )
                 then
                 on
                 the
                 limb
                 the
                 thread
                 cuts
                 the
                 degree
                 of
                 altitude
                 ,
                 if
                 you
                 reckon
                 from
                 0
                 /
                 60
                 on
                 the
                 loose
                 piece
                 toward
                 the
                 head
                 of
                 the
                 movable
                 piece
                 .
              
            
             
               
               
                 PROBL.
                 2.
                 
              
               
                 
                   The
                   day
                   of
                   the
                   Moneth
                   given
                   ,
                   to
                   find
                   the
                   Suns
                   place
                   ,
                   declination
                   ,
                   ascensional
                   difference
                   ,
                   or
                   time
                   of
                   rising
                   and
                   setting
                   ,
                   with
                   his
                   right
                   ascension
                   .
                
              
               
                 The
                 thread
                 laid
                 to
                 the
                 day
                 of
                 the
                 Moneth
                 gives
                 the
                 Suns
                 place
                 in
                 the
                 line
                 of
                 signs
                 ,
                 reckoning
                 according
                 to
                 the
                 order
                 of
                 the
                 Moneths
                 
                   (
                   viz.
                
                 forward
                 from
                 March
                 the
                 10th
                 .
                 to
                 June
                 ,
                 then
                 backward
                 to
                 December
                 ,
                 and
                 forward
                 again
                 to
                 March
                 10.
                 )
                 In
                 the
                 limb
                 you
                 have
                 the
                 Suns
                 declination
                 ,
                 reckoning
                 from
                 60
                 /
                 0
                 on
                 the
                 movable
                 piece
                 towards
                 the
                 head
                 for
                 North
                 ,
                 toward
                 the
                 end
                 for
                 South
                 declination
                 .
                 Again
                 ,
                 on
                 the
                 line
                 of
                 right
                 ascensions
                 ,
                 the
                 thread
                 shews
                 the
                 Suns
                 right
                 ascension
                 ,
                 in
                 degrees
                 ,
                 or
                 hours
                 ,
                 (
                 according
                 to
                 the
                 making
                 of
                 your
                 line
                 )
                 counting
                 from
                 Aries
                 toward
                 the
                 head
                 ,
                 and
                 so
                 back
                 again
                 according
                 to
                 the
                 course
                 of
                 the
                 signs
                 unto
                 24
                 hours
                 ,
                 or
                 360
                 degrees
                 .
                 Lastly
                 on
                 the
                 line
                 of
                 hours
                 you
                 have
                 the
                 time
                 of
                 Sun
                 rising
                 and
                 setting
                 ,
                 which
                 turned
                 into
                 degrees
                 (
                 for
                 the
                 time
                 from
                 six
                 )
                 gives
                 the
                 ascensional
                 difference
                 .
                 Ex.
                 gr
                 .
                 in
                 lat
                 .
                 52.
                 deg
                 .
                 30
                 min.
                 for
                 which
                 latitude
                 I
                 shall
                 make
                 all
                 the
                 examples
                 .
                 
                 The
                 22
                 day
                 of
                 March
                 I
                 lay
                 the
                 thread
                 to
                 the
                 day
                 in
                 the
                 Moneths
                 ,
                 and
                 find
                 it
                 cut
                 in
                 the
                 Signs
                 12
                 deg
                 .
                 20
                 min.
                 for
                 the
                 Suns
                 place
                 ,
                 on
                 the
                 limb
                 4
                 deg
                 .
                 43
                 min.
                 for
                 the
                 Suns
                 declination
                 North.
                 In
                 the
                 line
                 of
                 right
                 ascensions
                 it
                 gives
                 46
                 min.
                 of
                 time
                 ,
                 or
                 11
                 deg
                 .
                 30
                 min.
                 of
                 the
                 circle
                 .
                 Lastly
                 ,
                 on
                 the
                 line
                 of
                 hours
                 it
                 shews
                 28
                 min.
                 before
                 six
                 for
                 the
                 Suns
                 rising
                 ;
                 or
                 which
                 is
                 all
                 ,
                 7
                 deg
                 .
                 for
                 his
                 ascensional
                 difference
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   The
                   declination
                   of
                   the
                   Sun
                   or
                   any
                   Star
                   given
                   to
                   find
                   their
                   amplitude
                   .
                
              
               
                 Take
                 the
                 declination
                 from
                 the
                 scale
                 of
                 altitudes
                 ,
                 with
                 this
                 distance
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 at
                 90
                 on
                 the
                 line
                 of
                 Azimuth
                 ,
                 apply
                 the
                 other
                 point
                 to
                 the
                 same
                 line
                 it
                 gives
                 the
                 amplitude
                 ,
                 counting
                 from
                 90
                 Ex.
                 gr
                 .
                 at
                 10
                 deg
                 .
                 declination
                 ,
                 the
                 amplitude
                 is
                 16
                 deg
                 .
                 30
                 min.
                 at
                 20
                 deg
                 .
                 declination
                 ,
                 the
                 amplitude
                 is
                 34
                 deg
                 .
              
            
             
               
               
                 PROBL.
                 4.
                 
              
               
                 
                   The
                   right
                   ascension
                   of
                   the
                   Sun
                   ,
                   with
                   his
                   ascensional
                   difference
                   ,
                   given
                   to
                   find
                   the
                   oblique
                   ascension
                   .
                
              
               
                 In
                 Northern
                 declination
                 ,
                 the
                 difference
                 betwixt
                 the
                 right
                 ascension
                 and
                 ascensional
                 difference
                 ,
                 is
                 the
                 oblique
                 ascension
                 .
                 In
                 Southern
                 declination
                 take
                 the
                 summ
                 of
                 them
                 for
                 the
                 oblique
                 ascension
                 ,
                 Ex.
                 gr
                 .
                 at
                 11
                 deg
                 .
                 30
                 sec.
                 right
                 ascension
                 ,
                 and
                 6
                 deg
                 .
                 30
                 sec.
                 ascensional
                 difference
                 .
                 In
                 Northern
                 declination
                 the
                 oblique
                 ascension
                 will
                 be
                 5
                 deg
                 .
                 in
                 Southern
                 18
                 deg
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   The
                   Suns
                   altitude
                   and
                   declination
                   ,
                   or
                   the
                   day
                   of
                   the
                   Moneth
                   given
                   to
                   find
                   the
                   hour
                   .
                
              
               
                 Take
                 the
                 Suns
                 altitude
                 from
                 the
                 Scale
                 of
                 altitudes
                 ,
                 and
                 laying
                 the
                 thread
                 to
                 the
                 declination
                 in
                 the
                 limb
                 (
                 or
                 which
                 is
                 all
                 one
                 ,
                 to
                 the
                 day
                 in
                 the
                 Moneths
                 )
                 move
                 one
                 point
                 of
                 the
                 compasses
                 along
                 the
                 line
                 of
                 hours
                 (
                 on
                 that
                 side
                 the
                 thread
                 next
                 the
                 end
                 )
                 until
                 the
                 other
                 point
                 just
                 touch
                 the
                 thread
                 ;
                 then
                 the
                 former
                 
                 point
                 shews
                 the
                 hour
                 ;
                 but
                 whether
                 it
                 be
                 before
                 or
                 after
                 noon
                 ,
                 is
                 left
                 to
                 your
                 judgment
                 to
                 determine
                 .
                 Ex.
                 gr
                 .
                 The
                 22
                 day
                 of
                 March
                 ,
                 or
                 4
                 deg
                 .
                 43
                 min.
                 North
                 declination
                 ,
                 and
                 20
                 deg
                 .
                 altitude
                 ,
                 the
                 hour
                 is
                 either
                 47
                 minutes
                 past
                 7
                 in
                 the
                 morning
                 ,
                 or
                 13
                 minutes
                 past
                 4
                 afternoon
                 .
              
            
             
               
                 PROBL.
                 6.
                 
              
               
                 
                   The
                   declination
                   of
                   the
                   Sun
                   ,
                   or
                   day
                   of
                   the
                   Moneth
                   ,
                   and
                   hour
                   given
                   to
                   find
                   the
                   altitude
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 day
                 or
                 declination
                 ,
                 and
                 take
                 the
                 least
                 distance
                 from
                 the
                 hour
                 to
                 the
                 thread
                 ,
                 this
                 applyed
                 to
                 the
                 line
                 of
                 altitudes
                 ,
                 gives
                 the
                 altitude
                 required
                 .
                 Ex.
                 gr
                 .
                 The
                 5
                 day
                 of
                 April
                 or
                 10
                 deg
                 .
                 declination
                 North
                 ,
                 at
                 7
                 in
                 the
                 morning
                 ,
                 or
                 5
                 afternoon
                 ,
                 the
                 altitude
                 will
                 be
                 17
                 deg
                 .
                 10
                 sec.
                 and
                 better
                 .
              
            
             
               
                 PROBL.
                 7.
                 
              
               
                 
                   The
                   declination
                   and
                   hour
                   of
                   the
                   night
                   ,
                   given
                   to
                   find
                   the
                   Suns
                   depression
                   under
                   the
                   horizon
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 declination
                 on
                 the
                 limb
                 ;
                 but
                 counted
                 the
                 contrary
                 way
                 ,
                 viz.
                 from
                 60
                 /
                 0
                 on
                 the
                 movable
                 piece
                 toward
                 the
                 
                 head
                 for
                 Southern
                 ;
                 and
                 toward
                 the
                 end
                 for
                 Northern
                 declination
                 .
                 This
                 done
                 take
                 the
                 nearest
                 distance
                 from
                 the
                 hour
                 to
                 the
                 thread
                 ,
                 and
                 applying
                 it
                 to
                 the
                 line
                 of
                 altitudes
                 ,
                 you
                 have
                 the
                 degrees
                 of
                 the
                 Suns
                 depression
                 .
                 Ex.
                 gr
                 .
                 at
                 5
                 deg
                 .
                 Northern
                 declination
                 ,
                 &
                 8
                 hours
                 afternoon
                 ,
                 the
                 depression
                 is
                 13
                 deg
                 .
                 30
                 min.
                 
              
            
             
               
                 PROBL.
                 8.
                 
              
               
                 
                   The
                   declination
                   given
                   to
                   find
                   the
                   beginning
                   and
                   end
                   of
                   twilight
                   ,
                   or
                   day-break
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 declination
                 counted
                 the
                 contrary
                 way
                 ,
                 as
                 in
                 the
                 last
                 Problem
                 ,
                 and
                 take
                 from
                 your
                 Scale
                 of
                 altitudes
                 18
                 deg
                 .
                 for
                 twilight
                 ,
                 and
                 13
                 deg
                 .
                 for
                 day-break
                 ,
                 or
                 clear
                 light
                 ;
                 with
                 this
                 run
                 one
                 point
                 of
                 the
                 Compasses
                 along
                 the
                 line
                 of
                 houres
                 (
                 on
                 that
                 side
                 next
                 the
                 end
                 )
                 until
                 the
                 other
                 will
                 just
                 touch
                 the
                 thread
                 ,
                 and
                 then
                 the
                 former
                 point
                 gives
                 the
                 respective
                 times
                 required
                 .
                 Ex.
                 gr
                 .
                 At
                 7
                 deg
                 .
                 North
                 declination
                 ,
                 day
                 breaks
                 8
                 minutes
                 before
                 4
                 :
                 but
                 twilight
                 is
                 3
                 houres
                 12
                 minutes
                 in
                 the
                 morning
                 ,
                 or
                 8
                 hours
                 52
                 minutes
                 afternoon
                 .
              
            
             
               
               
                 PROBL.
                 9.
                 
              
               
                 
                   The
                   declination
                   and
                   altitude
                   of
                   the
                   Sun
                   or
                   any
                   Star
                   ,
                   given
                   to
                   find
                   their
                   Azimuth
                   in
                   Northern
                   declination
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 altitude
                 numbred
                 on
                 the
                 limb
                 of
                 the
                 moveable
                 piece
                 from
                 60
                 /
                 0
                 toward
                 the
                 end
                 (
                 and
                 when
                 occasion
                 requires
                 ,
                 continue
                 your
                 numbring
                 forward
                 upon
                 the
                 loose
                 piece
                 )
                 and
                 take
                 the
                 declination
                 from
                 your
                 line
                 of
                 altitude
                 ;
                 with
                 this
                 distance
                 run
                 one
                 point
                 of
                 your
                 Compasses
                 along
                 the
                 line
                 of
                 Azimuths
                 (
                 on
                 that
                 side
                 the
                 thread
                 next
                 the
                 head
                 )
                 until
                 the
                 other
                 just
                 touch
                 the
                 thread
                 ,
                 then
                 the
                 former
                 point
                 gives
                 the
                 Azimuth
                 from
                 South
                 .
                 Ex.
                 gr
                 .
                 at
                 10
                 deg
                 .
                 declination
                 North
                 ,
                 and
                 30
                 deg
                 .
                 altitude
                 ,
                 the
                 Azimuth
                 from
                 South
                 is
                 64
                 ,
                 deg
                 .
                 40
                 min.
                 
              
            
             
               
                 PROBL.
                 10.
                 
              
               
                 
                   The
                   Suns
                   altitude
                   given
                   to
                   find
                   his
                   Azimuth
                   in
                   the
                   aequator
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 altitude
                 in
                 the
                 limb
                 ,
                 counted
                 from
                 60
                 /
                 0
                 on
                 the
                 loose
                 piece
                 toward
                 the
                 end
                 ,
                 and
                 on
                 the
                 line
                 of
                 Azimuths
                 
                 it
                 cuts
                 the
                 Azimuth
                 from
                 South
                 .
                 Ex.
                 gr
                 .
                 at
                 25
                 deg
                 .
                 altitude
                 the
                 Azimuth
                 is
                 53
                 deg
                 .
                 At
                 30
                 deg
                 .
                 altitude
                 the
                 Azimuth
                 is
                 41
                 deg
                 .
                 30
                 min.
                 fere
                 .
              
            
             
               
                 PROBL.
                 11.
                 
              
               
                 
                   The
                   declination
                   and
                   altitude
                   of
                   the
                   Sun
                   ,
                   or
                   any
                   Star
                   given
                   to
                   find
                   the
                   Azimuth
                   in
                   Southern
                   declination
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 altitude
                 numbred
                 on
                 the
                 limb
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 toward
                 the
                 end
                 ,
                 and
                 take
                 the
                 declination
                 from
                 the
                 Scale
                 of
                 altitudes
                 ;
                 then
                 carry
                 one
                 point
                 of
                 your
                 Compasses
                 on
                 the
                 line
                 of
                 Azimuths
                 (
                 on
                 that
                 side
                 the
                 thread
                 next
                 the
                 end
                 )
                 until
                 the
                 other
                 just
                 touch
                 the
                 thread
                 ,
                 which
                 done
                 ,
                 the
                 former
                 point
                 gives
                 the
                 Azimuth
                 from
                 South
                 .
                 Ex.
                 gr
                 .
                 at
                 15
                 deg
                 .
                 altitude
                 and
                 6
                 deg
                 .
                 South
                 declination
                 the
                 Azimuth
                 is
                 58
                 deg
                 .
                 30
                 min.
                 
              
            
             
               
                 PROBL.
                 12.
                 
              
               
                 
                   The
                   declination
                   given
                   to
                   find
                   the
                   Suns
                   altitude
                   at
                   East
                   or
                   West
                   in
                   North
                   declination
                   ,
                   and
                   by
                   consequent
                   his
                   depression
                   in
                   South
                   declination
                   .
                
              
               
                 Take
                 the
                 declination
                 given
                 from
                 the
                 Scale
                 of
                 altitudes
                 ,
                 and
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 90
                 on
                 the
                 line
                 of
                 Azimuths
                 ,
                 lay
                 the
                 thread
                 to
                 the
                 other
                 point
                 (
                 on
                 that
                 side
                 
                 90
                 next
                 the
                 head
                 )
                 on
                 the
                 limb
                 it
                 cuts
                 the
                 altitude
                 ,
                 counting
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 .
                 Ex.
                 gr
                 .
                 at
                 10
                 deg
                 .
                 declination
                 the
                 altitude
                 is
                 12
                 deg
                 .
                 40
                 min.
                 
              
            
             
               
                 PROBL.
                 13.
                 
              
               
                 
                   The
                   declination
                   and
                   Azimuth
                   given
                   to
                   find
                   the
                   altitude
                   of
                   the
                   Sun
                   or
                   any
                   Star.
                   
                
              
               
                 Take
                 the
                 declination
                 from
                 the
                 Scale
                 of
                 altitudes
                 ;
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Azimuth
                 given
                 ,
                 then
                 in
                 North
                 declinanation
                 turn
                 the
                 other
                 point
                 toward
                 the
                 head
                 ,
                 in
                 South
                 toward
                 the
                 end
                 ;
                 and
                 thereto
                 laying
                 the
                 thread
                 ,
                 on
                 the
                 limb
                 you
                 have
                 the
                 altitude
                 ,
                 numbring
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 toward
                 the
                 end
                 .
                 Ex.
                 gr
                 .
                 At
                 7
                 deg
                 .
                 North
                 declination
                 ,
                 and
                 48
                 deg
                 .
                 Azimuth
                 from
                 South
                 ,
                 the
                 altitude
                 is
                 35
                 deg
                 .
                 but
                 at
                 7
                 deg
                 .
                 declination
                 South
                 ,
                 and
                 50
                 deg
                 .
                 Azimuth
                 the
                 altitude
                 is
                 onely
                 18
                 deg
                 .
                 30
                 min.
                 
              
            
             
               
                 PROBL.
                 14.
                 
              
               
                 
                   The
                   altitude
                   ,
                   declination
                   ,
                   and
                   right
                   ascension
                   of
                   any
                   Star
                   with
                   the
                   right
                   ascension
                   of
                   the
                   Sun
                   given
                   ,
                   to
                   find
                   the
                   hour
                   of
                   the
                   night
                   .
                
              
               
                 Take
                 the
                 Stars
                 altitude
                 from
                 the
                 Scale
                 of
                 
                 altitudes
                 ,
                 and
                 laying
                 the
                 thread
                 to
                 his
                 declination
                 in
                 the
                 limb
                 ,
                 find
                 his
                 hour
                 from
                 the
                 last
                 Meridian
                 he
                 was
                 upon
                 ,
                 as
                 you
                 did
                 for
                 the
                 Sun
                 by
                 Probl.
                 5.
                 
                 If
                 the
                 Star
                 be
                 past
                 the
                 South
                 ,
                 this
                 is
                 an
                 afternoon
                 hour
                 ;
                 if
                 not
                 come
                 to
                 the
                 South
                 ,
                 a
                 morning
                 hour
                 ;
                 which
                 keep
                 .
              
               
                 Then
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Suns
                 right
                 ascension
                 (
                 numbred
                 upon
                 the
                 line
                 twice
                 12
                 or
                 24
                 next
                 the
                 outward
                 ledge
                 on
                 the
                 fixed
                 piece
                 )
                 extend
                 the
                 other
                 point
                 to
                 the
                 right
                 ascension
                 of
                 the
                 Star
                 numbred
                 upon
                 the
                 same
                 line
                 ,
                 observing
                 which
                 way
                 you
                 turned
                 the
                 point
                 of
                 your
                 Compasses
                 ,
                 viz.
                 toward
                 the
                 head
                 or
                 end
                 .
                 With
                 this
                 distance
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Stars
                 hour
                 before
                 found
                 counted
                 on
                 the
                 same
                 line
                 ,
                 and
                 turning
                 the
                 other
                 point
                 the
                 same
                 way
                 ,
                 as
                 you
                 did
                 for
                 the
                 right
                 ascensions
                 ,
                 it
                 gives
                 the
                 true
                 hour
                 of
                 the
                 night
                 .
                 Ex.
                 gr
                 .
                 The
                 22
                 of
                 March
                 I
                 find
                 the
                 altitude
                 of
                 the
                 Lions
                 heart
                 45
                 deg
                 .
                 his
                 declination
                 13
                 d.
                 40
                 min.
                 then
                 by
                 Probl.
                 5.
                 
                 I
                 find
                 his
                 hour
                 from
                 the
                 last
                 Meridian
                 10
                 houres
                 ,
                 5
                 min.
                 
                 The
                 right
                 ascension
                 of
                 the
                 Sun
                 is
                 46
                 m.
                 of
                 time
                 ,
                 or
                 11
                 d.
                 30
                 m.
                 of
                 the
                 Circle
                 ,
                 the
                 right
                 ascension
                 of
                 the
                 Lions
                 heart
                 ,
                 is
                 9
                 hour
                 51
                 m.
                 fere
                 ,
                 of
                 time
                 ,
                 or
                 147
                 deg
                 .
                 43
                 m.
                 of
                 a
                 circle
                 ;
                 then
                 by
                 a
                 line
                 of
                 twice
                 12
                 ,
                 you
                 may
                 find
                 the
                 true
                 hour
                 of
                 the
                 night
                 ,
                 7
                 hour
                 13
                 min.
                 
              
            
             
               
               
                 PROBL.
                 15.
                 
              
               
                 
                   The
                   right
                   ascension
                   and
                   declination
                   of
                   any
                   Star
                   ,
                   with
                   the
                   right
                   ascension
                   of
                   the
                   Sun
                   and
                   time
                   of
                   night
                   given
                   ,
                   to
                   find
                   the
                   altitude
                   of
                   that
                   Star
                   with
                   his
                   Azimuth
                   from
                   South
                   ,
                   and
                   by
                   consequent
                   to
                   find
                   the
                   Star
                   ,
                   although
                   before
                   you
                   knew
                   it
                   not
                   .
                
              
               
                 This
                 is
                 no
                 more
                 than
                 unravelling
                 the
                 last
                 Problem
                 .
                 1
                 Therefore
                 upon
                 the
                 line
                 of
                 twice
                 12
                 or
                 24
                 ,
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 right
                 ascension
                 of
                 the
                 Star
                 ,
                 extending
                 the
                 other
                 to
                 the
                 right
                 ascension
                 of
                 the
                 Sun
                 upon
                 the
                 same
                 line
                 ,
                 that
                 distance
                 laid
                 the
                 same
                 way
                 upon
                 the
                 same
                 line
                 ,
                 from
                 the
                 hour
                 of
                 the
                 night
                 ,
                 gives
                 the
                 Stars
                 hour
                 from
                 the
                 last
                 Meridian
                 he
                 was
                 upon
                 .
                 This
                 found
                 by
                 Probl.
                 5.
                 find
                 his
                 altitude
                 as
                 you
                 did
                 for
                 the
                 Sun.
                 Lastly
                 ,
                 having
                 now
                 his
                 declination
                 and
                 altitude
                 by
                 Probl.
                 8.
                 or
                 10.
                 according
                 to
                 his
                 declination
                 ,
                 you
                 will
                 soon
                 get
                 his
                 Azimuth
                 from
                 South
                 .
                 This
                 needs
                 not
                 an
                 example
                 .
              
               
                 By
                 help
                 of
                 this
                 Problem
                 the
                 Instrument
                 might
                 be
                 so
                 contrived
                 ,
                 as
                 to
                 be
                 one
                 of
                 the
                 best
                 Tutors
                 for
                 knowing
                 of
                 the
                 Stars
                 .
              
            
             
               
               
                 PROBL.
                 16.
                 
              
               
                 
                   The
                   altitude
                   and
                   Azimuth
                   of
                   any
                   Star
                   given
                   to
                   find
                   his
                   declination
                   .
                
              
               
                 Lay
                 the
                 thread
                 to
                 the
                 altitude
                 counted
                 on
                 the
                 limb
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 toward
                 the
                 end
                 ,
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Azimuth
                 ,
                 take
                 the
                 nearest
                 distance
                 to
                 the
                 thread
                 ;
                 this
                 applyed
                 to
                 the
                 Scale
                 of
                 altitudes
                 gives
                 the
                 declination
                 .
                 If
                 the
                 Azimuth
                 given
                 be
                 on
                 that
                 side
                 the
                 thread
                 toward
                 the
                 end
                 ,
                 the
                 declination
                 is
                 South
                 ;
                 when
                 on
                 that
                 side
                 toward
                 the
                 head
                 ,
                 its
                 North.
                 
              
            
             
               
                 PROBL.
                 17.
                 
              
               
                 
                   The
                   altitude
                   and
                   declination
                   of
                   any
                   Star
                   ,
                   with
                   the
                   right
                   ascension
                   of
                   the
                   Sun
                   ,
                   and
                   hour
                   of
                   night
                   given
                   to
                   find
                   the
                   Stars
                   right
                   ascension
                   .
                
              
               
                 By
                 Probl.
                 5.
                 or
                 14.
                 find
                 the
                 Stars
                 hour
                 from
                 the
                 Meridian
                 .
                 Then
                 on
                 the
                 line
                 twice
                 12
                 ,
                 or
                 24
                 ,
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Stars
                 hour
                 (
                 thus
                 found
                 )
                 and
                 extend
                 the
                 other
                 to
                 the
                 hour
                 of
                 the
                 night
                 .
                 Upon
                 the
                 same
                 line
                 with
                 this
                 distance
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 right
                 ascension
                 of
                 the
                 
                 Sun
                 ,
                 and
                 turning
                 the
                 other
                 point
                 the
                 same
                 way
                 ,
                 as
                 you
                 did
                 for
                 the
                 hour
                 ,
                 it
                 gives
                 the
                 Stars
                 right
                 ascension
                 .
              
            
             
               
                 PROBL.
                 18.
                 
              
               
                 
                   The
                   Meridian
                   altitude
                   given
                   to
                   find
                   the
                   time
                   of
                   Sunrise
                   and
                   Sunset
                   .
                
              
               
                 Take
                 the
                 Meridian
                 altitude
                 from
                 your
                 particular
                 Scale
                 ,
                 and
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 upon
                 the
                 point
                 12
                 on
                 the
                 line
                 of
                 hours
                 (
                 that
                 is
                 the
                 pin
                 at
                 the
                 end
                 )
                 lay
                 the
                 thread
                 to
                 the
                 other
                 point
                 ,
                 and
                 on
                 the
                 line
                 of
                 hours
                 the
                 thread
                 gives
                 the
                 time
                 required
                 .
              
            
             
               
                 PROBL.
                 19.
                 
              
               
                 
                   To
                   find
                   any
                   latitude
                   your
                   particular
                   Scale
                   is
                   made
                   for
                   .
                
              
               
                 Take
                 the
                 distance
                 from
                 90
                 ,
                 on
                 the
                 line
                 of
                 Azimuth
                 unto
                 the
                 pin
                 at
                 the
                 end
                 of
                 that
                 line
                 ,
                 or
                 the
                 point
                 12
                 :
                 this
                 applyed
                 to
                 the
                 particular
                 Scale
                 ,
                 gives
                 the
                 complement
                 of
                 that
                 latitude
                 the
                 Instrument
                 was
                 made
                 for
                 .
              
            
             
               
               
                 PROBL.
                 20.
                 
              
               
                 
                   To
                   find
                   the
                   angles
                   of
                   the
                   substile
                   ,
                   stile
                   ,
                   inclination
                   of
                   Meridians
                   ,
                   and
                   six
                   and
                   twelve
                   ,
                   for
                   exact
                   declining
                   plains
                   ,
                   in
                   that
                   latitude
                   your
                   Scale
                   of
                   altitudes
                   is
                   made
                   for
                   .
                
              
               
                 
                   Sect.
                   1.
                   
                
                 
                   
                     To
                     find
                     the
                     distance
                     of
                     the
                     substile
                     from
                     12
                     ,
                     or
                     the
                     plains
                     perpendicular
                     .
                  
                
                 
                   Lay
                   the
                   thread
                   to
                   the
                   complement
                   of
                   declination
                   counted
                   on
                   the
                   line
                   of
                   Azimuths
                   ,
                   and
                   on
                   the
                   limb
                   it
                   gives
                   the
                   substile
                   counting
                   from
                   60
                   /
                   0
                   on
                   the
                   moveable
                   piece
                   .
                
              
               
                 
                   Sect.
                   2.
                   
                
                 
                   
                     To
                     find
                     the
                     angle
                     of
                     the
                     Stile
                     's
                     height
                     .
                  
                
                 
                   On
                   the
                   line
                   of
                   Azimuths
                   take
                   the
                   distance
                   from
                   the
                   Plains
                   declination
                   to
                   90.
                   
                   This
                   applyed
                   to
                   the
                   Scale
                   of
                   altitudes
                   gives
                   the
                   angle
                   of
                   the
                   stile
                   .
                
              
               
                 
                 
                   Sect.
                   3.
                   
                
                 
                   
                     The
                     angle
                     of
                     the
                     Substile
                     given
                     to
                     find
                     the
                     inclination
                     of
                     Meridians
                     .
                  
                
                 
                   Take
                   the
                   angle
                   of
                   the
                   substile
                   from
                   the
                   Scale
                   of
                   altitudes
                   ,
                   and
                   applying
                   it
                   from
                   90
                   on
                   the
                   Azimuth
                   line
                   toward
                   the
                   end
                   ;
                   the
                   figures
                   shew
                   the
                   complement
                   of
                   inclination
                   of
                   Meridians
                   .
                
              
               
                 
                   Sect.
                   4.
                   
                
                 
                   
                     To
                     find
                     the
                     angle
                     betwixt
                     6
                     and
                     12.
                     
                  
                
                 
                   Take
                   the
                   declination
                   from
                   the
                   Scale
                   of
                   altitudes
                   ,
                   and
                   setting
                   one
                   point
                   of
                   your
                   Compasses
                   in
                   90
                   on
                   the
                   line
                   of
                   Azimuths
                   ,
                   lay
                   the
                   thread
                   to
                   the
                   other
                   point
                   and
                   on
                   the
                   limb
                   it
                   gives
                   the
                   complement
                   of
                   the
                   angle
                   sought
                   ,
                   numbring
                   from
                   60
                   /
                   0
                   on
                   the
                   moveable
                   piece
                   toward
                   the
                   end
                   .
                
                 
                   This
                   last
                   rule
                   is
                   not
                   exact
                   ,
                   nor
                   is
                   it
                   here
                   worth
                   the
                   labour
                   to
                   rectifie
                   it
                   by
                   another
                   sine
                   added
                   ;
                   sith
                   you
                   have
                   an
                   exact
                   proportion
                   for
                   the
                   Problem
                   in
                   the
                   Treatise
                   of
                   Dialling
                   Chap.
                   2
                   .
                   Sect.
                   5.
                   
                   Paragr
                   .
                   4.
                   
                
              
            
          
           
             
             
               CHAP.
               III.
               
            
             
               
                 Some
                 uses
                 of
                 the
                 Line
                 of
                 natural
                 signs
                 on
                 the
                 Quadrantal
                 side
                 of
                 the
                 fixed
                 piece
                 .
              
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   How
                   to
                   adde
                   one
                   sign
                   to
                   another
                   on
                   the
                   Line
                   of
                   Natural
                   Sines
                   .
                
              
               
                 TO
                 adde
                 one
                 sine
                 to
                 another
                 ,
                 is
                 to
                 augment
                 the
                 line
                 of
                 one
                 sine
                 by
                 the
                 line
                 of
                 the
                 other
                 sine
                 to
                 be
                 added
                 to
                 it
                 .
                 Ex.
                 gr
                 .
                 To
                 adde
                 the
                 sine
                 15
                 to
                 the
                 sine
                 20
                 ,
                 I
                 take
                 the
                 distance
                 from
                 the
                 beginning
                 of
                 the
                 line
                 of
                 sines
                 unto
                 15
                 ,
                 and
                 setting
                 one
                 point
                 of
                 the
                 Compasses
                 in
                 20
                 ,
                 upon
                 the
                 same
                 line
                 ,
                 turn
                 the
                 other
                 toward
                 90
                 ,
                 which
                 I
                 finde
                 touch
                 in
                 37.
                 
                 So
                 that
                 in
                 this
                 case
                 (
                 for
                 we
                 regard
                 not
                 the
                 Arithmetical
                 ,
                 but
                 proportional
                 aggregate
                 )
                 15
                 added
                 to
                 20
                 ,
                 upon
                 the
                 line
                 of
                 natural
                 sines
                 ,
                 is
                 the
                 sine
                 37
                 upon
                 that
                 line
                 ,
                 and
                 from
                 the
                 beginning
                 of
                 the
                 line
                 to
                 37
                 is
                 the
                 distance
                 I
                 am
                 to
                 take
                 for
                 the
                 summe
                 of
                 20
                 and
                 15
                 sines
                 .
              
            
             
               
               
                 PROBL.
                 2.
                 
              
               
                 
                   How
                   to
                   substract
                   one
                   sine
                   from
                   another
                   upon
                   the
                   line
                   of
                   natural
                   sines
                   .
                
              
               
                 The
                 substracting
                 of
                 one
                 sine
                 from
                 another
                 ,
                 is
                 no
                 more
                 than
                 taking
                 the
                 distance
                 from
                 the
                 lesser
                 to
                 the
                 greater
                 on
                 the
                 line
                 of
                 sines
                 ,
                 and
                 that
                 distance
                 applyed
                 to
                 the
                 line
                 from
                 the
                 beginning
                 ,
                 gives
                 the
                 residue
                 or
                 remainer
                 .
                 Ex.
                 gr
                 .
                 To
                 substract
                 20
                 from
                 37
                 I
                 take
                 the
                 distance
                 from
                 20
                 to
                 37
                 that
                 applyed
                 to
                 the
                 line
                 from
                 the
                 beginning
                 gives
                 15
                 for
                 the
                 sine
                 remaining
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   To
                   work
                   proportions
                   in
                   sines
                   alone
                   .
                
              
               
                 Here
                 are
                 four
                 Cases
                 that
                 include
                 all
                 proportions
                 in
                 sines
                 alone
                 .
              
               
                 
                   CASE
                   1.
                   
                
                 
                   
                     When
                     the
                     first
                     term
                     is
                     Radius
                     ,
                     or
                     the
                     Sine
                     90.
                     
                  
                
                 
                   Lay
                   the
                   thread
                   to
                   the
                   second
                   term
                   counted
                   on
                   the
                   degrees
                   upon
                   the
                   movaeble
                   piece
                   from
                   the
                   head
                   toward
                   the
                   end
                   ,
                   then
                   numbring
                   
                   the
                   third
                   on
                   the
                   line
                   of
                   sines
                   ,
                   take
                   the
                   nearest
                   distance
                   from
                   thence
                   to
                   the
                   thread
                   ,
                   and
                   that
                   applyed
                   to
                   the
                   Scale
                   from
                   the
                   beginning
                   gives
                   the
                   fourth
                   term
                   .
                   Ex.
                   gr
                   .
                
                 
                   As
                   the
                   Radius
                   90
                   is
                   to
                   the
                   sine
                   20
                   ,
                   so
                   is
                   the
                   sine
                   30
                   to
                   the
                   sine
                   10.
                   
                
              
               
                 
                   CASE
                   2.
                   
                
                 
                   
                     When
                     the
                     Radius
                     is
                     the
                     third
                     term
                     .
                  
                
                 
                   Take
                   the
                   sine
                   of
                   the
                   second
                   term
                   in
                   your
                   Compasses
                   ,
                   and
                   enter
                   it
                   in
                   the
                   first
                   term
                   upon
                   the
                   line
                   of
                   sines
                   ,
                   and
                   laying
                   the
                   thread
                   to
                   the
                   nearest
                   distance
                   ,
                   on
                   the
                   limb
                   the
                   thread
                   gives
                   the
                   fourth
                   term
                   .
                   Ex.
                   gr
                   .
                
                 
                   As
                   the
                   sine
                   30
                   is
                   to
                   the
                   sine
                   20
                   ,
                   so
                   is
                   the
                   Radius
                   to
                   the
                   sine
                   43.
                   30.
                   min.
                   
                
              
               
                 
                   CASE
                   3.
                   
                
                 
                   
                     When
                     the
                     Radius
                     is
                     the
                     second
                     term
                     .
                  
                
                 
                   Provided
                   the
                   third
                   term
                   be
                   not
                   greater
                   than
                   the
                   first
                   ,
                   transpose
                   the
                   terms
                   .
                   The
                   method
                   of
                   transposition
                   in
                   this
                   case
                   is
                   ,
                   as
                   the
                   first
                   term
                   is
                   to
                   the
                   third
                   ,
                   so
                   is
                   the
                   second
                   to
                   the
                   fourth
                   ,
                   and
                   then
                   the
                   work
                   will
                   be
                   the
                   same
                   as
                   in
                   the
                   second
                   case
                   .
                   Ex.
                   gr
                   .
                
                 
                   As
                   the
                   sine
                   30
                   is
                   to
                   the
                   radius
                   or
                   sine
                   90
                   ,
                   so
                   is
                   the
                   sine
                   20
                   to
                   what
                   sine
                   ;
                   which
                   transposed
                   is
                
                 
                 
                   As
                   the
                   sine
                   30
                   is
                   to
                   the
                   sine
                   20
                   ,
                   so
                   is
                   the
                   radius
                   to
                   a
                   fourth
                   sine
                   ,
                   which
                   will
                   be
                   found
                   43
                   ,
                   30
                   min.
                   as
                   before
                   .
                
              
               
                 
                   CASE
                   4.
                   
                
                 
                   
                     When
                     the
                     Radius
                     is
                     none
                     of
                     the
                     three
                     terms
                     given
                     .
                  
                
                 
                   In
                   this
                   case
                   when
                   both
                   the
                   middle
                   terms
                   are
                   less
                   than
                   the
                   first
                   ,
                   enter
                   the
                   sine
                   of
                   the
                   second
                   term
                   in
                   the
                   first
                   ,
                   and
                   laying
                   the
                   thread
                   to
                   the
                   nearest
                   distance
                   ,
                   take
                   the
                   nearest
                   extent
                   from
                   the
                   third
                   to
                   the
                   thread
                   :
                   this
                   distance
                   applyed
                   to
                   the
                   scale
                   from
                   the
                   beginning
                   gives
                   the
                   fourth
                   .
                   Ex.
                   gr
                   .
                
                 
                   As
                   the
                   sine
                   20
                   to
                   the
                   sine
                   10
                   ,
                   so
                   is
                   the
                   sine
                   30
                   to
                   the
                   sine
                   15.
                   
                
                 
                   When
                   only
                   the
                   second
                   term
                   is
                   greater
                   than
                   the
                   first
                   ,
                   transpose
                   the
                   terms
                   and
                   work
                   as
                   before
                   .
                
                 
                   But
                   when
                   both
                   the
                   middle
                   tearms
                   be
                   greater
                   than
                   the
                   first
                   ,
                   this
                   proportion
                   will
                   not
                   be
                   performed
                   by
                   this
                   line
                   without
                   a
                   paralel
                   entrance
                   or
                   double
                   radius
                   ;
                   which
                   inconveniency
                   shall
                   be
                   remedied
                   in
                   its
                   proper
                   place
                   ,
                   when
                   we
                   shew
                   how
                   to
                   work
                   proportions
                   by
                   the
                   lines
                   of
                   natural
                   sines
                   on
                   the
                   proportional
                   or
                   sector
                   side
                   .
                
                 
                   These
                   four
                   cases
                   comprizing
                   the
                   method
                   of
                   working
                   all
                   proportions
                   by
                   natural
                   sines
                   
                   alone
                   ,
                   I
                   shall
                   propose
                   some
                   examples
                   for
                   the
                   exercise
                   of
                   young
                   practitioners
                   ,
                   and
                   therewith
                   conclude
                   this
                   Chapter
                   .
                
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   To
                   finde
                   the
                   Suns
                   amplitude
                   in
                   any
                   Latitude
                   .
                
              
               
                 As
                 the
                 cosine
                 of
                 the
                 Latitude
                 is
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 amplitude
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   finde
                   the
                   hour
                   in
                   any
                   Latitude
                   in
                   Northern
                   Declination
                   .
                
              
               
                 Proport
                 .
                 1.
                 
                 As
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 latitude
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 altitude
                 at
                 six
                 .
                 By
                 Probl.
                 2.
                 substract
                 this
                 altitude
                 at
                 six
                 from
                 the
                 present
                 altitude
                 ,
                 and
                 take
                 the
                 difference
                 .
                 Then
              
               
                 Proport
                 .
                 2.
                 
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 is
                 to
                 that
                 difference
                 ,
                 so
                 is
                 the
                 radius
                 to
                 a
                 fourth
                 sine
                 .
                 Again
              
               
                 Proport
                 .
                 3.
                 
                 As
                 the
                 cosine
                 of
                 the
                 declination
                 to
                 that
                 fourth
                 sine
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 hour
                 from
                 six
                 .
              
            
             
               
               
                 PROBL.
                 6.
                 
              
               
                 
                   To
                   finde
                   the
                   hour
                   in
                   any
                   Latitude
                   when
                   the
                   Sun
                   is
                   in
                   the
                   Equinoctial
                   .
                
              
               
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 is
                 to
                 the
                 sine
                 of
                 altitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 hour
                 from
                 six
                 .
              
            
             
               
                 PROBL.
                 7.
                 
              
               
                 
                   To
                   finde
                   the
                   hour
                   in
                   any
                   latitude
                   in
                   Southern
                   Declination
                   .
                
              
               
                 Proport
                 .
                 1.
                 
                 As
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 latitude
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 depression
                 at
                 six
                 ;
                 adde
                 the
                 sine
                 of
                 depression
                 to
                 the
                 present
                 altitude
                 by
                 Probl.
                 1.
                 Then
              
               
                 Proport
                 .
                 2.
                 
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 is
                 to
                 that
                 summe
                 ,
                 so
                 is
                 the
                 radius
                 to
                 a
                 fourth
                 sine
                 .
                 Again
                 ,
              
               
                 Proport
                 .
                 3.
                 
                 As
                 the
                 cosine
                 of
                 declination
                 is
                 to
                 the
                 fourth
                 sine
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 hour
                 from
                 six
                 .
              
            
             
               
               
                 PROBL.
                 8.
                 
              
               
                 
                   To
                   finde
                   the
                   Suns
                   Azimuth
                   in
                   any
                   latitude
                   in
                   Northern
                   Declination
                   .
                
              
               
                 Proport
                 .
                 1.
                 
                 As
                 the
                 sine
                 of
                 the
                 latitude
                 to
                 the
                 sine
                 of
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 altitude
                 at
                 East
                 ,
                 or
                 West
                 .
                 By
                 Probl.
                 2.
                 substract
                 this
                 from
                 the
                 present
                 altitude
                 ,
                 then
                 ,
              
               
                 Proport
                 .
                 2.
                 
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 is
                 to
                 that
                 residue
                 ,
                 so
                 is
                 the
                 radius
                 to
                 a
                 fourth
                 sine
                 .
                 Again
                 ,
              
               
                 Proport
                 .
                 3.
                 
                 As
                 the
                 cosine
                 of
                 the
                 altitude
                 is
                 to
                 that
                 fourth
                 sine
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 Azimuth
                 from
                 East
                 or
                 West
                 .
              
            
             
               
                 PROBL.
                 9.
                 
              
               
                 
                   To
                   finde
                   the
                   Azimuth
                   for
                   any
                   latitude
                   when
                   the
                   Sun
                   is
                   in
                   the
                   Equator
                   .
                
              
               
                 Proport
                 .
                 1.
                 
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 to
                 the
                 sine
                 of
                 altitude
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 latitude
                 to
                 a
                 fourth
                 sine
                 .
              
               
                 Proport
                 .
                 2.
                 
                 As
                 the
                 cosine
                 of
                 altitude
                 to
                 that
                 fourth
                 sine
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 Azimuth
                 from
                 East
                 ,
                 or
                 West
                 .
              
            
             
               
               
                 PROBL.
                 10.
                 
              
               
                 
                   To
                   finde
                   the
                   Azimuth
                   for
                   any
                   latitude
                   in
                   Southern
                   Declination
                   .
                
              
               
                 Proport
                 .
                 1.
                 
                 As
                 the
                 cosine
                 of
                 the
                 latitude
                 to
                 the
                 sine
                 of
                 altitude
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 latitude
                 to
                 a
                 fourth
                 .
                 Having
                 by
                 Probl.
                 4.
                 found
                 the
                 Suns
                 amplitude
                 ,
                 adde
                 it
                 to
                 this
                 fourth
                 sine
                 by
                 Probl.
                 1.
                 and
                 say
              
               
                 As
                 the
                 cosine
                 of
                 the
                 altitude
                 is
                 to
                 the
                 sum
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 Azimuth
                 from
                 East
                 or
                 West
                 .
                 The
                 terms
                 mentioned
                 in
                 the
                 5th
                 .
                 7th
                 .
                 8th
                 .
                 10th
                 .
                 Problems
                 are
                 appropriated
                 unto
                 us
                 that
                 live
                 on
                 the
                 North
                 side
                 the
                 Equator
                 .
                 In
                 case
                 they
                 be
                 applyed
                 to
                 such
                 latitudes
                 as
                 lie
                 on
                 the
                 South
                 side
                 the
                 Equator
                 .
                 Then
                 what
                 is
                 now
                 called
                 Northern
                 declination
                 ,
                 name
                 Southern
                 ,
                 and
                 what
                 is
                 here
                 styled
                 Southern
                 declination
                 ,
                 term
                 Northern
                 ,
                 and
                 all
                 the
                 proportion
                 with
                 the
                 operation
                 is
                 the
                 same
                 .
              
               
                 These
                 proportions
                 to
                 finde
                 the
                 hour
                 and
                 Azimuth
                 ,
                 may
                 be
                 all
                 readily
                 wrought
                 by
                 the
                 lines
                 of
                 artificial
                 sines
                 ,
                 only
                 the
                 addition
                 and
                 substraction
                 must
                 alwayes
                 be
                 wrought
                 upon
                 the
                 line
                 of
                 natural
                 sines
                 .
              
            
          
           
             
             
               CHAP.
               IV.
               
            
             
               
                 Some
                 uses
                 of
                 the
                 Lines
                 on
                 the
                 proportional
                 side
                 of
                 the
                 Instrument
                 ,
                 viz.
                 the
                 Lines
                 of
                 natural
                 Sines
                 ,
                 Tangents
                 ,
                 and
                 Secants
                 .
              
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   To
                   lay
                   down
                   any
                   Sine
                   ,
                   Tangent
                   ,
                   or
                   Secant
                   to
                   a
                   Radius
                   given
                   .
                   See
                   Fig.
                   1.
                   
                
              
               
                 IF
                 you
                 be
                 to
                 lay
                 down
                 a
                 Sine
                 ,
                 enter
                 the
                 Radius
                 given
                 in
                 90
                 ,
                 and
                 90
                 upon
                 the
                 lines
                 of
                 Sines
                 ,
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 set
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 the
                 Sine
                 required
                 upon
                 one
                 line
                 ,
                 and
                 extend
                 the
                 other
                 point
                 to
                 the
                 same
                 Sine
                 upon
                 the
                 other
                 Line
                 :
                 This
                 distance
                 is
                 the
                 length
                 of
                 the
                 Sine
                 required
                 to
                 the
                 given
                 Radius
                 .
                 Ex.
                 gr
                 .
                 Suppose
                 A.
                 B.
                 the
                 Radius
                 given
                 ,
                 and
                 I
                 require
                 the
                 Sine
                 40.
                 proportional
                 to
                 that
                 Radius
                 .
                 Enter
                 A.
                 B.
                 in
                 90
                 ,
                 and
                 90
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 I
                 take
                 the
                 distance
                 ,
                 twixt
                 40
                 on
                 one
                 side
                 ,
                 to
                 40
                 on
                 the
                 other
                 ,
                 that
                 is
                 ,
                 C.
                 D.
                 the
                 Sine
                 required
                 .
              
               
                 The
                 work
                 is
                 the
                 same
                 ,
                 to
                 lay
                 down
                 a
                 Tangent
                 to
                 any
                 Radius
                 given
                 ,
                 provided
                 you
                 enter
                 
                 the
                 given
                 Radius
                 in
                 45
                 ,
                 and
                 45
                 ,
                 on
                 the
                 line
                 of
                 Tangents
                 .
                 Only
                 observe
                 if
                 the
                 Tangent
                 required
                 be
                 less
                 than
                 45.
                 you
                 must
                 enter
                 the
                 Radius
                 in
                 45.
                 and
                 45
                 next
                 the
                 end
                 of
                 the
                 Rule
                 .
                 But
                 when
                 the
                 Tangent
                 required
                 exceeds
                 45.
                 enter
                 the
                 Radius
                 given
                 in
                 45
                 ,
                 and
                 45
                 'twixt
                 the
                 center
                 and
                 end
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 Gage
                 ,
                 take
                 out
                 the
                 Tangent
                 required
                 .
                 This
                 is
                 so
                 plain
                 ,
                 there
                 needs
                 no
                 example
                 .
              
               
                 To
                 lay
                 down
                 a
                 Secant
                 to
                 any
                 Radius
                 given
                 ,
                 is
                 no
                 more
                 than
                 to
                 enter
                 the
                 Radius
                 in
                 the
                 two
                 pins
                 at
                 the
                 beginning
                 of
                 the
                 line
                 of
                 Secants
                 ;
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 Gage
                 ,
                 take
                 the
                 distance
                 from
                 the
                 number
                 of
                 the
                 Secant
                 required
                 on
                 one
                 side
                 ,
                 to
                 the
                 same
                 number
                 on
                 the
                 other
                 side
                 ,
                 and
                 that
                 is
                 the
                 Secant
                 sought
                 at
                 the
                 Radius
                 given
                 .
              
               
                 The
                 use
                 of
                 this
                 Problem
                 will
                 be
                 sufficiently
                 seen
                 in
                 delineating
                 Dyals
                 ,
                 and
                 projecting
                 the
                 Sphere
                 .
              
            
             
               
                 PROBL.
                 2.
                 
              
               
                 
                   To
                   lay
                   down
                   any
                   Angle
                   required
                   by
                   the
                   Lines
                   of
                   Sines
                   ,
                   Tangents
                   ,
                   and
                   Secants
                   .
                   See
                   Fig.
                   2.
                   
                
              
               
                 There
                 are
                 two
                 wayes
                 of
                 protracting
                 an
                 Angle
                 by
                 the
                 Line
                 of
                 Sines
                 ,
                 First
                 if
                 you
                 use
                 the
                 Sines
                 
                 in
                 manner
                 of
                 Chords
                 .
                 Then
                 having
                 drawn
                 the
                 line
                 A
                 B
                 at
                 any
                 distance
                 of
                 your
                 Compass
                 ,
                 set
                 one
                 point
                 in
                 B
                 ,
                 and
                 draw
                 a
                 mark
                 to
                 intersect
                 the
                 Line
                 B
                 A
                 ,
                 as
                 E
                 F.
                 Enter
                 this
                 distance
                 B
                 F
                 in
                 30
                 ,
                 and
                 30
                 upon
                 the
                 Lines
                 of
                 Sines
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 Gage
                 ,
                 take
                 out
                 the
                 Sine
                 of
                 half
                 the
                 Angle
                 required
                 ,
                 and
                 setting
                 one
                 point
                 where
                 F
                 intersects
                 B
                 A
                 ,
                 turn
                 the
                 other
                 toward
                 E
                 ,
                 and
                 make
                 the
                 mark
                 E
                 ,
                 with
                 a
                 ruler
                 draw
                 B
                 E
                 and
                 the
                 Angle
                 E
                 B
                 F
                 is
                 the
                 Angle
                 required
                 ,
                 which
                 here
                 is
                 40.
                 d.
                 
              
               
                 A
                 second
                 method
                 by
                 the
                 lines
                 of
                 Sines
                 is
                 thus
                 ,
                 Enter
                 B
                 A
                 Radius
                 in
                 the
                 Lines
                 of
                 Sines
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 Gage
                 ,
                 take
                 out
                 the
                 Sine
                 of
                 your
                 Angle
                 required
                 with
                 that
                 distance
                 ,
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 A
                 ,
                 sweep
                 the
                 ark
                 D
                 ,
                 a
                 line
                 drawn
                 from
                 B
                 by
                 the
                 connexity
                 of
                 the
                 Ark
                 D
                 ,
                 makes
                 the
                 Angle
                 A
                 B
                 C
                 40
                 d.
                 as
                 before
                 .
              
               
                 To
                 protract
                 an
                 Angle
                 by
                 the
                 Lines
                 of
                 Tangents
                 is
                 easily
                 done
                 ,
                 draw
                 B
                 A
                 the
                 Radius
                 upon
                 A
                 ,
                 erect
                 a
                 perpendicular
                 ,
                 A
                 C
                 ,
                 enter
                 B
                 A
                 in
                 45
                 ,
                 and
                 45
                 on
                 the
                 Lines
                 of
                 Tangents
                 ,
                 and
                 taking
                 out
                 the
                 Tangent
                 required
                 (
                 as
                 here
                 40
                 )
                 set
                 it
                 from
                 A
                 to
                 C.
                 Lastly
                 ,
                 draw
                 B
                 C
                 ,
                 and
                 the
                 Angle
                 C
                 B
                 A
                 is
                 40
                 d.
                 as
                 before
                 .
              
               
                 In
                 case
                 you
                 would
                 protract
                 an
                 Angle
                 by
                 the
                 Lines
                 of
                 Secants
                 .
                 Draw
                 B
                 A
                 ,
                 and
                 upon
                 A
                 erect
                 the
                 perpendicular
                 A
                 C
                 ,
                 enter
                 A
                 B
                 in
                 
                 the
                 beginning
                 of
                 the
                 Lines
                 of
                 Secants
                 ,
                 and
                 take
                 out
                 the
                 Secant
                 of
                 the
                 Angle
                 ,
                 with
                 that
                 distance
                 ,
                 setting
                 one
                 point
                 of
                 your
                 Compasses
                 in
                 B
                 ,
                 with
                 the
                 other
                 cross
                 the
                 perpendicular
                 A
                 C
                 ,
                 as
                 in
                 C.
                 This
                 done
                 ,
                 lay
                 a
                 Ruler
                 to
                 B
                 ,
                 and
                 the
                 point
                 of
                 intersection
                 ,
                 and
                 draw
                 the
                 Line
                 B
                 C.
                 So
                 have
                 you
                 again
                 the
                 Angle
                 C
                 B
                 A.
                 40.
                 d.
                 by
                 another
                 projection
                 .
                 These
                 varieties
                 are
                 here
                 inserted
                 only
                 to
                 satisfie
                 a
                 friend
                 ,
                 and
                 recreate
                 the
                 young
                 practitioner
                 in
                 trying
                 the
                 truth
                 of
                 his
                 projection
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   To
                   work
                   proportions
                   in
                   Sines
                   alone
                   ,
                   by
                   the
                   Lines
                   of
                   natural
                   Sines
                   on
                   the
                   proportional
                   side
                   of
                   the
                   Instrument
                   .
                
              
               
                 The
                 general
                 rule
                 is
                 this
                 .
                 Account
                 the
                 first
                 term
                 upon
                 the
                 Lines
                 of
                 Sines
                 from
                 the
                 Center
                 ,
                 and
                 enter
                 the
                 second
                 term
                 in
                 the
                 first
                 so
                 accounted
                 ,
                 keeping
                 the
                 Sector
                 at
                 that
                 Gage
                 ,
                 account
                 the
                 third
                 term
                 on
                 both
                 lines
                 from
                 the
                 Center
                 ,
                 and
                 taking
                 the
                 distance
                 from
                 the
                 third
                 term
                 on
                 one
                 line
                 to
                 the
                 third
                 term
                 on
                 the
                 other
                 line
                 ,
                 measure
                 it
                 upon
                 the
                 line
                 of
                 Sines
                 from
                 the
                 beginning
                 ,
                 and
                 you
                 have
                 the
                 fourth
                 term
                 .
                 Ex.
                 gr
                 .
              
               
               
                 As
                 the
                 Radius
                 is
                 to
                 the
                 Sine
                 30
                 ,
                 so
                 is
                 the
                 Sine
                 40
                 to
                 the
                 Sine
                 18.
                 45.
                 
              
               
                 There
                 is
                 but
                 one
                 exception
                 in
                 this
                 Rule
                 ,
                 and
                 that
                 is
                 when
                 the
                 second
                 term
                 is
                 greater
                 than
                 the
                 first
                 ;
                 yet
                 the
                 third
                 lesser
                 than
                 the
                 first
                 ,
                 and
                 in
                 this
                 case
                 transpose
                 the
                 terms
                 ,
                 by
                 Chap
                 3.
                 
                 Probl.
                 3.
                 
                 Case
                 3.
                 
              
               
                 But
                 when
                 the
                 second
                 term
                 is
                 not
                 twice
                 the
                 length
                 of
                 the
                 first
                 ,
                 it
                 may
                 be
                 wrought
                 by
                 the
                 general
                 Rule
                 without
                 any
                 transposition
                 of
                 terms
                 .
                 Ex.
                 gr
                 .
              
               
                 As
                 the
                 Sine
                 30
                 is
                 to
                 the
                 Sine
                 50
                 ,
                 so
                 is
                 the
                 Sine
                 20
                 to
                 the
                 Sine
                 31.
                 30.
                 min.
                 
              
               
                 And
                 by
                 consequent
                 ,
                 when
                 the
                 third
                 term
                 is
                 greater
                 than
                 the
                 first
                 ,
                 provided
                 it
                 be
                 not
                 upon
                 the
                 line
                 ,
                 double
                 the
                 length
                 thereof
                 ,
                 it
                 may
                 be
                 wrought
                 by
                 transposing
                 the
                 terms
                 ,
                 although
                 the
                 second
                 was
                 twice
                 the
                 length
                 of
                 the
                 first
                 .
                 Ex.
                 gr
                 .
              
               
                 As
                 the
                 Sine
                 20
                 is
                 to
                 the
                 Sine
                 60
                 ,
                 so
                 is
                 the
                 Sine
                 42
                 ,
                 to
                 what
                 Sine
                 ?
                 which
                 transposed
                 is
                 ,
              
               
                 As
                 the
                 Sine
                 20
                 is
                 to
                 the
                 Sine
                 42
                 ,
                 so
                 is
                 the
                 Sine
                 60
                 to
                 the
                 Sine
                 35.
                 30.
                 
              
               
                 This
                 case
                 will
                 remove
                 the
                 inconveniency
                 mentioned
                 ,
                 Chap.
                 3.
                 
                 Probl.
                 3.
                 
                 Case
                 4.
                 of
                 a
                 double
                 Radius
                 .
                 I
                 intended
                 there
                 to
                 have
                 adjoyned
                 the
                 method
                 of
                 working
                 proportions
                 by
                 natural
                 Tangents
                 alone
                 ,
                 and
                 by
                 natural
                 Sines
                 ,
                 and
                 Tangents
                 ,
                 conjunctly
                 :
                 But
                 considering
                 
                 the
                 multiplicity
                 of
                 proportions
                 when
                 the
                 Tangents
                 exceed
                 45.
                 
                 I
                 suppose
                 it
                 too
                 troublesome
                 for
                 beginners
                 ,
                 and
                 a
                 needless
                 variety
                 for
                 those
                 that
                 are
                 already
                 Mathematicians
                 .
                 Sith
                 ,
                 both
                 may
                 be
                 eased
                 by
                 the
                 artificial
                 Sines
                 and
                 Tangents
                 on
                 the
                 outward
                 ledge
                 ,
                 where
                 I
                 intend
                 to
                 treat
                 of
                 those
                 Cases
                 at
                 large
                 ,
                 and
                 shall
                 in
                 this
                 place
                 only
                 annex
                 some
                 proportions
                 in
                 Sines
                 alone
                 ,
                 for
                 the
                 exercise
                 of
                 young
                 beginners
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   By
                   the
                   Lines
                   of
                   Natural
                   Sines
                   to
                   lay
                   down
                   any
                   Tangent
                   ,
                   or
                   Secant
                   required
                   to
                   a
                   Radius
                   given
                   .
                
              
               
                 In
                 some
                 Cases
                 ,
                 especially
                 for
                 Dyalling
                 ,
                 your
                 Instrument
                 may
                 be
                 defective
                 of
                 a
                 Tangent
                 ,
                 or
                 Secant
                 for
                 your
                 purpose
                 ,
                 Ex.
                 gr
                 .
                 when
                 the
                 Tangent
                 exceeds
                 76
                 ,
                 or
                 the
                 Secant
                 is
                 more
                 than
                 60.
                 
                 In
                 these
                 extremities
                 use
                 the
                 following
                 Remedies
                 .
                 First
                 ,
                 for
                 a
                 Tangent
                 .
              
               
                 As
                 the
                 cosine
                 of
                 the
                 Ark
                 is
                 to
                 the
                 Radius
                 given
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 Ark
                 to
                 the
                 length
                 of
                 the
                 Tangent
                 required
                 .
                 Secondly
                 ,
                 for
                 a
                 Secant
                 .
              
               
                 As
                 the
                 cosine
                 of
                 the
                 Ark
                 is
                 to
                 the
                 Radius
                 given
                 ,
                 so
                 is
                 the
                 Sine
                 90
                 to
                 the
                 length
                 of
                 the
                 Secant
                 required
                 .
              
            
             
               
               
                 PROBL.
                 5.
                 
              
               
                 
                   The
                   distance
                   from
                   the
                   next
                   Equinoctial
                   Point
                   given
                   to
                   finde
                   the
                   Suns
                   declination
                   .
                
              
               
                 As
                 the
                 Radius
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 greatest
                 declination
                 ,
                 so
                 is
                 the
                 sine
                 of
                 his
                 distance
                 from
                 the
                 next
                 Equinoctial
                 Point
                 to
                 the
                 sine
                 of
                 his
                 present
                 declination
                 .
              
            
             
               
                 PROBL.
                 6.
                 
              
               
                 
                   The
                   declination
                   given
                   to
                   finde
                   the
                   Suns
                   Equinoctial
                   Distance
                   .
                
              
               
                 As
                 the
                 sine
                 of
                 the
                 greatest
                 declination
                 is
                 to
                 the
                 sine
                 of
                 the
                 present
                 declination
                 ,
                 so
                 is
                 the
                 Radius
                 to
                 the
                 sine
                 of
                 his
                 Equinoctial
                 Distance
                 .
              
            
             
               
                 PROBL.
                 7.
                 
              
               
                 
                   The
                   Altitude
                   ,
                   Declination
                   ,
                   and
                   Distance
                   of
                   the
                   Sun
                   from
                   the
                   Meridian
                   given
                   to
                   finde
                   his
                   Azimuth
                   .
                
              
               
                 As
                 the
                 cosine
                 of
                 the
                 altitude
                 ,
                 to
                 the
                 cosine
                 of
                 the
                 hour
                 from
                 the
                 Meridian
                 ,
                 so
                 is
                 
                 the
                 cosine
                 of
                 declination
                 to
                 the
                 sine
                 of
                 the
                 Azimuth
                 .
              
            
          
           
             
               CHAP.
               V.
               
            
             
               
                 Some
                 uses
                 of
                 the
                 Lines
                 of
                 the
                 Lines
                 ,
                 on
                 the
                 proportional
                 side
                 of
                 the
                 Instrument
                 .
              
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   To
                   divide
                   a
                   Line
                   given
                   into
                   any
                   Number
                   of
                   equal
                   parts
                   .
                   See
                   Fig.
                   3.
                   
                
              
               
                 SUppose
                 A
                 B
                 a
                 Line
                 given
                 to
                 be
                 divided
                 into
                 nine
                 equal
                 parts
                 .
                 Enter
                 A
                 B
                 in
                 9
                 ,
                 and
                 9
                 on
                 the
                 lines
                 of
                 lines
                 ,
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 take
                 the
                 distance
                 from
                 8
                 ,
                 on
                 one
                 side
                 ,
                 to
                 8
                 on
                 the
                 other
                 ,
                 and
                 apply
                 it
                 from
                 A
                 upon
                 the
                 line
                 A
                 B
                 ,
                 which
                 reacheth
                 to
                 C
                 ;
                 then
                 is
                 C
                 B
                 a
                 ninth
                 part
                 of
                 the
                 line
                 A
                 B.
                 By
                 this
                 means
                 you
                 may
                 divide
                 any
                 line
                 (
                 that
                 is
                 not
                 more
                 than
                 the
                 Instrument
                 in
                 length
                 )
                 into
                 as
                 many
                 parts
                 as
                 you
                 please
                 ,
                 viz.
                 10
                 ,
                 20
                 ,
                 30
                 ,
                 40
                 ,
                 50
                 ,
                 100
                 ,
                 500
                 ,
                 &c.
                 parts
                 according
                 to
                 your
                 reckoning
                 the
                 divisions
                 upon
                 the
                 lines
                 ,
                 Ex.
                 gr
                 .
                 The
                 line
                 is
                 actually
                 divided
                 into
                 200
                 parts
                 ,
                 viz.
                 first
                 into
                 10
                 ,
                 marked
                 with
                 Figures
                 ,
                 and
                 each
                 of
                 those
                 into
                 
                 twenty
                 parts
                 more
                 .
                 Again
                 ,
                 if
                 the
                 line
                 represents
                 a
                 1000
                 ,
                 then
                 every
                 figured
                 division
                 is
                 100
                 ,
                 the
                 second
                 or
                 shorter
                 division
                 is
                 10
                 ,
                 and
                 the
                 third
                 or
                 shortest
                 division
                 is
                 5.
                 
                 In
                 case
                 the
                 whole
                 line
                 was
                 2000
                 ,
                 then
                 every
                 figured
                 division
                 is
                 200
                 ,
                 every
                 smaller
                 or
                 second
                 division
                 is
                 20
                 ,
                 every
                 third
                 or
                 smallest
                 division
                 is
                 10
                 ,
                 &c.
                 
                 Suppose
                 I
                 have
                 any
                 line
                 given
                 ,
                 which
                 is
                 the
                 base
                 of
                 a
                 Triangle
                 ,
                 whose
                 content
                 is
                 2000
                 poles
                 ,
                 and
                 I
                 demand
                 so
                 much
                 of
                 the
                 Base
                 as
                 may
                 answer
                 1750
                 poles
                 .
                 Enter
                 the
                 whole
                 line
                 in
                 10
                 ,
                 and
                 10
                 at
                 the
                 end
                 of
                 the
                 lines
                 of
                 lines
                 ,
                 and
                 keep
                 the
                 Sector
                 at
                 that
                 gage
                 .
                 Now
                 the
                 whole
                 line
                 representing
                 2000
                 poles
                 ,
                 every
                 figured
                 division
                 is
                 200
                 ;
                 therefore
                 1700
                 is
                 eight
                 and
                 an
                 half
                 of
                 the
                 figured
                 divisions
                 ,
                 and
                 50
                 is
                 five
                 of
                 the
                 smallest
                 divisions
                 more
                 (
                 for
                 in
                 this
                 case
                 every
                 smallest
                 division
                 is
                 10
                 ,
                 as
                 was
                 before
                 expressed
                 )
                 wherefore
                 setting
                 one
                 point
                 of
                 the
                 Compasses
                 in
                 15
                 of
                 the
                 smallest
                 divisions
                 beyond
                 8
                 on
                 the
                 Rule
                 .
                 I
                 extend
                 the
                 other
                 point
                 to
                 the
                 same
                 division
                 upon
                 the
                 line
                 on
                 the
                 other
                 side
                 ,
                 and
                 that
                 distance
                 is
                 1750
                 poles
                 in
                 the
                 base
                 of
                 the
                 Triangle
                 proposed
                 .
                 How
                 ready
                 this
                 is
                 to
                 set
                 out
                 a
                 just
                 quantity
                 in
                 any
                 plat
                 of
                 ground
                 ,
                 I
                 shall
                 shew
                 in
                 a
                 Scheam
                 ,
                 Chap.
                 12.
                 
              
            
             
               
               
                 PROBL.
                 2.
                 
              
               
                 
                   To
                   work
                   proportions
                   in
                   Lines
                   ,
                   or
                   Numbers
                   ,
                   or
                   the
                   Rule
                   of
                   three
                   direct
                   by
                   the
                   Lines
                   of
                   Lines
                   .
                
              
               
                 Enter
                 the
                 second
                 term
                 in
                 the
                 first
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 take
                 the
                 distance
                 'twixt
                 the
                 third
                 on
                 one
                 line
                 ,
                 to
                 the
                 third
                 on
                 the
                 other
                 line
                 ,
                 that
                 distance
                 is
                 the
                 fourth
                 in
                 lines
                 ,
                 or
                 measured
                 upon
                 the
                 line
                 from
                 the
                 centre
                 ,
                 gives
                 the
                 fourth
                 in
                 numbers
                 ,
                 Ex.
                 gr
                 .
              
               
                 As
                 7
                 is
                 to
                 3
                 ,
                 so
                 is
                 21
                 to
                 9.
                 
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   To
                   work
                   the
                   Rule
                   of
                   Three
                   inverse
                   ,
                   or
                   the
                   back
                   Rule
                   of
                   Three
                   by
                   the
                   Lines
                   of
                   Lines
                   .
                
              
               
                 In
                 these
                 proportions
                 there
                 are
                 alwayes
                 three
                 terms
                 given
                 to
                 finde
                 a
                 fourth
                 ,
                 and
                 of
                 the
                 three
                 given
                 terms
                 two
                 are
                 of
                 one
                 denomination
                 (
                 which
                 for
                 distinction
                 sake
                 I
                 call
                 the
                 double
                 denomination
                 ;
                 )
                 and
                 the
                 third
                 term
                 is
                 of
                 a
                 different
                 denomination
                 from
                 those
                 two
                 ,
                 which
                 I
                 therefore
                 call
                 the
                 single
                 denomination
                 ,
                 of
                 which
                 the
                 fourth
                 term
                 sought
                 must
                 also
                 be
                 .
                 Now
                 to
                 bring
                 these
                 
                 into
                 a
                 direct
                 proportion
                 ,
                 the
                 rule
                 is
                 this
                 .
                 When
                 the
                 fourth
                 term
                 sought
                 is
                 to
                 be
                 greater
                 than
                 the
                 single
                 denomination
                 (
                 which
                 you
                 may
                 know
                 by
                 sight
                 of
                 the
                 terms
                 given
                 )
                 say
                 ,
              
               
                 As
                 the
                 lesser
                 double
                 denomination
                 is
                 to
                 the
                 greater
                 double
                 denomination
                 ,
                 so
                 is
                 the
                 single
                 denomination
                 to
                 the
                 fourth
                 term
                 sought
                 .
                 The
                 work
                 is
                 by
                 Probl.
                 2.
                 
              
               
                 If
                 60
                 men
                 do
                 a
                 work
                 in
                 5
                 dayes
                 ,
                 how
                 long
                 will
                 30
                 men
                 be
                 about
                 it
                 ?
              
               
                 As
                 30
                 is
                 to
                 60
                 ,
                 so
                 is
                 5
                 to
                 10.
                 
                 The
                 number
                 of
                 dayes
                 for
                 30
                 men
                 in
                 the
                 work
                 .
              
               
                 Again
                 ,
                 when
                 the
                 fourth
                 term
                 is
                 to
                 be
                 less
                 than
                 the
                 single
                 denomination
                 ,
                 say
                 ,
              
               
                 As
                 the
                 greater
                 double
                 denomination
                 is
                 to
                 the
                 lesser
                 double
                 denomination
                 ,
                 so
                 is
                 the
                 single
                 denomination
                 to
                 the
                 fourth
                 term
                 sought
                 .
              
               
                 If
                 30
                 men
                 do
                 a
                 work
                 in
                 5
                 dayes
                 ,
                 how
                 long
                 shall
                 60
                 be
                 doing
                 of
                 it
                 ?
              
               
                 As
                 60
                 is
                 to
                 30
                 ,
                 so
                 is
                 5
                 to
                 2½
                 .
                 The
                 time
                 for
                 60
                 men
                 in
                 the
                 work
                 .
              
            
             
               
               
                 PROBL.
                 4.
                 
              
               
                 
                   The
                   length
                   of
                   any
                   perpendicular
                   ,
                   with
                   the
                   length
                   of
                   the
                   shadow
                   thereof
                   given
                   ,
                   to
                   finde
                   the
                   Suns
                   Altitude
                   .
                
              
               
                 At
                 the
                 length
                 of
                 the
                 shadow
                 upon
                 the
                 lines
                 of
                 lines
                 ,
                 is
                 to
                 the
                 Tangent
                 45
                 ,
                 so
                 is
                 the
                 length
                 of
                 the
                 perpendicular
                 numbred
                 upon
                 the
                 lines
                 of
                 lines
                 ,
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 altitude
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   finde
                   the
                   Altitude
                   of
                   any
                   Tree
                   ,
                   Steeple
                   ,
                   &c.
                   at
                   one
                   station
                   .
                
              
               
                 At
                 any
                 distance
                 from
                 the
                 object
                 (
                 provided
                 the
                 ground
                 be
                 level
                 )
                 with
                 your
                 Instrument
                 ,
                 look
                 to
                 the
                 top
                 of
                 the
                 object
                 along
                 the
                 outward
                 ledge
                 of
                 the
                 fixed
                 piece
                 ,
                 and
                 take
                 the
                 angle
                 of
                 its
                 altitude
                 .
                 This
                 done
                 ,
                 measure
                 by
                 feet
                 or
                 yards
                 ,
                 the
                 distance
                 from
                 your
                 standing
                 to
                 the
                 bottom
                 of
                 the
                 object
                 .
                 Then
                 say
                 ,
              
               
                 As
                 the
                 cosine
                 of
                 the
                 altitude
                 is
                 to
                 the
                 measured
                 distance
                 numbred
                 upon
                 the
                 lines
                 of
                 lines
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 altitude
                 to
                 a
                 fourth
                 
                 number
                 of
                 feet
                 or
                 yards
                 (
                 according
                 to
                 the
                 measure
                 you
                 meeted
                 the
                 distance
                 )
                 to
                 this
                 fourth
                 ,
                 adde
                 the
                 height
                 of
                 your
                 eye
                 from
                 the
                 ground
                 ,
                 and
                 that
                 sum
                 gives
                 the
                 number
                 of
                 feet
                 or
                 yards
                 in
                 the
                 altitude
                 .
              
            
          
           
             
               CHAP.
               VI.
               
            
             
               
                 How
                 to
                 work
                 proportions
                 in
                 Numbers
                 ,
                 Sines
                 ,
                 or
                 Tangents
                 ,
                 by
                 the
                 Artificial
                 Lines
                 thereof
                 on
                 the
                 outward
                 ledge
                 .
              
            
             
               THe
               general
               rule
               for
               all
               of
               these
               ,
               is
               to
               extend
               the
               Compasses
               from
               the
               first
               term
               to
               the
               second
               (
               and
               observing
               whether
               that
               extent
               was
               upward
               or
               downward
               )
               with
               the
               same
               distance
               ,
               set
               one
               point
               in
               the
               third
               term
               ,
               and
               turning
               the
               other
               point
               the
               same
               way
               ,
               as
               at
               first
               ,
               it
               gives
               the
               fourth
               .
               But
               in
               Tangents
               when
               any
               of
               the
               terms
               exceeds
               45
               ,
               there
               may
               be
               excursions
               ,
               which
               in
               their
               due
               place
               I
               shall
               remove
               .
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   Numeration
                   by
                   the
                   Line
                   of
                   Numbers
                   .
                
              
               
                 The
                 whole
                 line
                 is
                 actually
                 divided
                 into
                 
                 100
                 proportional
                 parts
                 ,
                 and
                 accordingly
                 distinguished
                 by
                 figures
                 ,
                 1
                 ,
                 2
                 ,
                 3
                 ,
                 4
                 ,
                 5
                 ,
                 6
                 ,
                 7
                 ,
                 8
                 ,
                 9
                 ,
                 and
                 then
                 ,
                 10
                 ,
                 20
                 ,
                 30
                 ,
                 40
                 ,
                 50
                 ,
                 60
                 ,
                 70
                 ,
                 80
                 ,
                 90
                 ,
                 100.
                 
                 So
                 that
                 for
                 any
                 number
                 under
                 100
                 ,
                 the
                 Figures
                 readily
                 direct
                 you
                 ,
                 Ex.
                 gr
                 .
                 To
                 finde
                 79
                 on
                 the
                 line
                 of
                 numbers
                 ,
                 count
                 9
                 of
                 the
                 small
                 divisions
                 beyond
                 70
                 ,
                 and
                 there
                 is
                 the
                 point
                 for
                 that
                 number
                 .
                 Now
                 as
                 the
                 whole
                 line
                 is
                 actually
                 divided
                 into
                 100
                 parts
                 ,
                 so
                 is
                 every
                 one
                 of
                 those
                 parts
                 subdivided
                 (
                 so
                 far
                 as
                 conveniency
                 will
                 permit
                 )
                 actually
                 into
                 ten
                 parts
                 more
                 ,
                 by
                 which
                 means
                 you
                 have
                 the
                 whole
                 line
                 actually
                 divided
                 into
                 1000
                 parts
                 .
                 For
                 reckoning
                 the
                 Figures
                 impressed
                 ,
                 1
                 ,
                 2
                 ,
                 3
                 ,
                 4
                 ,
                 5
                 ,
                 6
                 ,
                 7
                 ,
                 8
                 ,
                 9
                 ,
                 to
                 be
                 10
                 ,
                 20
                 ,
                 30
                 ,
                 40
                 ,
                 50
                 ,
                 60
                 ,
                 70
                 ,
                 80
                 ,
                 90
                 ,
                 and
                 the
                 other
                 figures
                 which
                 are
                 stamped
                 10
                 ,
                 20
                 ,
                 30
                 ,
                 40
                 ,
                 50
                 ,
                 60
                 ,
                 70
                 ,
                 80
                 ,
                 90
                 ,
                 100
                 ,
                 to
                 be
                 100
                 ,
                 200
                 ,
                 300
                 ,
                 400
                 ,
                 500
                 ,
                 600
                 ,
                 700
                 ,
                 800
                 ,
                 900
                 ,
                 1000.
                 
                 You
                 may
                 enter
                 any
                 number
                 under
                 1000
                 upon
                 the
                 line
                 ,
                 according
                 to
                 the
                 former
                 directions
                 .
                 And
                 any
                 numbers
                 whose
                 product
                 surmount
                 not
                 1000
                 ,
                 may
                 be
                 wrought
                 upon
                 this
                 line
                 ;
                 but
                 where
                 the
                 product
                 exceeds
                 1000
                 ,
                 this
                 line
                 will
                 do
                 nothing
                 accurately
                 :
                 Wherefore
                 I
                 shall
                 willingly
                 omit
                 many
                 Problems
                 mentioned
                 by
                 some
                 Writers
                 to
                 be
                 wrought
                 by
                 this
                 line
                 ,
                 as
                 squaring
                 ,
                 and
                 cubing
                 of
                 numbers
                 ,
                 &c.
                 
                 Sith
                 they
                 have
                 only
                 nicety
                 ,
                 and
                 nothing
                 of
                 exactness
                 in
                 them
                 .
              
            
             
               
               
                 PROBL.
                 2.
                 
              
               
                 
                   To
                   multiply
                   two
                   numbers
                   ,
                   given
                   by
                   the
                   Line
                   of
                   Numbers
                   .
                   The
                   proportion
                   is
                   this
                   .
                
              
               
                 As
                 1
                 on
                 the
                 line
                 is
                 to
                 the
                 multiplicator
                 ,
                 so
                 is
                 the
                 multiplicand
                 to
                 the
                 product
                 .
                 Ex.
                 gr
                 .
              
               
                 As
                 1
                 is
                 to
                 4
                 ,
                 so
                 is
                 7
                 ,
                 to
                 what
                 ?
              
               
                 Extend
                 the
                 Compasses
                 from
                 the
                 first
                 term
                 ,
                 viz.
                 I
                 unto
                 the
                 second
                 term
                 ,
                 viz.
                 4.
                 with
                 that
                 distance
                 ,
                 setting
                 one
                 point
                 in
                 7
                 the
                 third
                 term
                 ,
                 turn
                 the
                 other
                 point
                 of
                 the
                 Compasses
                 toward
                 the
                 same
                 end
                 of
                 the
                 rule
                 ,
                 as
                 at
                 first
                 ,
                 and
                 you
                 have
                 the
                 fourth
                 ,
                 viz.
                 28.
                 
                 There
                 is
                 only
                 one
                 difficulty
                 remaining
                 in
                 this
                 Problem
                 ,
                 and
                 that
                 is
                 to
                 determin
                 the
                 number
                 of
                 places
                 ,
                 or
                 figures
                 in
                 the
                 product
                 ,
                 which
                 may
                 be
                 resolved
                 by
                 this
                 general
                 rule
                 .
                 The
                 product
                 alwayes
                 contains
                 as
                 many
                 figures
                 as
                 are
                 in
                 the
                 multiplicand
                 ,
                 and
                 multiplicator
                 both
                 ,
                 unless
                 the
                 two
                 first
                 figures
                 of
                 the
                 product
                 be
                 greater
                 than
                 the
                 two
                 first
                 figures
                 in
                 the
                 multiplicator
                 ,
                 and
                 then
                 the
                 product
                 must
                 have
                 one
                 figure
                 less
                 than
                 are
                 in
                 the
                 multiplicator
                 ,
                 and
                 multiplicand
                 both
                 .
                 Ex.
                 gr
                 .
                 47
                 multiplied
                 by
                 25
                 ,
                 is
                 2175
                 ,
                 consisting
                 of
                 four
                 figures
                 ;
                 but
                 16
                 multiplied
                 by
                 16
                 ,
                 is
                 240
                 ,
                 consisting
                 of
                 no
                 more
                 than
                 three
                 places
                 ,
                 for
                 the
                 reason
                 before
                 mentioned
                 .
                 
                 I
                 here
                 (
                 for
                 distinction
                 sake
                 )
                 call
                 the
                 multiplicator
                 the
                 lesser
                 of
                 the
                 two
                 numbers
                 ,
                 although
                 it
                 may
                 be
                 either
                 of
                 them
                 at
                 pleasure
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   To
                   work
                   Division
                   by
                   the
                   Line
                   of
                   Numbers
                   .
                
              
               
                 As
                 the
                 divisor
                 is
                 to
                 1
                 ,
                 so
                 is
                 the
                 dividend
                 to
                 the
                 quotient
                 .
              
               
                 Suppose
                 800
                 to
                 be
                 divided
                 by
                 20
                 ,
                 the
                 quotient
                 is
                 40.
                 
                 For
                 ,
              
               
                 As
                 20
                 is
                 to
                 1
                 ,
                 so
                 is
                 800
                 to
                 40.
                 
              
               
                 To
                 know
                 how
                 many
                 figures
                 you
                 shall
                 have
                 in
                 the
                 quotient
                 ,
                 take
                 this
                 rule
                 .
                 Note
                 the
                 difference
                 of
                 the
                 numbers
                 of
                 places
                 or
                 figures
                 in
                 the
                 dividend
                 and
                 divisor
                 .
                 Then
                 in
                 case
                 the
                 quantity
                 of
                 the
                 two
                 first
                 figures
                 to
                 the
                 left
                 hand
                 in
                 your
                 divisor
                 be
                 less
                 than
                 the
                 quantity
                 of
                 the
                 two
                 first
                 figures
                 to
                 the
                 left
                 hand
                 in
                 your
                 dividend
                 ,
                 the
                 quotient
                 shall
                 have
                 one
                 figure
                 more
                 than
                 the
                 number
                 of
                 difference
                 :
                 But
                 where
                 the
                 quantity
                 of
                 the
                 two
                 first
                 figures
                 of
                 the
                 divisor
                 is
                 greater
                 than
                 the
                 quantity
                 of
                 the
                 two
                 first
                 figures
                 of
                 the
                 dividend
                 ,
                 the
                 quotient
                 will
                 have
                 only
                 that
                 number
                 of
                 figures
                 noted
                 by
                 the
                 difference
                 .
                 Ex.
                 gr
                 .
                 245
                 divided
                 by
                 15.
                 will
                 have
                 two
                 figures
                 in
                 the
                 quotient
                 ;
                 
                 but
                 16
                 divided
                 〈…〉
                 ●●ve
                 only
                 one
                 figure
                 in
                 the
                 quotient
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   To
                   finde
                   a
                   mean
                   proportional
                   'twixt
                   two
                   Numbers
                   given
                   by
                   the
                   line
                   of
                   Numbers
                   .
                
              
               
                 Divide
                 the
                 space
                 betwixt
                 them
                 upon
                 the
                 line
                 of
                 numbers
                 into
                 two
                 equal
                 parts
                 ,
                 and
                 the
                 middle
                 point
                 is
                 the
                 mean
                 proportional
                 .
                 Ex.
                 gr
                 .
                 betwixt
                 4
                 and
                 16
                 ,
                 the
                 mean
                 proportional
                 is
                 8.
                 
                 If
                 you
                 were
                 to
                 finde
                 two
                 mean
                 proportionals
                 ,
                 divide
                 the
                 space
                 'twixt
                 the
                 given
                 numbers
                 into
                 three
                 parts
                 .
                 If
                 four
                 mean
                 proportionals
                 divide
                 it
                 into
                 five
                 parts
                 ,
                 and
                 the
                 several
                 points
                 'twixt
                 the
                 two
                 given
                 numbers
                 ,
                 will
                 show
                 the
                 respective
                 mean
                 proportionals
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   work
                   proportions
                   in
                   Sines
                   alone
                   ,
                   by
                   the
                   Artificial
                   Line
                   of
                   Sines
                   .
                
              
               
                 Extend
                 the
                 Compasses
                 from
                 the
                 first
                 term
                 to
                 the
                 second
                 ,
                 with
                 that
                 distance
                 set
                 one
                 point
                 in
                 the
                 third
                 term
                 ,
                 and
                 the
                 other
                 point
                 gives
                 the
                 fourth
                 .
                 Only
                 observe
                 that
                 if
                 the
                 second
                 term
                 be
                 less
                 than
                 the
                 first
                 ,
                 the
                 fourth
                 
                 must
                 be
                 less
                 〈…〉
                 or
                 if
                 the
                 second
                 term
                 exceed
                 〈…〉
                 fourth
                 will
                 be
                 greater
                 than
                 the
                 third
                 .
                 This
                 may
                 direct
                 you
                 in
                 all
                 proportions
                 of
                 sines
                 and
                 tangents
                 singly
                 or
                 conjunctly
                 ,
                 to
                 which
                 end
                 of
                 the
                 rule
                 to
                 turn
                 the
                 point
                 of
                 your
                 Compasses
                 ,
                 for
                 finding
                 the
                 fourth
                 term
                 .
                 Ex.
                 gr
                 .
              
               
                 As
                 the
                 sine
                 60
                 is
                 to
                 the
                 sine
                 40
                 ,
                 so
                 is
                 the
                 sine
                 20
                 to
                 the
                 sine
                 14.
                 40.
                 
                 Again
                 ,
              
               
                 As
                 the
                 sine
                 10
                 is
                 to
                 the
                 sine
                 20
                 ,
                 so
                 is
                 the
                 sine
                 30
                 to
                 the
                 sine
                 80.
                 
              
            
             
               
                 PROBL.
                 6.
                 
              
               
                 
                   To
                   work
                   proportions
                   in
                   Tangents
                   alone
                   by
                   the
                   Artificial
                   Line
                   of
                   Tangents
                   .
                
              
               
                 For
                 this
                 purpose
                 the
                 artificial
                 line
                 of
                 tangents
                 must
                 be
                 imagined
                 twice
                 the
                 length
                 of
                 the
                 rules
                 ,
                 and
                 therefore
                 for
                 the
                 greater
                 conveniency
                 ,
                 it
                 is
                 doubly
                 numbred
                 ,
                 viz.
                 First
                 from
                 1
                 to
                 45
                 ,
                 which
                 is
                 the
                 radius
                 ,
                 or
                 equal
                 to
                 the
                 sine
                 90
                 :
                 In
                 which
                 account
                 every
                 division
                 hath
                 (
                 as
                 to
                 its
                 length
                 on
                 the
                 rule
                 )
                 a
                 proportional
                 decrease
                 .
                 Secondly
                 ,
                 it
                 s
                 numbred
                 back
                 again
                 from
                 45
                 to
                 89
                 ,
                 in
                 which
                 account
                 every
                 division
                 hath
                 (
                 as
                 to
                 its
                 length
                 on
                 the
                 line
                 )
                 a
                 proportional
                 encrease
                 .
                 So
                 that
                 the
                 tangent
                 60
                 you
                 must
                 imagine
                 the
                 whole
                 
                 length
                 of
                 the
                 Rule
                 ;
                 and
                 so
                 much
                 more
                 as
                 the
                 distance
                 from
                 45
                 unto
                 30
                 or
                 60
                 is
                 .
                 This
                 well
                 observed
                 ,
                 all
                 proportions
                 in
                 tangents
                 are
                 wrought
                 after
                 the
                 same
                 manner
                 of
                 extending
                 the
                 Compasses
                 from
                 the
                 first
                 term
                 to
                 the
                 second
                 ,
                 and
                 that
                 distance
                 set
                 in
                 the
                 third
                 ,
                 gives
                 the
                 fourth
                 ,
                 as
                 was
                 for
                 sines
                 and
                 numbers
                 .
                 But
                 for
                 the
                 remedying
                 of
                 excursions
                 ,
                 sith
                 the
                 line
                 is
                 no
                 more
                 than
                 half
                 the
                 length
                 ,
                 we
                 must
                 imagine
                 it
                 .
                 I
                 shall
                 lay
                 down
                 these
                 Cases
                 .
              
               
                 
                   CASE
                   1.
                   
                
                 
                   
                     When
                     the
                     fourth
                     term
                     is
                     a
                     tangent
                     exceeding
                     45
                     ,
                     or
                     the
                     Radius
                     .
                     Ex.
                     gr
                     .
                  
                
                 
                   As
                   the
                   tangent
                   10
                   is
                   to
                   the
                   tangent
                   30
                   ,
                   so
                   is
                   the
                   tangent
                   20
                   ,
                   to
                   what
                   ?
                
                 
                   Extending
                   the
                   Compasses
                   from
                   10
                   on
                   the
                   line
                   of
                   tangents
                   to
                   30
                   ,
                   with
                   that
                   distance
                   I
                   set
                   one
                   point
                   in
                   20
                   ,
                   and
                   finde
                   the
                   other
                   point
                   reach
                   beyond
                   45
                   ,
                   which
                   tells
                   me
                   the
                   fourth
                   term
                   exceeds
                   45
                   ,
                   or
                   the
                   radius
                   ;
                   wherefore
                   with
                   the
                   former
                   extent
                   ,
                   I
                   set
                   one
                   point
                   in
                   45
                   ;
                   and
                   turning
                   the
                   other
                   toward
                   the
                   beginning
                   of
                   the
                   line
                   ,
                   I
                   mark
                   where
                   it
                   toucheth
                   ,
                   and
                   from
                   thence
                   taking
                   the
                   distance
                   to
                   the
                   third
                   term
                   ,
                   I
                   have
                   the
                   excess
                   of
                   the
                   fourth
                   
                   term
                   above
                   45
                   in
                   my
                   compass
                   :
                   wherefore
                   with
                   this
                   last
                   distance
                   setting
                   one
                   point
                   in
                   45
                   ,
                   I
                   turn
                   the
                   other
                   upon
                   the
                   line
                   ,
                   and
                   it
                   reacheth
                   to
                   50
                   ,
                   the
                   tangent
                   sought
                   .
                
              
               
                 
                   CASE
                   2.
                   
                
                 
                   
                     When
                     the
                     first
                     term
                     is
                     a
                     tangent
                     exceeding
                     45
                     ,
                     or
                     the
                     Radius
                     .
                     Ex.
                     gr
                     .
                  
                
                 
                   As
                   the
                   tangent
                   50
                   is
                   to
                   the
                   tangent
                   20
                   ,
                   so
                   is
                   the
                   tangent
                   30
                   ,
                   to
                   what
                   ?
                
                 
                   Because
                   the
                   second
                   term
                   is
                   less
                   than
                   the
                   first
                   ,
                   I
                   know
                   the
                   fourth
                   must
                   be
                   less
                   than
                   the
                   third
                   .
                   All
                   the
                   difficulty
                   is
                   to
                   get
                   the
                   true
                   extent
                   from
                   the
                   tangent
                   50
                   to
                   20.
                   
                   To
                   do
                   this
                   ,
                   take
                   the
                   distance
                   from
                   45
                   to
                   50
                   ,
                   and
                   setting
                   one
                   point
                   in
                   20
                   ,
                   the
                   second
                   term
                   ,
                   turn
                   the
                   other
                   toward
                   the
                   beginning
                   of
                   the
                   line
                   ,
                   marking
                   where
                   it
                   toucheth
                   ,
                   extend
                   the
                   Compasses
                   from
                   the
                   point
                   where
                   it
                   toucheth
                   to
                   45
                   ,
                   and
                   you
                   will
                   have
                   the
                   same
                   distance
                   in
                   your
                   Compasses
                   as
                   from
                   50
                   to
                   20
                   ,
                   if
                   the
                   line
                   had
                   been
                   continued
                   at
                   length
                   unto
                   89
                   tangents
                   ,
                   with
                   this
                   distance
                   ,
                   set
                   one
                   point
                   in
                   30
                   the
                   third
                   term
                   ,
                   and
                   turn
                   the
                   other
                   toward
                   the
                   beginning
                   (
                   because
                   you
                   know
                   the
                   fourth
                   must
                   be
                   less
                   )
                   and
                   it
                   gives
                   10
                   the
                   tangent
                   sought
                   .
                
              
               
                 
                 
                   CASE
                   .
                   3.
                   
                
                 
                   
                     When
                     the
                     third
                     term
                     is
                     a
                     Tangent
                     exceeding
                     45
                     ,
                     or
                     the
                     Radius
                     .
                  
                
                 
                   As
                   the
                   tangent
                   40
                   is
                   to
                   the
                   tangent
                   12
                   ,
                   40
                   min.
                   so
                   is
                   the
                   tangent
                   65
                   ,
                   to
                   what
                   ?
                
                 
                   Extend
                   the
                   Compasses
                   from
                   40
                   to
                   12
                   d.
                   40
                   min.
                   with
                   distance
                   ,
                   setting
                   one
                   point
                   in
                   65.
                   turn
                   the
                   other
                   toward
                   45
                   ,
                   and
                   you
                   will
                   finde
                   it
                   reach
                   beyond
                   it
                   ,
                   which
                   assures
                   you
                   the
                   fourth
                   term
                   will
                   be
                   less
                   than
                   45.
                   
                   Therefore
                   lay
                   the
                   extent
                   from
                   45
                   toward
                   the
                   beginning
                   ,
                   and
                   mark
                   where
                   it
                   toucheth
                   ,
                   take
                   the
                   distance
                   from
                   that
                   point
                   to
                   65
                   ,
                   and
                   laying
                   that
                   distance
                   from
                   45
                   toward
                   the
                   beginning
                   it
                   gives
                   30
                   ,
                   the
                   tangent
                   sought
                   .
                
                 
                   These
                   Cases
                   are
                   sufficient
                   to
                   remove
                   all
                   difficulties
                   .
                   For
                   when
                   the
                   second
                   term
                   exceeds
                   the
                   Radius
                   ,
                   you
                   may
                   transpose
                   them
                   ,
                   saying
                   ,
                   as
                   the
                   first
                   term
                   is
                   to
                   the
                   third
                   ;
                   so
                   is
                   the
                   second
                   to
                   the
                   fourth
                   ,
                   and
                   then
                   it
                   s
                   wrought
                   by
                   the
                   third
                   Case
                   .
                
                 
                   I
                   suppose
                   it
                   needless
                   to
                   adde
                   any
                   thing
                   about
                   working
                   proportions
                   by
                   sines
                   and
                   tangents
                   conjunctly
                   ,
                   sith
                   ,
                   enough
                   hath
                   been
                   already
                   said
                   of
                   both
                   of
                   them
                   apart
                   ,
                   in
                   these
                   two
                   last
                   Problems
                   ;
                   and
                   the
                   work
                   is
                   the
                   same
                   
                   when
                   they
                   are
                   intermixed
                   .
                   Only
                   some
                   proportions
                   I
                   shall
                   adjoyn
                   ,
                   and
                   leave
                   to
                   the
                   practice
                   of
                   the
                   young
                   beginner
                   ,
                   with
                   the
                   directions
                   in
                   the
                   former
                   Cases
                   .
                
              
            
             
               
                 PROBL.
                 7.
                 
              
               
                 
                   To
                   finde
                   the
                   Suns
                   ascensional
                   difference
                   in
                   any
                   Latitude
                   .
                
              
               
                 As
                 the
                 co-tangent
                 of
                 the
                 latitude
                 is
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 ascensional
                 difference
                 .
              
            
             
               
                 PROBL.
                 8.
                 
              
               
                 
                   To
                   finde
                   at
                   what
                   hour
                   the
                   Sun
                   will
                   be
                   East
                   ,
                   or
                   West
                   in
                   any
                   Latitude
                   .
                
              
               
                 As
                 the
                 tangent
                 of
                 the
                 latitude
                 is
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 cosine
                 of
                 the
                 hour
                 from
                 noon
                 .
              
            
             
               
                 PROBL.
                 9.
                 
              
               
                 
                   The
                   Latitude
                   ,
                   Declination
                   of
                   the
                   Sun
                   ,
                   and
                   his
                   Azimuth
                   from
                   South
                   ,
                   given
                   to
                   finde
                   the
                   Suns
                   Altitude
                   at
                   that
                   Azimuth
                   .
                
              
               
                 As
                 the
                 radius
                 to
                 the
                 cosine
                 of
                 the
                 Azimuth
                 from
                 south
                 ,
                 so
                 is
                 the
                 co-tangent
                 of
                 the
                 latitude
                 ,
                 
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 altitude
                 in
                 the
                 equator
                 at
                 the
                 Azimuth
                 given
                 .
                 Again
                 ,
              
               
                 As
                 the
                 sine
                 of
                 the
                 latitude
                 is
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 cosine
                 of
                 the
                 Suns
                 altitude
                 in
                 the
                 equator
                 (
                 at
                 the
                 same
                 Azimuth
                 from
                 East
                 or
                 West
                 )
                 to
                 a
                 fourth
                 ark
                 .
              
               
                 When
                 the
                 Azimuth
                 is
                 under
                 90
                 ,
                 and
                 the
                 latitude
                 and
                 declination
                 is
                 under
                 the
                 same
                 pole
                 ,
                 adde
                 this
                 fourth
                 ark
                 to
                 the
                 altitude
                 in
                 the
                 equator
                 .
                 In
                 Azimuths
                 exceeding
                 90
                 ,
                 when
                 the
                 latitude
                 and
                 declination
                 is
                 under
                 the
                 same
                 pole
                 ,
                 take
                 the
                 equator
                 altitude
                 out
                 of
                 the
                 fourth
                 ark
                 .
                 Lastly
                 ,
                 when
                 the
                 latitude
                 and
                 declination
                 respect
                 different
                 poles
                 ,
                 take
                 the
                 fourth
                 ark
                 out
                 of
                 the
                 equator
                 altitude
                 ,
                 and
                 you
                 have
                 the
                 altitude
                 sought
                 .
              
            
             
               
                 PROBL.
                 10.
                 
              
               
                 
                   The
                   Azimuth
                   ,
                   Altitude
                   ,
                   and
                   Declination
                   of
                   the
                   Sun
                   ,
                   given
                   to
                   finde
                   the
                   hour
                   .
                
              
               
                 As
                 the
                 cosine
                 of
                 declination
                 is
                 to
                 the
                 sine
                 of
                 the
                 Suns
                 Azimuth
                 ,
                 so
                 is
                 the
                 cosine
                 of
                 the
                 altitude
                 to
                 the
                 fine
                 of
                 the
                 hour
                 from
                 the
                 Meridian
                 .
              
               
                 Proportions
                 may
                 be
                 varied
                 eight
                 several
                 wayes
                 in
                 this
                 manner
                 following
                 .
                 
                   
                     1.
                     
                     As
                     the
                     first
                     term
                     is
                     to
                     the
                     second
                     ,
                     so
                     is
                     the
                     third
                     to
                     the
                     fourth
                     .
                  
                   
                     
                     2.
                     
                     As
                     the
                     second
                     term
                     is
                     to
                     the
                     first
                     ,
                     so
                     is
                     the
                     fourth
                     to
                     the
                     third
                     .
                  
                   
                     3.
                     
                     As
                     the
                     third
                     term
                     is
                     to
                     the
                     first
                     ,
                     so
                     is
                     the
                     fourth
                     to
                     the
                     second
                     .
                  
                   
                     4.
                     
                     As
                     the
                     fourth
                     term
                     is
                     to
                     the
                     second
                     ,
                     so
                     is
                     the
                     third
                     to
                     the
                     first
                     .
                  
                   
                     5.
                     
                     As
                     the
                     second
                     term
                     is
                     to
                     the
                     fourth
                     ,
                     so
                     is
                     the
                     first
                     to
                     the
                     third
                     .
                  
                   
                     6.
                     
                     As
                     the
                     first
                     term
                     is
                     to
                     the
                     third
                     ,
                     so
                     is
                     the
                     second
                     to
                     the
                     fourth
                     .
                  
                   
                     7.
                     
                     As
                     the
                     third
                     term
                     is
                     to
                     the
                     fourth
                     ,
                     so
                     is
                     the
                     first
                     to
                     the
                     second
                     .
                  
                   
                     8.
                     
                     As
                     the
                     fourth
                     term
                     is
                     to
                     the
                     third
                     ,
                     so
                     is
                     the
                     second
                     to
                     the
                     first
                     .
                  
                
              
               
                 By
                 thesse
                 any
                 one
                 may
                 vary
                 the
                 former
                 proportions
                 ,
                 and
                 make
                 the
                 Problems
                 three
                 times
                 the
                 number
                 here
                 inserted
                 .
                 Ex.
                 gr
                 .
                 To
                 finde
                 the
                 ascensional
                 difference
                 in
                 Problem
                 10
                 ,
                 of
                 this
                 Chapter
                 ,
                 which
                 runs
                 thus
                 .
              
               
                 As
                 the
                 co-tangent
                 of
                 the
                 latitude
                 is
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 ascensional
                 difference
                 .
                 Then
                 by
                 the
                 third
                 variety
                 you
                 may
                 make
                 another
                 Problem
                 ,
                 viz.
                 
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 co-tangent
                 of
                 the
                 latitude
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 Suns
                 ascensional
                 difference
                 to
                 the
                 sine
                 of
                 his
                 declination
                 .
                 Again
                 ,
                 by
                 the
                 fourth
                 variety
                 you
                 may
                 make
                 a
                 third
                 Problem
                 ,
                 thus
                 ,
              
               
               
                 As
                 the
                 sine
                 of
                 the
                 Suns
                 ascensional
                 difference
                 is
                 to
                 the
                 tangent
                 of
                 the
                 Suns
                 declination
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 co-tangent
                 of
                 the
                 latitude
                 .
              
               
                 By
                 this
                 Artifice
                 many
                 have
                 stuffed
                 their
                 Books
                 with
                 bundles
                 of
                 Problems
                 .
              
            
          
           
             
               CHAP.
               VII
               .
            
             
               
                 Some
                 uses
                 of
                 the
                 Lines
                 of
                 Circumference
                 ,
                 Diameter
                 ,
                 Square
                 Equal
                 ,
                 and
                 Square
                 Inscribed
                 .
              
            
             
               ALL
               these
               are
               lines
               of
               equal
               parts
               ,
               bearing
               such
               proportion
               to
               each
               other
               ,
               as
               the
               things
               signified
               by
               their
               names
               .
               Their
               use
               is
               this
               ,
               Any
               one
               of
               them
               given
               in
               inches
               or
               feet
               ,
               &c.
               to
               finde
               how
               much
               any
               of
               the
               other
               three
               are
               in
               the
               same
               measure
               .
               Suppose
               I
               have
               the
               circumference
               of
               a
               Circle
               ,
               Tree
               ,
               or
               Cylinder
               given
               in
               inches
               ,
               I
               take
               the
               same
               number
               of
               parts
               (
               as
               the
               Circle
               is
               inches
               )
               from
               the
               line
               of
               circumference
               ,
               and
               applying
               that
               distance
               to
               the
               respective
               lines
               ,
               I
               have
               immediately
               the
               square
               equal
               ,
               square
               inscribed
               ,
               and
               diameter
               ,
               in
               inches
               ,
               and
               the
               like
               ,
               if
               any
               of
               those
               were
               given
               to
               finde
               the
               circumference
               .
               This
               needs
               no
               example
               .
            
             
               The
               conveniency
               of
               this
               line
               any
               one
               may
               
               experiment
               in
               standing
               timber
               ;
               for
               taking
               the
               girth
               ,
               or
               circumference
               with
               a
               line
               ,
               finde
               the
               diameter
               ;
               from
               that
               diameter
               abate
               twice
               the
               thickness
               of
               the
               bark
               ,
               and
               you
               have
               the
               true
               diameter
               ,
               when
               it
               s
               barked
               ,
               and
               by
               Chap.
               9.
               
               Probl.
               5.
               you
               will
               guess
               very
               near
               at
               the
               quantity
               of
               timber
               in
               any
               standing
               Tree
               .
            
          
           
             
               CHAP.
               VIII
               .
            
             
               
                 To
                 measure
                 any
                 kinde
                 of
                 Superficies
                 ,
                 as
                 Board
                 ,
                 Glass
                 ,
                 Pavement
                 ,
                 Walnscot
                 ,
                 Hangings
                 ,
                 Walling
                 ,
                 Slating
                 ,
                 or
                 Tyling
                 ,
                 by
                 the
                 line
                 of
                 Numbers
                 on
                 the
                 outward
                 ledge
                 .
              
            
             
               THe
               way
               of
               accounting
               any
               number
               upon
               ,
               or
               working
               proportions
               by
               ,
               the
               line
               of
               numbers
               ,
               is
               sufficiently
               shewn
               already
               ,
               Chap.
               6.
               which
               I
               shall
               not
               here
               repeat
               ,
               only
               propose
               the
               proportions
               for
               these
               Problems
               ,
               and
               refer
               you
               to
               those
               directions
               .
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   The
                   breadth
                   of
                   a
                   Board
                   given
                   in
                   inches
                   ,
                   to
                   finde
                   how
                   many
                   inches
                   in
                   length
                   make
                   a
                   foot
                   at
                   that
                   breadth
                   ,
                   say
                   ,
                
              
               
                 As
                 the
                 breadth
                 in
                 inches
                 is
                 to
                 12
                 ,
                 so
                 
                 is
                 12
                 to
                 the
                 length
                 in
                 inches
                 for
                 a
                 foot
                 at
                 that
                 breadth
                 .
                 Ex.
                 gr
                 .
                 At
                 8
                 inches
                 breadth
                 you
                 must
                 have
                 18
                 inches
                 in
                 length
                 for
                 a
                 foot
                 .
              
            
             
               
                 PROBL.
                 2.
                 
              
               
                 
                   The
                   breadth
                   and
                   length
                   of
                   a
                   Board
                   given
                   to
                   finde
                   the
                   content
                   .
                
              
               
                 As
                 12
                 is
                 to
                 the
                 length
                 in
                 feet
                 and
                 inches
                 ,
                 so
                 is
                 the
                 breadth
                 in
                 inches
                 to
                 the
                 content
                 in
                 feet
                 .
                 Ex.
                 gr
                 .
                 at
                 15
                 inches
                 breadth
                 ,
                 and
                 20
                 foot
                 length
                 ,
                 you
                 have
                 25
                 foot
                 of
                 Board
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   A
                   speedy
                   way
                   to
                   measure
                   any
                   quantity
                   of
                   Board
                   .
                
              
               
                 The
                 two
                 former
                 Problems
                 are
                 sufficient
                 to
                 measure
                 small
                 parcels
                 of
                 Board
                 .
                 When
                 you
                 have
                 occasion
                 to
                 measure
                 greater
                 quantities
                 ,
                 as
                 100
                 foot
                 ,
                 or
                 more
                 ,
                 lay
                 all
                 the
                 boards
                 of
                 one
                 length
                 together
                 ,
                 and
                 when
                 the
                 length
                 of
                 the
                 boards
                 exceeds
                 12
                 foot
                 ,
                 use
                 this
                 proportion
                 .
              
               
                 As
                 the
                 length
                 in
                 feet
                 and
                 inches
                 is
                 to
                 12
                 ,
                 so
                 is
                 100
                 to
                 the
                 breadth
                 in
                 inches
                 for
                 an
                 100
                 foot
                 .
                 Ex.
                 gr
                 .
                 At
                 30
                 foot
                 in
                 length
                 40
                 inches
                 in
                 breadth
                 ,
                 make
                 an
                 100
                 foot
                 of
                 board
                 (
                 reckoning
                 five
                 score
                 to
                 the
                 hundred
                 .
                 )
                 This
                 
                 found
                 with
                 a
                 rule
                 or
                 line
                 ,
                 measure
                 40
                 inches
                 at
                 both
                 ends
                 in
                 breadth
                 ,
                 and
                 you
                 have
                 100
                 foot
                 .
                 When
                 one
                 end
                 is
                 broader
                 than
                 another
                 ,
                 you
                 may
                 take
                 the
                 breadth
                 of
                 the
                 over-plus
                 of
                 100
                 foot
                 at
                 both
                 ends
                 ,
                 and
                 taking
                 half
                 that
                 sum
                 for
                 the
                 true
                 breadth
                 of
                 the
                 over-plus
                 by
                 Probl.
                 2.
                 finde
                 the
                 content
                 thereof
                 .
              
               
                 When
                 your
                 boards
                 are
                 under
                 12
                 foot
                 in
                 length
                 ,
                 say
                 ,
              
               
                 As
                 the
                 length
                 in
                 feet
                 and
                 inches
                 is
                 to
                 12
                 ,
                 so
                 is
                 50
                 unto
                 the
                 breadth
                 in
                 inches
                 for
                 50
                 foot
                 of
                 board
                 ,
                 and
                 then
                 you
                 need
                 only
                 double
                 that
                 breadth
                 to
                 measure
                 100
                 foot
                 as
                 before
                 .
                 In
                 like
                 manner
                 you
                 may
                 measure
                 two
                 ,
                 three
                 ,
                 four
                 ,
                 five
                 ,
                 a
                 hundred
                 ,
                 &c.
                 foot
                 of
                 board
                 speedily
                 ,
                 as
                 your
                 occasion
                 requires
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   To
                   measure
                   Wainscot
                   ,
                   Hangings
                   ,
                   Plaister
                   ,
                   &c.
                   
                
              
               
                 These
                 are
                 usually
                 computed
                 by
                 the
                 yard
                 ,
                 and
                 then
                 the
                 proportion
                 is
                 .
              
               
                 As
                 nine
                 to
                 the
                 length
                 in
                 feet
                 ,
                 and
                 inches
                 ,
                 so
                 is
                 the
                 breadth
                 or
                 depth
                 in
                 feet
                 and
                 inches
                 to
                 the
                 content
                 in
                 yards
                 ,
                 Ex.
                 gr
                 .
                 At
                 18
                 foot
                 in
                 length
                 ,
                 and
                 two
                 foot
                 in
                 breadth
                 ,
                 you
                 have
                 four
                 yards
                 .
              
            
             
               
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   measure
                   Masons
                   ,
                   or
                   Slaters
                   Work
                   ,
                   as
                   Walling
                   ,
                   Tyling
                   ,
                   &c.
                   
                
              
               
                 The
                 common
                 account
                 of
                 these
                 is
                 by
                 the
                 rood
                 ,
                 which
                 is
                 eighteen
                 foot
                 square
                 ,
                 that
                 is
                 324
                 square
                 foot
                 in
                 one
                 rood
                 ,
                 and
                 then
                 the
                 proportion
                 is
                 .
              
               
                 As
                 324
                 to
                 the
                 length
                 in
                 feet
                 ,
                 so
                 is
                 the
                 breadth
                 in
                 feet
                 to
                 the
                 content
                 in
                 roods
                 .
              
               
                 Ex.
                 gr
                 .
                 At
                 30
                 foot
                 in
                 length
                 ,
                 and
                 15
                 foot
                 in
                 breadth
                 ,
                 you
                 have
                 1
                 rood
                 3
                 /
                 10
                 and
                 better
                 ,
                 or
                 one
                 rood
                 126
                 /
                 124
                 parts
                 of
                 a
                 rood
                 .
              
            
          
           
             
               CHAP.
               IX
               .
            
             
               
                 The
                 mensuration
                 of
                 Solids
                 ,
                 as
                 Timber
                 ,
                 Stone
                 ,
                 &c.
                 by
                 the
                 lines
                 on
                 the
                 proportional
                 side
                 of
                 the
                 loose
                 piece
                 .
              
            
             
               THese
               two
               lines
               meeting
               upon
               one
               line
               in
               the
               midst
               betwixt
               them
               (
               for
               distinction
               sake
               )
               I
               call
               one
               the
               right
               ,
               the
               other
               the
               left
               line
               ,
               which
               are
               known
               by
               the
               hand
               they
               stand
               toward
               when
               you
               hold
               up
               the
               piece
               in
               the
               right
               way
               to
               read
               the
               Figures
               .
            
             
             
               The
               right
               line
               hath
               two
               figured
               partitions
               .
               The
               first
               partition
               is
               from
               3
               ,
               at
               the
               beginning
               to
               the
               letters
               Sq.
               every
               figured
               division
               representing
               an
               inch
               ,
               and
               each
               subdivision
               quarters
               of
               an
               inch
               .
            
             
               The
               next
               partition
               is
               from
               the
               letters
               Sq.
               unto
               12
               ,
               at
               the
               end
               ,
               every
               figured
               division
               signifying
               a
               foot
               ,
               and
               each
               sub-division
               the
               inches
               in
               a
               foot
               .
               The
               letters
               T
               R
               ,
               and
               T
               D
               ,
               are
               for
               the
               circumference
               and
               Diameter
               in
               the
               next
               measuring
               of
               Cylinders
               .
               The
               letters
               R
               and
               D.
               for
               the
               measuring
               of
               Timber
               ,
               according
               to
               the
               vulgar
               allowance
               ,
               when
               the
               fourth
               part
               of
               the
               girt
               is
               taken
               ,
               &c.
               
               The
               letters
               A
               and
               W
               are
               the
               gauger
               points
               for
               Ale
               and
               Wine
               Measures
               .
               Lastly
               ,
               the
               figures
               12
               'twixt
               D
               and
               T
               D.
               are
               for
               an
               use
               ,
               expressed
               Probl.
               2.
               
            
             
               The
               left
               line
               also
               hath
               two
               figured
               partitions
               ,
               proceeding
               first
               from
               1
               at
               the
               beginning
               to
               one
               foot
               ,
               or
               12
               inches
               ,
               each
               whereof
               is
               sub-divided
               into
               quarters
               .
               From
               thence
               again
               to
               100
               ,
               each
               whereof
               to
               10
               foot
               ,
               is
               sub-divided
               into
               inches
               ,
               &c.
               and
               every
               foot
               is
               figured
               .
               But
               from
               10
               foot
               to
               100
               ,
               only
               every
               tenth
               foot
               is
               figured
               ;
               the
               sub-divisions
               representing
               feet
               .
               The
               method
               of
               working
               proportions
               by
               these
               lines
               (
               only
               observing
               the
               sides
               )
               is
               the
               same
               as
               by
               the
               line
               of
               Numbers
               ,
               viz.
               extending
               from
               the
               first
               to
               the
               second
               ,
               &c.
               
            
             
               
               
                 PROBL.
                 1.
                 
              
               
                 
                   To
                   reduce
                   Timber
                   of
                   unequal
                   breadth
                   and
                   depth
                   to
                   a
                   true
                   Square
                   .
                
              
               
                 As
                 the
                 breadth
                 on
                 the
                 left
                 is
                 to
                 the
                 breadth
                 on
                 the
                 right
                 ,
                 so
                 is
                 the
                 depth
                 on
                 the
                 left
                 ,
                 to
                 the
                 square
                 on
                 the
                 right
                 line
                 .
              
               
                 At
                 7
                 inches
                 breadth
                 ,
                 and
                 18
                 inches
                 depth
                 ,
                 you
                 have
                 11
                 inches
                 ¼
                 and
                 better
                 for
                 the
                 true
                 square
                 .
              
            
             
               
                 PROBL.
                 2.
                 
              
               
                 
                   The
                   square
                   of
                   a
                   Piece
                   of
                   Timber
                   given
                   in
                   Inches
                   ,
                   or
                   Feet
                   ,
                   and
                   Inches
                   to
                   finde
                   how
                   much
                   in
                   length
                   makes
                   a
                   Foot.
                   
                
              
               
                 As
                 the
                 square
                 in
                 feet
                 and
                 inches
                 on
                 the
                 right
                 is
                 to
                 one
                 foot
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 point
                 Sq.
                 on
                 the
                 right
                 to
                 the
                 number
                 of
                 feet
                 and
                 inches
                 on
                 the
                 left
                 for
                 a
                 foot
                 square
                 of
                 Timber
                 .
              
               
                 At
                 18
                 inches
                 square
                 ,
                 5
                 inches
                 ¼
                 and
                 almost
                 half
                 a
                 quarter
                 in
                 length
                 makes
                 a
                 foot
                 .
                 When
                 your
                 Timber
                 (
                 if
                 it
                 be
                 proper
                 to
                 call
                 such
                 pieces
                 by
                 that
                 name
                 )
                 is
                 under
                 3
                 inches
                 square
                 ,
                 account
                 the
                 figured
                 divisions
                 on
                 the
                 right
                 line
                 from
                 the
                 letters
                 Sq.
                 to
                 the
                 end
                 ,
                 for
                 
                 inches
                 ,
                 and
                 each
                 sub-division
                 twelve
                 parts
                 of
                 an
                 inch
                 .
                 So
                 that
                 every
                 three
                 of
                 them
                 makes
                 a
                 quarter
                 of
                 an
                 inch
                 .
                 Then
                 the
                 proportion
                 is
                 ,
                 as
                 the
                 inches
                 and
                 quarters
                 square
                 on
                 the
                 right
                 is
                 to
                 100
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 point
                 12
                 'twixt
                 D
                 ,
                 and
                 T
                 D
                 ,
                 on
                 the
                 right
                 ,
                 to
                 the
                 number
                 of
                 feet
                 in
                 length
                 on
                 the
                 left
                 to
                 make
                 a
                 foot
                 of
                 Timber
                 .
              
               
                 As
                 2
                 inches
                 ½
                 square
                 you
                 must
                 have
                 23
                 foot
                 6
                 inches
                 ,
                 and
                 somewhat
                 better
                 for
                 the
                 length
                 of
                 a
                 foot
                 of
                 Timber
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   The
                   square
                   and
                   length
                   of
                   a
                   plece
                   of
                   Timber
                   given
                   to
                   finde
                   the
                   content
                   .
                
              
               
                 As
                 the
                 point
                 Sq.
                 on
                 the
                 right
                 is
                 to
                 the
                 length
                 in
                 feet
                 and
                 inches
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 square
                 in
                 feet
                 and
                 inches
                 on
                 the
                 right
                 to
                 the
                 content
                 in
                 feet
                 on
                 the
                 left
                 .
              
               
                 At
                 30
                 foot
                 in
                 length
                 ,
                 and
                 15
                 inches
                 square
                 ,
                 you
                 have
                 46
                 foot
                 ½
                 of
                 Timber
                 .
              
               
                 At
                 20
                 foot
                 in
                 length
                 ,
                 and
                 11
                 inches
                 square
                 you
                 have
                 16
                 foot
                 ,
                 and
                 almost
                 ¼
                 of
                 Timber
                 .
              
               
                 When
                 you
                 have
                 a
                 great
                 piece
                 of
                 Timber
                 exceeding
                 100
                 foot
                 (
                 which
                 you
                 may
                 easily
                 see
                 by
                 the
                 excursion
                 upon
                 the
                 rule
                 )
                 then
                 take
                 the
                 true
                 square
                 ,
                 and
                 half
                 the
                 length
                 ,
                 sinde
                 the
                 
                 content
                 thereof
                 by
                 the
                 former
                 proportion
                 ,
                 and
                 doubling
                 that
                 content
                 ,
                 you
                 have
                 the
                 whole
                 content
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   The
                   Circumference
                   ,
                   or
                   girth
                   ,
                   of
                   a
                   round
                   piece
                   of
                   Timber
                   ,
                   being
                   given
                   ,
                   together
                   with
                   the
                   length
                   ,
                   to
                   finde
                   the
                   content
                   .
                
              
               
                 As
                 the
                 point
                 R.
                 on
                 the
                 right
                 ,
                 is
                 to
                 the
                 length
                 in
                 feet
                 and
                 inches
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 circumference
                 in
                 feet
                 and
                 inches
                 on
                 the
                 right
                 ,
                 to
                 the
                 content
                 in
                 feet
                 on
                 the
                 left
                 .
              
               
                 At
                 20
                 foot
                 in
                 length
                 ,
                 and
                 7
                 foot
                 in
                 girth
                 ,
                 you
                 have
                 60
                 foot
                 of
                 Timber
                 for
                 the
                 content
                 .
                 This
                 is
                 after
                 the
                 common
                 allowance
                 for
                 the
                 waste
                 in
                 squaring
                 ;
                 and
                 although
                 some
                 are
                 pleased
                 to
                 quarrel
                 with
                 the
                 allowancer
                 ,
                 as
                 wronging
                 the
                 seller
                 ,
                 and
                 giving
                 the
                 quantity
                 less
                 ,
                 than
                 in
                 truth
                 it
                 is
                 ;
                 yet
                 I
                 presume
                 when
                 they
                 buy
                 it
                 themselves
                 ,
                 they
                 scarcely
                 judge
                 those
                 Chips
                 worth
                 the
                 hewing
                 ,
                 and
                 have
                 as
                 low
                 thoughts
                 of
                 the
                 over-plus
                 ,
                 as
                 others
                 have
                 of
                 that
                 their
                 admonition
                 .
              
               
                 If
                 it
                 be
                 a
                 Cylinder
                 that
                 you
                 would
                 take
                 exact
                 content
                 of
                 ,
                 then
                 say
                 ,
              
               
                 As
                 the
                 point
                 T
                 R
                 ,
                 on
                 the
                 right
                 ,
                 is
                 to
                 the
                 length
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 girt
                 on
                 the
                 right
                 
                 to
                 the
                 exact
                 content
                 on
                 the
                 left
                 .
              
               
                 At
                 15
                 foot
                 in
                 length
                 ,
                 and
                 7
                 foot
                 in
                 girt
                 ,
                 you
                 have
                 59
                 foot
                 of
                 solid
                 measure
                 .
              
               
                 The
                 Diameter
                 of
                 any
                 Cylinder
                 given
                 ,
                 you
                 may
                 by
                 the
                 same
                 proportion
                 finde
                 the
                 content
                 ,
                 placing
                 the
                 point
                 D
                 ,
                 instead
                 of
                 R
                 ,
                 in
                 the
                 proportion
                 for
                 the
                 usual
                 allowance
                 ,
                 and
                 the
                 point
                 T
                 D
                 ,
                 for
                 the
                 exact
                 compute
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   measure
                   tapered
                   Timber
                   .
                
              
               
                 Take
                 the
                 square
                 or
                 girt
                 at
                 both
                 ends
                 ,
                 and
                 note
                 the
                 sum
                 and
                 difference
                 of
                 them
                 .
                 Then
                 for
                 round
                 Timber
                 ,
                 as
                 the
                 point
                 R.
                 on
                 the
                 right
                 ,
                 is
                 to
                 the
                 length
                 on
                 the
                 left
                 ,
                 so
                 is
                 half
                 the
                 sum
                 of
                 the
                 girt
                 at
                 both
                 ends
                 on
                 the
                 right
                 ,
                 to
                 a
                 number
                 of
                 feet
                 on
                 the
                 left
                 .
                 Keep
                 this
                 number
                 ,
                 and
                 say
                 again
                 ,
              
               
                 As
                 the
                 point
                 R.
                 on
                 the
                 right
                 ,
                 is
                 to
                 the
                 third
                 part
                 of
                 the
                 former
                 length
                 on
                 the
                 left
                 ,
                 so
                 is
                 half
                 the
                 difference
                 of
                 the
                 girts
                 on
                 the
                 right
                 ,
                 to
                 a
                 number
                 of
                 feet
                 on
                 the
                 left
                 ;
                 which
                 number
                 added
                 to
                 the
                 former
                 ,
                 gives
                 the
                 true
                 content
                 .
                 The
                 same
                 way
                 you
                 may
                 use
                 for
                 square
                 Timber
                 ,
                 only
                 setting
                 the
                 feet
                 and
                 inches
                 square
                 instead
                 of
                 the
                 girt
                 ,
                 and
                 the
                 point
                 Sq.
                 instead
                 of
                 the
                 point
                 R.
                 At
                 30
                 foot
                 in
                 length
                 ,
                 7
                 
                 foot
                 at
                 one
                 end
                 ,
                 and
                 5
                 at
                 the
                 other
                 in
                 girt
                 ,
                 half
                 the
                 sum
                 of
                 the
                 girts
                 is
                 6
                 foot
                 ,
                 or
                 72
                 inches
                 ;
                 the
                 first
                 number
                 of
                 feet
                 found
                 67
                 ,
                 half
                 the
                 difference
                 of
                 the
                 girts
                 is
                 1
                 foot
                 ,
                 or
                 12
                 inches
                 ,
                 the
                 third
                 part
                 of
                 the
                 length
                 10
                 foot
                 ;
                 then
                 the
                 second
                 number
                 found
                 will
                 be
                 7
                 foot
                 ,
                 one
                 quarter
                 and
                 half
                 a
                 quarter
                 .
                 The
                 sum
                 of
                 both
                 (
                 or
                 true
                 content
                 )
                 74
                 foot
                 ,
                 one
                 quarter
                 ,
                 and
                 half
                 a
                 quarter
                 .
              
               
                 For
                 standing
                 Timber
                 ,
                 take
                 the
                 girt
                 about
                 a
                 yard
                 from
                 the
                 bottom
                 ,
                 and
                 at
                 5
                 foot
                 from
                 the
                 bottom
                 ,
                 by
                 Chap.
                 7
                 ,
                 set
                 down
                 these
                 two
                 diameters
                 without
                 the
                 bark
                 ;
                 and
                 likewise
                 the
                 difference
                 'twixt
                 them
                 .
                 Again
                 ,
                 by
                 Chap.
                 6.
                 
                 Probl.
                 4.
                 finde
                 the
                 altitude
                 of
                 the
                 tree
                 ,
                 so
                 far
                 as
                 it
                 bears
                 Timber
                 (
                 or
                 as
                 we
                 commonly
                 phrase
                 it
                 ,
                 to
                 the
                 collar
                 )
                 this
                 done
                 ,
                 you
                 may
                 very
                 near
                 proportion
                 the
                 girt
                 at
                 the
                 collar
                 and
                 content
                 of
                 the
                 tree
                 ,
                 before
                 it
                 falls
                 .
              
               
                 In
                 case
                 any
                 make
                 choice
                 of
                 the
                 hollow
                 contrivance
                 mentioned
                 ,
                 Chap.
                 1.
                 they
                 need
                 no
                 compasses
                 in
                 the
                 mensuration
                 of
                 any
                 solid
                 ;
                 provided
                 the
                 lines
                 for
                 solid
                 measure
                 ,
                 and
                 gauging
                 vessels
                 ,
                 be
                 doubly
                 impressed
                 (
                 only
                 in
                 a
                 reverted
                 order
                 ,
                 one
                 pair
                 of
                 lines
                 proceeding
                 from
                 the
                 head
                 toward
                 the
                 end
                 ,
                 and
                 the
                 other
                 pair
                 from
                 the
                 end
                 toward
                 the
                 head
                 )
                 upon
                 the
                 sliding
                 cover
                 ,
                 and
                 its
                 adjacent
                 ledges
                 .
              
               
               
                 This
                 done
                 ,
                 the
                 method
                 of
                 performing
                 any
                 of
                 the
                 Problems
                 mentioned
                 in
                 this
                 Chapter
                 ,
                 is
                 easie
                 .
                 For
                 whereas
                 you
                 are
                 before
                 directed
                 to
                 extend
                 the
                 Compasses
                 from
                 the
                 first
                 term
                 to
                 the
                 second
                 ;
                 and
                 with
                 that
                 distance
                 setting
                 one
                 point
                 in
                 the
                 third
                 term
                 ,
                 the
                 other
                 point
                 gave
                 the
                 fourth
                 ,
                 or
                 term
                 sought
                 .
                 So
                 here
                 ,
                 observing
                 the
                 lines
                 as
                 before
                 ,
                 slide
                 the
                 cover
                 until
                 the
                 first
                 term
                 stand
                 directly
                 against
                 the
                 second
                 ;
                 then
                 looking
                 for
                 the
                 third
                 on
                 its
                 proper
                 line
                 ,
                 it
                 stands
                 exactly
                 against
                 the
                 fourth
                 term
                 ,
                 or
                 term
                 sought
                 on
                 the
                 other
                 line
                 .
              
               
                 Only
                 note
                 ,
                 that
                 when
                 the
                 second
                 term
                 is
                 greater
                 than
                 the
                 first
                 ,
                 it
                 s
                 performed
                 by
                 that
                 pair
                 of
                 lines
                 proceeding
                 from
                 the
                 head
                 toward
                 the
                 end
                 :
                 But
                 when
                 the
                 first
                 term
                 is
                 greater
                 than
                 the
                 second
                 ,
                 it
                 is
                 resolved
                 by
                 that
                 pair
                 of
                 lines
                 which
                 is
                 numbred
                 from
                 the
                 end
                 toward
                 the
                 head
                 .
              
            
          
           
             
             
               CHAP.
               X.
               
            
             
               
                 The
                 Gauge
                 Vessels
                 ,
                 either
                 for
                 Wine
                 ,
                 or
                 Ale
                 ,
                 Measure
                 .
              
            
             
               
                 PROBL.
                 1
                 .
              
               
                 
                   The
                   Diameter
                   at
                   Head
                   ,
                   and
                   Diameter
                   at
                   Boung
                   ,
                   given
                   in
                   Inches
                   ,
                   and
                   tenth
                   parts
                   of
                   an
                   Inch
                   ,
                   to
                   finde
                   the
                   mean
                   Diameter
                   in
                   like
                   measure
                   .
                
              
               
                 TAke
                 the
                 difference
                 in
                 inches
                 ,
                 and
                 tenth
                 parts
                 of
                 an
                 inch
                 ,
                 between
                 the
                 two
                 diameters
                 .
                 Then
                 say
                 by
                 the
                 line
                 of
                 numbers
                 ,
              
               
                 As
                 1
                 is
                 to
                 7
                 ,
                 so
                 is
                 the
                 difference
                 to
                 a
                 fourth
                 number
                 of
                 inches
                 ,
                 and
                 tenth
                 parts
                 of
                 an
                 inch
                 .
                 This
                 added
                 to
                 the
                 Diameter
                 at
                 head
                 ,
                 gives
                 the
                 mean
                 Diameter
                 .
                 Ex.
                 gr
                 .
              
               
                 At
                 27
                 inches
                 the
                 boung
                 ,
                 and
                 19
                 inches
                 two
                 tenths
                 at
                 the
                 head
                 ,
                 the
                 difference
                 is
                 7
                 inches
                 ,
                 8
                 tenths
                 .
                 The
                 fourth
                 number
                 found
                 by
                 the
                 proportion
                 will
                 be
                 5
                 inches
                 ,
                 4
                 tenths
                 ,
                 and
                 one
                 half
                 ,
                 which
                 added
                 to
                 the
                 diameter
                 at
                 the
                 head
                 ,
                 gives
                 24
                 inches
                 ,
                 6
                 tenths
                 ,
                 and
                 one
                 half
                 tenth
                 of
                 an
                 inch
                 for
                 the
                 mean
                 diameter
                 .
              
            
             
               
               
                 PROBL.
                 2.
                 
              
               
                 
                   The
                   length
                   of
                   the
                   Vessel
                   ,
                   and
                   the
                   mean
                   Diameter
                   given
                   in
                   Inches
                   ,
                   and
                   tenth
                   parts
                   of
                   an
                   Inch
                   ,
                   to
                   finde
                   the
                   content
                   in
                   Gallons
                   ,
                   either
                   in
                   Wine
                   or
                   Ale
                   measure
                   .
                
              
               
                 Note
                 first
                 ,
                 that
                 the
                 point
                 A.
                 on
                 the
                 right
                 is
                 the
                 gauge
                 point
                 for
                 Ale
                 measure
                 ,
                 and
                 the
                 point
                 W.
                 on
                 the
                 right
                 ,
                 is
                 the
                 gauge
                 point
                 for
                 Wine
                 measure
                 .
                 Then
                 say
                 ,
              
               
                 As
                 the
                 gauge
                 point
                 on
                 the
                 right
                 to
                 the
                 length
                 in
                 inches
                 ,
                 and
                 tenth
                 parts
                 of
                 an
                 inch
                 on
                 the
                 left
                 ,
                 so
                 is
                 the
                 mean
                 diameter
                 in
                 inches
                 and
                 tenth
                 parts
                 of
                 an
                 inch
                 on
                 the
                 right
                 ,
                 to
                 the
                 content
                 in
                 gallons
                 on
                 the
                 left
                 .
              
            
          
           
             
               CHAP.
               XI
               .
            
             
               
                 Some
                 uses
                 of
                 the
                 Lines
                 on
                 the
                 inward
                 ledge
                 of
                 the
                 moveable
                 piece
                 .
              
            
             
               THe
               Line
               of
               inches
               and
               foot
               measure
               do
               by
               inspection
               (
               only
               )
               reduce
               either
               of
               the
               measures
               into
               the
               other
               .
               Ex.
               gr
               .
               Three
               on
               the
               line
               of
               inches
               stands
               directly
               against
               25
               cents
               of
               foot
               measure
               .
               Or
               75
               cents
               of
               
               foot
               measure
               is
               directly
               against
               9
               on
               the
               inches
               .
            
             
               Another
               use
               of
               these
               lines
               (
               welcome
               perhaps
               to
               them
               that
               delight
               in
               instrumental
               computations
               )
               is
               to
               know
               the
               price
               of
               carriage
               for
               any
               quantity
               ,
               &c.
               by
               inspection
               .
            
             
               For
               this
               purpose
               ,
               the
               line
               of
               inches
               represents
               the
               price
               of
               a
               pound
               ,
               every
               inch
               being
               a
               penny
               ,
               and
               every
               quarter
               a
               farthing
               .
               The
               prickt
               line
               is
               the
               price
               of
               an
               hundred
               pound
               at
               five
               score
               and
               twelve
               to
               the
               hundred
               ,
               every
               division
               signifying
               a
               shilling
               .
            
             
               The
               line
               of
               foot
               measure
               is
               the
               price
               of
               an
               hundred
               pound
               at
               five
               score
               to
               the
               hundred
               ,
               every
               division
               standing
               for
               a
               shilling
               .
               Ex.
               gr
               .
            
             
               At
               3
               pence
               the
               pound
               on
               the
               inches
               ,
               is
               28
               shillings
               on
               the
               prickt
               line
               ,
               and
               25
               shillings
               on
               the
               foot
               measure
               ,
               for
               the
               price
               of
               an
               hundred
               ,
               the
               like
               for
               the
               converse
               .
               Otherwise
               ,
               the
               price
               of
               a
               pound
               being
               given
               ,
               the
               rate
               of
               an
               hundred
               is
               readily
               computed
               without
               the
               rule
               .
               For
               considering
               the
               number
               of
               farthings
               is
               in
               the
               price
               of
               a
               pound
               ,
               twice
               that
               number
               of
               shillings
               ,
               and
               once
               that
               number
               of
               pence
               ,
               is
               the
               price
               of
               an
               hundred
               ,
               reckoning
               five
               score
               to
               the
               hundred
               ;
               of
               twice
               that
               number
               of
               shillings
               ,
               and
               once
               
               that
               number
               of
               groats
               ,
               is
               the
               price
               of
               an
               hundred
               ,
               at
               five
               score
               and
               twelve
               to
               the
               hundred
               .
               Ex.
               gr
               .
               At
               three
               half
               pence
               the
               pound
               ,
               the
               number
               of
               farthings
               is
               six
               .
               Therefore
               ,
               twice
               six
               shillings
               ,
               and
               once
               six
               pence
               (
               that
               is
               125.
               6d
               .
               )
               is
               the
               price
               of
               an
               hundred
               ,
               at
               five
               score
               to
               the
               hundred
               .
               Again
               ,
               twice
               six
               shillings
               ,
               and
               once
               six
               groats
               (
               that
               is
               14s
               .
               )
               is
               the
               price
               of
               an
               hundred
               ,
               at
               five
               score
               and
               twelve
               to
               the
               hundred
               .
               But
               of
               this
               enough
               ,
               if
               not
               too
               much
               already
               .
            
          
           
             
               CHAP.
               XII
               .
            
             
               
                 To
                 divide
                 a
                 plot
                 of
                 Ground
                 into
                 any
                 proposed
                 quantities
                 .
                 See
                 Fig.
                 4
              
            
             
               SUppose
               A
               B
               C
               D
               E
               F
               G
               H
               I
               K
               L
               ,
               a
               plot
               of
               ground
               ,
               containing
               54
               acr
               .
               2
               roods
               ,
               28
               poles
               ,
               from
               the
               point
               A
               ,
               I
               am
               required
               to
               shut
               off
               18
               acres
               ,
               next
               the
               side
               B
               C.
               Draw
               A
               D.
               and
               measure
               the
               figure
               A
               B
               C
               D
               ,
               which
               is
               14
               acr
               :
               2
               r.
               3
               p.
               that
               is
               3
               acr
               .
               1.
               r.
               37
               p.
               or
               557
               poles
               too
               little
               .
               Draw
               again
               A
               F
               ,
               and
               measure
               A
               D
               F
               ,
               which
               is
               1309.
               poles
               .
               Then
               by
               Chap.
               5.
               
               Probl.
               1.
               entring
               D
               F
               ,
               the
               base
               of
               the
               triangle
               in
               1309
               on
               the
               lines
               of
               lines
               ,
               and
               taking
               out
               557
               ,
               set
               it
               from
               
               D
               to
               E
               ,
               and
               draw
               A
               E.
               So
               have
               you
               the
               figure
               A
               B
               C
               D
               E
               18
               acres
               .
               Again
               ,
               from
               the
               point
               G
               ,
               I
               would
               set
               off
               20
               acres
               next
               the
               side
               A
               E
               ,
               draw
               A
               G
               ,
               and
               measure
               A
               E
               F
               G
               15
               acr
               .
               or
               .
               2
               pol.
               whereof
               want
               4
               acr
               .
               or
               .
               38
               pol.
               that
               678
               poles
               ,
               then
               draw
               G
               K
               ,
               and
               measure
               G
               A
               K
               1113
               poles
               .
               Lastly
               ,
               by
               Chap.
               5.
               
               Probl.
               I.
               enter
               A
               K
               ,
               the
               base
               of
               the
               triangle
               in
               1113
               upon
               the
               lines
               of
               lines
               ,
               and
               taking
               out
               678
               ,
               set
               it
               from
               A
               ,
               and
               draw
               G
               L.
               So
               have
               you
               the
               figure
               A
               E
               F
               G
               L
               ,
               twenty
               acres
               .
            
             
               How
               ready
               the
               instrument
               would
               be
               for
               surveying
               with
               the
               help
               of
               a
               staff
               ,
               Ballsocket
               ,
               and
               Needle
               ,
               is
               obvious
               to
               any
               one
               that
               considers
               its
               graduated
               into
               180
               degrees
               .
            
          
           
             
               CHAP.
               XIII
               .
            
             
               
                 So
                 much
                 of
                 Geography
                 as
                 concerns
                 finding
                 the
                 distance
                 of
                 any
                 two
                 places
                 upon
                 the
                 Terrestrial
                 Globe
                 .
              
            
             
               HEre
               are
               three
               Cases
               ,
               and
               each
               of
               those
               contains
               the
               same
               number
               of
               propositions
               .
            
             
               
               
                 CASE
                 1.
                 
              
               
                 
                   When
                   the
                   two
                   places
                   differ
                   in
                   latitude
                   only
                   .
                
              
               
                 
                   PROP.
                   1.
                   
                
                 
                   
                     When
                     one
                     place
                     lies
                     under
                     the
                     Equator
                     ,
                     having
                     no
                     Latitude
                     .
                  
                
                 
                   The
                   latitude
                   of
                   the
                   other
                   place
                   turned
                   into
                   miles
                   ,
                   (
                   reckoning
                   60
                   miles
                   ,
                   the
                   usual
                   compute
                   ,
                   for
                   a
                   degree
                   )
                   is
                   the
                   distance
                   sought
                   .
                
              
               
                 
                   PROP.
                   2.
                   
                
                 
                   
                     When
                     both
                     places
                     have
                     the
                     same
                     pole
                     elevated
                     ,
                     viz.
                     North
                     ,
                     or
                     South
                     .
                  
                
                 
                   Take
                   the
                   difference
                   of
                   their
                   Latitudes
                   ,
                   and
                   reckoning
                   60
                   miles
                   for
                   a
                   degree
                   (
                   as
                   before
                   )
                   you
                   have
                   their
                   distance
                   .
                
              
               
                 
                 
                   PROP.
                   3.
                   
                
                 
                   
                     When
                     the
                     two
                     places
                     have
                     different
                     poles
                     elevated
                     ,
                     viz.
                     one
                     North
                     ,
                     the
                     other
                     South
                     .
                  
                
                 
                   Adde
                   two
                   latitudes
                   together
                   ,
                   and
                   that
                   sum
                   turned
                   into
                   miles
                   is
                   the
                   distance
                   .
                
              
            
             
               
                 CASE
                 2.
                 
              
               
                 
                   When
                   the
                   two
                   places
                   differ
                   in
                   Longitude
                   only
                   .
                
              
               
                 
                   PROP.
                   1.
                   
                
                 
                   
                     When
                     neither
                     of
                     them
                     have
                     any
                     Latitude
                     ,
                     but
                     lie
                     both
                     under
                     the
                     Equator
                     .
                  
                
                 
                   Their
                   difference
                   of
                   Longitude
                   turned
                   into
                   miles
                   (
                   as
                   before
                   )
                   is
                   their
                   distance
                   .
                
              
               
                 
                   PROP.
                   2.
                   
                
                 
                   
                     When
                     the
                     two
                     places
                     have
                     the
                     same
                     Pole
                     elevated
                     .
                  
                
                 
                   The
                   proportion
                   is
                   thus
                   .
                
                 
                   As
                   the
                   Radius
                   is
                   to
                   the
                   number
                   60
                   ,
                   so
                   is
                   the
                   cosine
                   of
                   the
                   common
                   latitude
                   to
                   the
                   number
                   of
                   miles
                   for
                   one
                   degree
                   of
                   longitude
                   .
                
                 
                 
                   Multiply
                   this
                   number
                   found
                   by
                   the
                   difference
                   of
                   longitude
                   ,
                   and
                   that
                   product
                   is
                   the
                   distance
                   in
                   miles
                   .
                
              
               
                 
                   PROP.
                   3.
                   
                
                 
                   
                     When
                     the
                     two
                     places
                     have
                     different
                     poles
                     elevated
                     .
                  
                
                 
                   As
                   the
                   Radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   common
                   latitude
                   ,
                   so
                   is
                   the
                   sine
                   of
                   half
                   the
                   difference
                   of
                   longitude
                   to
                   the
                   sine
                   of
                   half
                   the
                   distance
                   .
                
                 
                   Wherefore
                   this
                   sine
                   of
                   half
                   the
                   distance
                   doubled
                   and
                   turned
                   into
                   miles
                   ,
                   is
                   the
                   true
                   distance
                   .
                
              
            
             
               
                 CASE
                 3.
                 
              
               
                 
                   When
                   the
                   two
                   places
                   differ
                   in
                   Longitude
                   and
                   Latitude
                   both
                   .
                
              
               
                 
                   PROP.
                   1.
                   
                
                 
                   
                     When
                     one
                     of
                     the
                     places
                     lies
                     under
                     the
                     equator
                     ,
                     having
                     no
                     Latitude
                     .
                  
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   difference
                   of
                   longitude
                   ,
                   so
                   is
                   the
                   cosine
                   of
                   the
                   latitude
                   to
                   the
                   cosine
                   of
                   the
                   distance
                   .
                
              
               
                 
                 
                   PROP.
                   2.
                   
                
                 
                   
                     When
                     both
                     the
                     places
                     have
                     the
                     same
                     pole
                     elevated
                     .
                  
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   difference
                   in
                   longitude
                   ,
                   so
                   is
                   the
                   co-tangent
                   of
                   the
                   lesser
                   altitude
                   ,
                   to
                   the
                   tangent
                   of
                   a
                   fourth
                   ark
                   .
                
                 
                   Subtract
                   this
                   fourth
                   ark
                   out
                   of
                   the
                   complement
                   of
                   the
                   lesser
                   latitude
                   ,
                   and
                   keep
                   the
                   remain
                   .
                   Then
                   ,
                
                 
                   As
                   the
                   cosine
                   of
                   the
                   fourth
                   ark
                   is
                   to
                   the
                   cosine
                   of
                   the
                   remain
                   ,
                   so
                   is
                   the
                   sine
                   of
                   the
                   lesser
                   latitude
                   to
                   the
                   cosine
                   of
                   the
                   distance
                   .
                
              
               
                 
                   PROP.
                   3.
                   
                
                 
                   
                     When
                     the
                     two
                     places
                     have
                     different
                     poles
                     elevated
                     .
                  
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   difference
                   in
                   longitude
                   ,
                   so
                   is
                   the
                   co-tangent
                   of
                   either
                   latitude
                   to
                   the
                   tangent
                   of
                   a
                   fourth
                   ark
                   .
                
                 
                   Subtract
                   the
                   fourth
                   out
                   of
                   the
                   latitude
                   not
                   taken
                   into
                   the
                   former
                   proportion
                   ,
                   and
                   note
                   the
                   difference
                   .
                   Then
                   ,
                
                 
                   As
                   the
                   cosine
                   of
                   the
                   fourth
                   ark
                   is
                   to
                   the
                   cosine
                   of
                   this
                   difference
                   ,
                   so
                   is
                   the
                   sine
                   of
                   the
                   latitude
                   first
                   taken
                   ,
                   to
                   the
                   cosine
                   of
                   the
                   distance
                   .
                
              
            
          
           
             
             
               CHAP.
               XIV
               .
            
             
               
                 Some
                 uses
                 of
                 the
                 Instrument
                 in
                 Navigation
                 ,
                 or
                 plain
                 Salling
                 .
              
            
             
               HEre
               it
               will
               be
               necessary
               to
               premise
               the
               explication
               of
               some
               terms
               ,
               and
               adjoyn
               two
               previous
               proportions
               .
            
             
               1.
               
               The
               Compass
               being
               a
               circle
               ,
               divided
               into
               32
               equal
               parts
               ,
               called
               rumbs
               ;
               one
               point
               or
               rumb
               is
               11
               d.
               15
               min.
               of
               a
               circle
               from
               the
               meridian
               :
               two
               points
               or
               rumbs
               is
               22
               d.
               30
               min.
               &c.
               of
               the
               rest
               .
            
             
               2.
               
               The
               angle
               which
               the
               needle
               ,
               or
               point
               of
               the
               compass
               under
               the
               needle
               ,
               makes
               with
               the
               meridian
               ,
               or
               North
               and
               South
               line
               is
               called
               the
               course
               or
               rumb
               ;
               but
               the
               angle
               which
               it
               makes
               with
               the
               East
               ,
               and
               West
               line
               ,
               or
               any
               parallel
               ,
               is
               named
               the
               complement
               of
               the
               course
               or
               rumb
               .
            
             
               3.
               
               The
               departure
               is
               the
               longitude
               of
               that
               Port
               from
               which
               you
               set
               sail
               .
            
             
               4.
               
               The
               distance
               run
               ,
               is
               the
               number
               of
               miles
               ,
               or
               leagues
               (
               turned
               into
               degrees
               )
               that
               you
               have
               sailed
               .
            
             
             
               5.
               
               When
               you
               are
               in
               North
               latitude
               ,
               and
               sail
               North-ward
               ,
               adde
               the
               difference
               of
               latitude
               to
               the
               latitude
               you
               sailed
               from
               ;
               and
               when
               you
               are
               in
               North
               latitude
               ,
               and
               sail
               Southward
               ,
               subtract
               the
               difference
               of
               latitude
               from
               the
               latitude
               you
               sailed
               ,
               and
               you
               have
               the
               latitude
               you
               are
               in
               .
            
             
               The
               same
               rule
               is
               to
               be
               observed
               in
               South
               latitude
               .
            
             
               6.
               
               To
               finde
               how
               many
               miles
               answer
               to
               one
               degree
               of
               longitude
               in
               any
               latitude
               .
            
             
               As
               the
               radius
               is
               to
               the
               number
               60
               ,
               so
               is
               the
               cosine
               of
               the
               latitude
               to
               the
               number
               of
               miles
               for
               one
               degree
               .
            
             
               7.
               
               To
               finde
               how
               many
               miles
               answer
               to
               one
               degree
               of
               latitude
               on
               any
               rumb
               .
            
             
               As
               the
               cosine
               of
               the
               rumb
               from
               the
               Meridian
               ,
               is
               to
               the
               number
               60
               ,
               so
               is
               the
               radius
               to
               the
               number
               of
               miles
               .
            
             
               The
               most
               material
               questions
               in
               Navigation
               are
               these
               four
               .
            
             
               First
               ,
               To
               finde
               the
               course
               .
               Secondly
               ,
               The
               distance
               run
               .
               Thirdly
               ,
               The
               difference
               of
               latitude
               .
               Fourthly
               ,
               The
               difference
               in
               longitude
               ;
               and
               any
               two
               of
               these
               being
               given
               ,
               the
               other
               two
               are
               readily
               found
               by
               the
               Square
               and
               Index
               .
            
             
               These
               two
               additional
               rulers
               were
               omitted
               in
               the
               first
               Chapter
               of
               this
               Treatise
               ;
               sith
               
               they
               are
               only
               for
               Navigation
               ,
               and
               large
               Instruments
               of
               two
               ,
               or
               three
               foot
               in
               length
               ,
               which
               made
               me
               judge
               their
               description
               most
               proper
               for
               this
               place
               :
               because
               ,
               such
               as
               intend
               the
               Instrument
               for
               a
               pocket
               companion
               ,
               will
               have
               no
               use
               of
               them
               .
            
             
               The
               square
               is
               a
               flat
               rule
               ,
               having
               a
               piece
               ,
               or
               plate
               fastened
               to
               the
               head
               ,
               that
               it
               may
               slide
               square
               ,
               or
               perpendicular
               to
               the
               outward
               ledge
               of
               the
               fixed
               piece
               .
               It
               hath
               the
               same
               line
               next
               either
               edge
               on
               the
               upper
               side
               ,
               which
               is
               a
               line
               of
               equal
               parts
               ,
               an
               hundred
               ,
               wherein
               is
               equal
               to
               the
               radius
               of
               the
               degrees
               on
               the
               outward
               limb
               of
               the
               moveable
               piece
               .
            
             
               The
               Index
               is
               a
               thin
               brass
               rule
               on
               one
               side
               ,
               having
               the
               same
               scale
               as
               the
               square
               .
               On
               the
               other
               side
               is
               a
               double
               line
               of
               tangents
               ,
               that
               next
               the
               left
               edge
               ,
               being
               to
               a
               smaller
               ;
               that
               next
               the
               right
               edge
               ,
               to
               a
               larger
               radius
               .
            
             
               For
               the
               use
               of
               these
               rulers
               ,
               you
               must
               have
               a
               line
               of
               equal
               parts
               adjoyning
               to
               the
               line
               of
               sines
               on
               the
               fixed
               piece
               ,
               divided
               into
               10
               parts
               ,
               stamped
               with
               figures
               ,
               each
               of
               those
               divided
               into
               10
               parts
               more
               ;
               so
               that
               the
               whole
               line
               is
               divided
               into
               100
               parts
               ,
               representing
               degrees
               .
               Lastly
               ,
               let
               each
               of
               those
               degrees
               be
               sub-divided
               into
               as
               many
               parts
               ,
               as
               the
               largeness
               of
               your
               scale
               will
               permit
               ,
               for
               computing
               the
               minutes
               of
               a
               degree
               .
            
             
             
               The
               Index
               is
               to
               move
               upon
               the
               pin
               on
               the
               fixed
               piece
               (
               where
               you
               hang
               the
               thread
               for
               taking
               altitudes
               )
               and
               that
               side
               of
               the
               Index
               (
               in
               any
               of
               the
               four
               former
               questions
               )
               must
               be
               upward
               ,
               which
               hath
               the
               scale
               of
               equal
               parts
               .
            
             
               The
               square
               is
               to
               be
               slided
               along
               the
               outward
               ledge
               of
               the
               fixed
               piece
               .
               Then
               the
               general
               rules
               are
               these
               .
            
             
               The
               difference
               of
               latitude
               is
               accounted
               on
               the
               line
               of
               equal
               parts
               ,
               adjoyning
               to
               the
               sines
               on
               the
               fixed
               piece
               .
            
             
               The
               difference
               of
               longitude
               is
               numbred
               on
               the
               square
               .
            
             
               The
               distance
               run
               is
               reckoned
               upon
               the
               Index
               .
            
             
               The
               course
               is
               computed
               upon
               the
               degrees
               on
               the
               limb
               from
               the
               head
               toward
               the
               end
               of
               the
               moveable
               piece
               .
            
             
               But
               when
               any
               would
               work
               these
               Problems
               in
               proportions
               ,
               let
               them
               note
               ,
            
             
               The
               distance
               run
               ,
               difference
               of
               longitude
               ,
               and
               difference
               of
               latitude
               ,
               are
               all
               accounted
               on
               the
               line
               of
               numbers
               ;
               the
               rumb
               or
               course
               is
               either
               a
               sine
               or
               tangent
               .
            
             
               This
               premised
               .
               I
               shall
               first
               shew
               how
               to
               resolve
               any
               Problem
               by
               the
               square
               and
               Index
               ;
               and
               next
               adjoyn
               the
               proportions
               for
               the
               use
               of
               such
               as
               have
               only
               pocket
               Instruments
               .
            
             
               
               
                 PROBL.
                 1.
                 
              
               
                 
                   The
                   course
                   and
                   distance
                   run
                   given
                   ,
                   to
                   finde
                   the
                   difference
                   of
                   latitude
                   ,
                   and
                   difference
                   of
                   longitude
                   .
                
              
               
                 Apply
                 the
                 Index
                 to
                 the
                 course
                 reckoned
                 on
                 the
                 limb
                 from
                 the
                 head
                 ,
                 and
                 slide
                 the
                 square
                 along
                 the
                 outward
                 ledge
                 of
                 the
                 fixed
                 piece
                 ,
                 until
                 the
                 fiducial
                 edge
                 intersect
                 the
                 distance
                 run
                 on
                 the
                 fiducial
                 edge
                 of
                 the
                 Index
                 .
                 Then
                 at
                 the
                 point
                 of
                 Intersection
                 you
                 have
                 the
                 difference
                 of
                 longitude
                 upon
                 the
                 square
                 ,
                 and
                 on
                 the
                 line
                 of
                 equal
                 parts
                 on
                 the
                 fixed
                 piece
                 ,
                 the
                 square
                 shows
                 the
                 difference
                 of
                 latitude
                 .
                 The
                 proportion
                 is
                 thus
                 ,
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 distance
                 run
                 ,
                 so
                 is
                 the
                 cosine
                 of
                 the
                 course
                 to
                 the
                 difference
                 of
                 latitude
                 .
                 Again
                 ,
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 distance
                 run
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 course
                 to
                 the
                 difference
                 in
                 longitude
                 .
              
            
             
               
                 PROBL.
                 2.
                 
              
               
                 
                   The
                   course
                   and
                   difference
                   of
                   latitude
                   given
                   to
                   find
                   the
                   distance
                   run
                   ,
                   and
                   difference
                   in
                   longitude
                   .
                
              
               
                 Slide
                 the
                 square
                 to
                 the
                 difference
                 of
                 latitude
                 
                 on
                 the
                 line
                 of
                 equal
                 parts
                 upon
                 the
                 fixed
                 piece
                 ,
                 and
                 set
                 the
                 Index
                 to
                 the
                 course
                 on
                 the
                 limb
                 .
                 Then
                 at
                 the
                 point
                 of
                 intersection
                 of
                 the
                 square
                 and
                 index
                 ,
                 on
                 the
                 square
                 is
                 the
                 difference
                 of
                 longitude
                 ,
                 on
                 the
                 index
                 the
                 distance
                 run
                 .
                 The
                 proportion
                 is
                 thus
                 .
              
               
                 As
                 the
                 cosine
                 of
                 the
                 course
                 is
                 to
                 the
                 difference
                 of
                 latitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 distance
                 run
                 .
                 Again
                 ,
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 sine
                 of
                 the
                 course
                 ,
                 so
                 is
                 the
                 distance
                 run
                 to
                 the
                 difference
                 of
                 longitude
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   The
                   course
                   and
                   difference
                   in
                   longitude
                   given
                   ,
                   to
                   finde
                   the
                   distance
                   run
                   ,
                   and
                   difference
                   of
                   latitude
                   .
                
              
               
                 Apply
                 the
                 Index
                 to
                 the
                 course
                 on
                 the
                 limb
                 ,
                 and
                 the
                 difference
                 of
                 longitude
                 on
                 the
                 square
                 to
                 the
                 fiducial
                 edge
                 of
                 the
                 Index
                 .
                 Then
                 at
                 the
                 point
                 of
                 intersection
                 you
                 have
                 distance
                 run
                 on
                 the
                 index
                 ,
                 and
                 upon
                 the
                 line
                 of
                 equal
                 parts
                 on
                 the
                 fixed
                 piece
                 ,
                 the
                 square
                 shows
                 the
                 difference
                 of
                 latitude
                 .
                 The
                 proportion
                 is
                 thus
                 .
              
               
                 As
                 the
                 sine
                 of
                 the
                 course
                 is
                 to
                 the
                 difference
                 of
                 longitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 distance
                 run
                 .
                 Again
                 ,
              
               
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 distance
                 run
                 ,
                 so
                 is
                 the
                 cosine
                 of
                 the
                 course
                 to
                 the
                 difference
                 of
                 latitude
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   The
                   distance
                   run
                   ,
                   and
                   difference
                   of
                   latitude
                   given
                   to
                   finde
                   the
                   course
                   ,
                   and
                   difference
                   in
                   longitude
                   .
                
              
               
                 Slide
                 the
                 square
                 to
                 the
                 difference
                 of
                 latitude
                 on
                 the
                 line
                 of
                 equal
                 parts
                 on
                 the
                 fixed
                 piece
                 ,
                 and
                 move
                 the
                 index
                 until
                 the
                 distance
                 run
                 numbred
                 thereon
                 ,
                 intersect
                 the
                 fiducial
                 edge
                 of
                 the
                 square
                 ;
                 then
                 at
                 the
                 point
                 of
                 intersection
                 you
                 have
                 the
                 difference
                 of
                 longitude
                 on
                 the
                 square
                 ,
                 and
                 the
                 fiducial
                 edge
                 of
                 the
                 Index
                 on
                 the
                 limb
                 shows
                 the
                 course
                 .
              
               
                 As
                 the
                 distance
                 run
                 is
                 to
                 the
                 difference
                 of
                 latitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 cosine
                 of
                 the
                 course
                 .
                 Again
                 ,
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 distance
                 run
                 ,
                 so
                 is
                 the
                 sine
                 of
                 the
                 course
                 to
                 the
                 difference
                 of
                 longitude
                 .
              
            
             
               
               
                 PROBL.
                 5.
                 
              
               
                 
                   The
                   distance
                   run
                   ,
                   and
                   difference
                   of
                   longitude
                   given
                   to
                   finde
                   the
                   course
                   ,
                   and
                   difference
                   of
                   latitude
                   .
                
              
               
                 Apply
                 the
                 distance
                 run
                 numbred
                 on
                 the
                 fiducial
                 edge
                 of
                 the
                 index
                 ,
                 to
                 the
                 difference
                 of
                 longitude
                 ,
                 reckoned
                 on
                 the
                 fiducial
                 edge
                 of
                 the
                 square
                 .
                 Then
                 on
                 the
                 line
                 of
                 equal
                 parts
                 upon
                 the
                 fixed
                 piece
                 ,
                 the
                 square
                 shows
                 the
                 difference
                 of
                 latitude
                 ,
                 and
                 the
                 index
                 shows
                 the
                 course
                 on
                 the
                 limb
                 .
                 The
                 proportion
                 is
                 thus
                 .
              
               
                 As
                 the
                 distance
                 run
                 is
                 to
                 the
                 difference
                 of
                 longitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 sine
                 of
                 the
                 course
                 .
                 Again
                 ,
              
               
                 As
                 the
                 radius
                 is
                 to
                 the
                 distance
                 run
                 ,
                 so
                 is
                 the
                 cosine
                 of
                 the
                 course
                 ,
                 to
                 the
                 difference
                 of
                 latitude
                 .
              
            
             
               
                 PROBL.
                 6.
                 
              
               
                 
                   The
                   difference
                   of
                   latitude
                   ,
                   and
                   difference
                   of
                   longitude
                   given
                   ,
                   to
                   finde
                   the
                   course
                   ,
                   and
                   distance
                   run
                   .
                
              
               
                 Apply
                 the
                 square
                 to
                 the
                 difference
                 of
                 latitude
                 
                 on
                 the
                 scale
                 of
                 equal
                 parts
                 upon
                 the
                 fixed
                 piece
                 ,
                 and
                 move
                 the
                 index
                 until
                 its
                 fiducial
                 edge
                 intersect
                 the
                 difference
                 of
                 longitude
                 ,
                 reckoned
                 on
                 the
                 square
                 .
                 Then
                 at
                 the
                 point
                 of
                 intersection
                 you
                 have
                 the
                 distance
                 run
                 upon
                 the
                 index
                 ,
                 and
                 the
                 fiducial
                 edge
                 of
                 the
                 index
                 upon
                 the
                 limb
                 shows
                 the
                 course
                 .
                 The
                 proportion
                 is
                 thus
                 .
              
               
                 As
                 the
                 difference
                 of
                 latitude
                 is
                 to
                 the
                 difference
                 of
                 longitude
                 ,
                 so
                 is
                 the
                 radius
                 to
                 the
                 tangent
                 of
                 the
                 course
                 .
                 Again
                 ,
              
               
                 As
                 the
                 sine
                 of
                 the
                 course
                 is
                 to
                 the
                 radius
                 ,
                 so
                 is
                 the
                 difference
                 of
                 longitude
                 to
                 the
                 distance
                 run
                 .
              
            
             
               
                 PROBL.
                 7.
                 
              
               
                 
                   Sailing
                   by
                   the
                   Ark
                   of
                   a
                   great
                   Circle
                   .
                
              
               
                 For
                 this
                 purpose
                 the
                 tangent
                 lines
                 on
                 the
                 index
                 will
                 be
                 a
                 ready
                 help
                 ,
                 using
                 the
                 lesser
                 for
                 small
                 ,
                 and
                 the
                 greater
                 tangent
                 line
                 ,
                 for
                 great
                 latitudes
                 .
                 The
                 way
                 is
                 thus
                 ,
              
               
                 Account
                 the
                 pont
                 60
                 /
                 0
                 ,
                 on
                 the
                 outward
                 limb
                 of
                 the
                 moveable
                 piece
                 to
                 be
                 the
                 point
                 ,
                 or
                 port
                 of
                 your
                 departure
                 ;
                 thereto
                 lay
                 the
                 fiducial
                 edge
                 of
                 the
                 index
                 ,
                 and
                 reckoning
                 the
                 latitude
                 of
                 the
                 Port
                 you
                 departed
                 from
                 upon
                 the
                 index
                 ,
                 strike
                 a
                 pin
                 directly
                 touching
                 
                 it
                 ,
                 into
                 the
                 table
                 your
                 instrument
                 lies
                 upon
                 .
                 This
                 pin
                 shall
                 represent
                 the
                 Port
                 of
                 your
                 departure
                 .
                 Therefore
                 hanging
                 a
                 thread
                 ,
                 or
                 hair
                 ,
                 on
                 the
                 center
                 ,
                 whereon
                 the
                 index
                 moves
                 ;
                 and
                 winding
                 it
                 about
                 this
                 pin
                 .
                 Count
                 the
                 difference
                 of
                 longitude
                 'twixt
                 the
                 port
                 of
                 your
                 departure
                 ,
                 and
                 the
                 Port
                 you
                 sail
                 toward
                 ,
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 toward
                 0
                 /
                 60
                 ,
                 on
                 the
                 loose
                 piece
                 ;
                 and
                 thereto
                 laying
                 the
                 same
                 fiducial
                 edge
                 of
                 the
                 index
                 ,
                 reckon
                 the
                 latitude
                 of
                 this
                 last
                 Port
                 upon
                 the
                 index
                 ,
                 directly
                 touching
                 of
                 it
                 ,
                 strike
                 down
                 another
                 pin
                 upon
                 the
                 table
                 ,
                 and
                 draw
                 the
                 thread
                 strait
                 about
                 this
                 pin
                 fastening
                 it
                 thereto
                 .
                 This
                 done
                 ,
                 the
                 thread
                 betwixt
                 the
                 two
                 pins
                 represents
                 the
                 ark
                 of
                 your
                 great
                 Circle
                 ;
                 and
                 laying
                 the
                 fiducial
                 edge
                 of
                 the
                 index
                 to
                 any
                 degree
                 of
                 difference
                 of
                 longitude
                 accounted
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 ,
                 the
                 thread
                 shows
                 upon
                 the
                 index
                 what
                 latitude
                 you
                 are
                 in
                 ,
                 and
                 how
                 much
                 you
                 have
                 raised
                 ,
                 or
                 depressed
                 the
                 pole
                 since
                 your
                 departure
                 .
                 On
                 the
                 contrary
                 ,
                 laying
                 the
                 latitude
                 you
                 are
                 in
                 (
                 numbred
                 upon
                 the
                 index
                 )
                 to
                 the
                 thread
                 ,
                 the
                 index
                 shows
                 the
                 difference
                 of
                 longitude
                 upon
                 the
                 limb
                 ;
                 counting
                 from
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 .
                 So
                 ,
                 that
                 were
                 it
                 possible
                 to
                 sail
                 exactly
                 by
                 the
                 ark
                 of
                 a
                 great
                 Circle
                 ,
                 it
                 would
                 be
                 no
                 difficulty
                 to
                 determine
                 
                 the
                 longitude
                 in
                 any
                 latitude
                 you
                 make
                 .
              
               
                 But
                 I
                 intend
                 not
                 a
                 treatise
                 of
                 Navigation
                 ;
                 wherefore
                 let
                 it
                 suffice
                 ,
                 that
                 I
                 have
                 already
                 shown
                 how
                 the
                 most
                 material
                 Problems
                 therein
                 ,
                 may
                 easily
                 ,
                 speedily
                 ,
                 and
                 (
                 if
                 the
                 instrument
                 be
                 large
                 )
                 exactly
                 ,
                 be
                 performed
                 by
                 the
                 instrument
                 without
                 the
                 trouble
                 of
                 Calculation
                 ,
                 or
                 Projection
                 .
              
            
          
           
             
               CHAP.
               XV.
               
            
             
               
                 The
                 Projection
                 and
                 Solution
                 of
                 the
                 sixteen
                 Cases
                 in
                 right
                 angled
                 Spherical
                 Triangles
                 by
                 five
                 Cases
                 .
                 See
                 Fig.
                 5.
                 
              
            
             
               THe
               fundamental
               Circle
               N
               B
               Z
               C
               ,
               is
               alwayes
               supposed
               ready
               drawn
               ,
               and
               crossed
               into
               quadrants
               ,
               and
               the
               diameters
               produced
               beyond
               the
               Circle
               .
            
             
               
                 CASE
                 1.
                 
              
               
                 
                   Given
                   both
                   the
                   sides
                   Z
                   D
                   ,
                   and
                   D
                   R
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 By
                 a
                 line
                 of
                 chords
                 ,
                 prick
                 off
                 Z
                 D
                 ,
                 upon
                 the
                 limb
                 ,
                 and
                 draw
                 the
                 diameter
                 D
                 A
                 E.
                 Again
                 ,
                 by
                 a
                 line
                 of
                 tangents
                 ,
                 set
                 half
                 co-tangent
                 
                 D
                 R
                 ,
                 upon
                 A
                 D
                 ,
                 from
                 A
                 to
                 R
                 ,
                 then
                 have
                 you
                 three
                 points
                 ,
                 viz.
                 N
                 R
                 Z
                 ,
                 to
                 draw
                 that
                 ark
                 ,
                 and
                 make
                 up
                 the
                 triangle
                 .
                 The
                 center
                 of
                 which
                 ark
                 always
                 lies
                 on
                 AC
                 ,
                 (
                 produced
                 beyond
                 C
                 ,
                 if
                 need
                 requires
                 )
                 and
                 is
                 found
                 by
                 the
                 intersection
                 of
                 the
                 two
                 arks
                 made
                 from
                 R
                 ,
                 and
                 Z.
                 
              
            
             
               
                 CASE
                 2.
                 
              
               
                 
                   Given
                   one
                   side
                   Z
                   D
                   ,
                   and
                   the
                   hypothenuse
                   Z
                   R
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Prick
                 off
                 Z
                 D
                 ,
                 and
                 draw
                 D
                 A
                 E
                 ,
                 by
                 Case
                 1.
                 
                 Again
                 ,
                 set
                 half
                 the
                 co-tangent
                 Z
                 R
                 ,
                 on
                 the
                 line
                 A
                 Z
                 ,
                 from
                 A
                 to
                 F
                 ,
                 and
                 the
                 tangent
                 Z
                 R
                 ,
                 set
                 from
                 F
                 to
                 P
                 ,
                 with
                 the
                 extent
                 F
                 P
                 ,
                 upon
                 the
                 center
                 P
                 ,
                 draw
                 the
                 ark
                 V
                 F
                 I
                 ,
                 and
                 where
                 it
                 intersects
                 the
                 diameter
                 D
                 A
                 E
                 ,
                 set
                 R
                 ;
                 then
                 have
                 you
                 three
                 points
                 N
                 R
                 Z
                 ,
                 to
                 draw
                 that
                 ark
                 ,
                 as
                 in
                 the
                 former
                 Case
                 .
              
            
             
               
                 CASE
                 3.
                 
              
               
                 
                   Given
                   the
                   Hypothenuse
                   Z
                   R
                   ,
                   and
                   the
                   Angle
                   D
                   Z
                   R
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Prick
                 half
                 the
                 co-tangent
                 D
                 Z
                 R
                 ,
                 from
                 A
                 to
                 S
                 ,
                 and
                 the
                 secant
                 D
                 Z
                 R
                 ,
                 from
                 S
                 to
                 T
                 ,
                 upon
                 the
                 center
                 T
                 ,
                 with
                 the
                 extent
                 T
                 S
                 ,
                 draw
                 the
                 ark
                 N
                 S
                 Z.
                 Again
                 ,
                 by
                 Case
                 2
                 ,
                 draw
                 the
                 
                 ark
                 H
                 F
                 I
                 ,
                 where
                 these
                 two
                 arks
                 intersect
                 each
                 other
                 ,
                 set
                 R.
                 Lastly
                 ,
                 lay
                 a
                 ruler
                 to
                 A
                 R
                 ,
                 and
                 draw
                 the
                 diameter
                 DRAE
                 ,
                 and
                 your
                 triangle
                 is
                 made
                 .
              
            
             
               
                 CASE
                 4.
                 
              
               
                 
                   Given
                   one
                   side
                   Z
                   D
                   ,
                   and
                   its
                   adjacent
                   Angle
                   D
                   Z
                   R
                   ,
                   to
                   project
                   the
                   triangle
                   .
                
              
               
                 Prick
                 off
                 Z
                 D
                 ,
                 and
                 draw
                 D
                 A
                 E
                 ,
                 by
                 Case
                 1.
                 
                 Again
                 ,
                 by
                 Case
                 3
                 ,
                 draw
                 the
                 ark
                 N
                 S
                 Z
                 ,
                 where
                 this
                 ark
                 intersects
                 the
                 diameter
                 DAE
                 ,
                 set
                 R
                 ,
                 and
                 your
                 triangle
                 is
                 made
                 .
                 Or
                 ,
              
               
                 Given
                 the
                 side
                 D
                 R
                 ,
                 and
                 its
                 opposite
                 angle
                 D
                 Z
                 R
                 ,
                 you
                 may
                 project
                 the
                 triangle
                 .
              
               
                 Draw
                 the
                 ark
                 N
                 S
                 Z
                 ,
                 as
                 before
                 .
                 Again
                 ,
                 take
                 half
                 the
                 co-tangent
                 D
                 R
                 ,
                 and
                 with
                 that
                 extent
                 upon
                 A
                 ,
                 the
                 center
                 ,
                 cross
                 the
                 ark
                 N
                 S
                 Z
                 ,
                 setting
                 R
                 ,
                 at
                 the
                 point
                 of
                 intersection
                 .
                 Lastly
                 ,
                 lay
                 a
                 ruler
                 to
                 A
                 R
                 ,
                 and
                 draw
                 the
                 diameter
                 D
                 R
                 A
                 E
                 ,
                 which
                 makes
                 up
                 your
                 triangle
                 .
              
               
                 The
                 triangle
                 projected
                 in
                 any
                 of
                 the
                 four
                 former
                 Cases
                 ,
                 to
                 measure
                 any
                 of
                 the
                 sides
                 ,
                 or
                 angles
                 ,
                 do
                 thus
                 .
              
               
                 First
                 ,
                 the
                 〈◊〉
                 Z
                 D
                 ,
                 is
                 found
                 by
                 applying
                 it
                 to
                 a
                 line
                 of
                 chords
                 .
                 Secondly
                 ,
                 R
                 A
                 ,
                 applyed
                 to
                 a
                 line
                 of
                 tangents
                 ,
                 is
                 half
                 the
                 co-tangent
                 D
                 R.
                 Thirdly
                 ,
                 A
                 S
                 ,
                 applyed
                 to
                 
                 a
                 line
                 of
                 tangents
                 is
                 the
                 co-tangent
                 D
                 Z
                 R.
                 Fourthly
                 ,
                 set
                 half
                 the
                 tangent
                 D
                 Z
                 R
                 ,
                 from
                 A
                 to
                 L
                 ,
                 then
                 is
                 L
                 the
                 pole
                 point
                 ,
                 and
                 laying
                 a
                 ruler
                 to
                 L
                 R
                 ,
                 it
                 cuts
                 the
                 limb
                 at
                 V
                 ,
                 and
                 the
                 ark
                 Z
                 V
                 ,
                 upon
                 the
                 line
                 of
                 chords
                 ,
                 gives
                 the
                 hypothenuse
                 Z
                 R.
                 Fifthly
                 ,
                 prick
                 Z
                 D
                 from
                 C
                 to
                 K
                 ,
                 a
                 ruler
                 laid
                 from
                 R
                 to
                 L
                 ,
                 cuts
                 the
                 limb
                 at
                 G
                 ,
                 then
                 G
                 K
                 ,
                 upon
                 a
                 line
                 of
                 chords
                 ,
                 is
                 the
                 quantity
                 of
                 Z
                 R
                 D.
                 
              
            
             
               
                 CASE
                 .
                 5.
                 
              
               
                 
                   The
                   two
                   oblique
                   angles
                   ,
                   D
                   Z
                   R
                   ,
                   and
                   Z
                   R
                   D
                   given
                   ,
                   to
                   present
                   the
                   Triangle
                   .
                   See
                   Fig.
                   6.
                   
                
              
               
                 This
                 is
                 no
                 more
                 than
                 turning
                 the
                 former
                 triangle
                 .
                 Thus
                 ,
              
               
                 Draw
                 the
                 ark
                 N
                 S
                 Z
                 ,
                 by
                 Case
                 1
                 ,
                 and
                 set
                 half
                 the
                 tangent
                 of
                 that
                 ark
                 from
                 A
                 to
                 L.
                 Again
                 ,
                 set
                 half
                 the
                 co-tangent
                 D
                 R
                 Z
                 ,
                 from
                 A
                 to
                 F
                 ,
                 and
                 the
                 secant
                 of
                 D
                 R
                 Z
                 ,
                 from
                 F
                 to
                 L
                 ,
                 upon
                 the
                 center
                 A
                 ,
                 with
                 the
                 extent
                 A
                 P.
                 Draw
                 the
                 ark
                 P
                 G
                 ,
                 and
                 with
                 the
                 extent
                 F
                 P
                 ,
                 from
                 L
                 ,
                 cross
                 the
                 ark
                 P
                 G
                 in
                 G.
                 Lastly
                 ,
                 upon
                 the
                 Center
                 G
                 ,
                 with
                 the
                 extent
                 G
                 L.
                 Draw
                 the
                 ark
                 R
                 D
                 F
                 L
                 ,
                 and
                 your
                 triangle
                 is
                 made
                 .
              
               
                 The
                 triangle
                 projected
                 you
                 may
                 measure
                 off
                 the
                 sides
                 and
                 hypothenuse
                 .
                 Thus
                 ,
              
               
                 First
                 ,
                 the
                 hypothenuse
                 Z
                 R
                 ,
                 is
                 measured
                 by
                 
                 a
                 line
                 of
                 chords
                 .
                 Secondly
                 ,
                 a
                 ruler
                 laid
                 to
                 L
                 D
                 ,
                 cuts
                 the
                 limb
                 at
                 H
                 ,
                 and
                 Z
                 H
                 ,
                 upon
                 a
                 line
                 of
                 chords
                 ,
                 is
                 the
                 measure
                 of
                 the
                 ark
                 Z
                 D.
                 Thirdly
                 ,
                 draw
                 A
                 G
                 ,
                 and
                 set
                 half
                 the
                 tangent
                 D
                 R
                 Z
                 ,
                 from
                 A
                 to
                 V
                 ,
                 apply
                 a
                 ruler
                 to
                 V
                 D
                 ,
                 it
                 cuts
                 the
                 limb
                 at
                 E
                 ,
                 then
                 R
                 E
                 ,
                 upon
                 a
                 line
                 of
                 chords
                 ,
                 measure
                 the
                 ark
                 R
                 D.
                 
              
               
                 Note
                 .
                 The
                 radius
                 to
                 all
                 the
                 chords
                 ,
                 tangents
                 ,
                 and
                 secants
                 ,
                 used
                 in
                 the
                 projection
                 ,
                 and
                 measuring
                 ,
                 any
                 ark
                 or
                 angle
                 ,
                 is
                 the
                 semidiameter
                 of
                 the
                 fundamental
                 circle
                 .
              
            
          
           
             
               CHAP.
               XVI
               .
            
             
               
                 The
                 projection
                 and
                 solution
                 of
                 the
                 12
                 Cases
                 in
                 oblique
                 angled
                 spherical
                 triangles
                 in
                 six
                 Cases
                 .
                 See
                 Fig.
                 7.
                 
              
            
             
               THe
               fundamental
               circle
               N
               H
               Z
               M
               ,
               is
               alwayes
               supposed
               ready
               drawn
               ,
               and
               crossed
               into
               Quadrants
               ,
               and
               the
               Diameters
               produced
               beyond
               the
               Circle
               .
            
             
               
                 CASE
                 1.
                 
              
               
                 
                   The
                   three
                   sides
                   ,
                   Z
                   P
                   ,
                   P
                   Z
                   ,
                   and
                   Z
                   S
                   ,
                   given
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 By
                 a
                 line
                 of
                 chords
                 prick
                 off
                 Z
                 P
                 ,
                 and
                 
                 draw
                 the
                 diameter
                 P
                 C
                 T
                 ,
                 crossing
                 it
                 at
                 right
                 angles
                 in
                 the
                 center
                 with
                 AE
                 C
                 E
                 ,
                 set
                 half
                 the
                 co-tangent
                 P
                 S
                 ,
                 from
                 C
                 to
                 G
                 ,
                 and
                 he
                 secant
                 P
                 S
                 from
                 C
                 to
                 R
                 ,
                 upon
                 the
                 center
                 R
                 ,
                 with
                 the
                 extent
                 R
                 G
                 ,
                 draw
                 the
                 the
                 ark
                 FGL
                 .
                 Again
                 ,
                 set
                 half
                 the
                 co-tangent
                 Z
                 S
                 ,
                 from
                 C
                 to
                 D
                 ,
                 and
                 the
                 tangent
                 Z
                 S
                 ,
                 from
                 D
                 to
                 O
                 ,
                 with
                 the
                 extent
                 O
                 D
                 ,
                 upon
                 the
                 center
                 O
                 ,
                 draw
                 the
                 ark
                 B
                 D
                 P
                 ,
                 mark
                 where
                 these
                 two
                 arks
                 intersect
                 each
                 other
                 as
                 at
                 S.
                 Then
                 have
                 you
                 three
                 points
                 T
                 S
                 P
                 ,
                 to
                 draw
                 that
                 ark
                 ,
                 and
                 the
                 three
                 points
                 N
                 S
                 P
                 ,
                 to
                 draw
                 that
                 ark
                 ,
                 which
                 make
                 up
                 your
                 triangle
                 .
              
            
             
               
                 CASE
                 2.
                 
              
               
                 
                   Given
                   two
                   sides
                   Z
                   S
                   ,
                   and
                   Z
                   P
                   ,
                   with
                   the
                   comprehended
                   Angle
                   P
                   Z
                   S
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Prick
                 off
                 Z
                 P
                 ,
                 and
                 draw
                 PCT
                 ,
                 and
                 AECE
                 ,
                 and
                 the
                 ark
                 B
                 D
                 P
                 ,
                 by
                 Case
                 1.
                 
                 Again
                 ,
                 set
                 the
                 tangent
                 of
                 half
                 the
                 excess
                 of
                 the
                 angle
                 P
                 Z
                 S
                 above
                 90
                 ,
                 from
                 C
                 to
                 W
                 ,
                 and
                 co-secant
                 of
                 that
                 excess
                 from
                 W
                 to
                 K
                 ,
                 upon
                 the
                 center
                 K
                 ,
                 with
                 the
                 extent
                 K
                 W
                 ;
                 draw
                 the
                 ark
                 N
                 W
                 Z
                 ,
                 which
                 cuts
                 the
                 ark
                 B
                 D
                 P
                 in
                 S.
                 Then
                 have
                 you
                 the
                 three
                 points
                 T
                 S
                 P
                 ,
                 to
                 draw
                 that
                 ark
                 which
                 makes
                 up
                 the
                 triangle
                 .
              
            
             
               
               
                 CASE
                 3.
                 
              
               
                 
                   Two
                   Angles
                   S
                   Z
                   P
                   ,
                   and
                   Z
                   P
                   S
                   ,
                   with
                   the
                   comprehended
                   side
                   Z
                   P
                   ,
                   given
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Prick
                 off
                 Z
                 P
                 ,
                 and
                 draw
                 the
                 lines
                 P
                 C
                 T
                 ,
                 and
                 AE
                 C
                 E
                 ,
                 by
                 Case
                 1
                 ,
                 and
                 the
                 angle
                 NWZ
                 .
                 by
                 Case
                 2.
                 
                 Lastly
                 ,
                 set
                 half
                 the
                 co-tangent
                 ZPS
                 from
                 C
                 to
                 X
                 ,
                 and
                 the
                 secant
                 Z
                 P
                 S
                 ,
                 from
                 X
                 to
                 V
                 ,
                 upon
                 the
                 center
                 V
                 ,
                 with
                 the
                 extent
                 V
                 X
                 ,
                 draw
                 the
                 ark
                 T
                 X
                 S
                 P
                 ,
                 and
                 the
                 triangle
                 is
                 made
                 .
              
            
             
               
                 CASE
                 4.
                 
              
               
                 
                   Two
                   sides
                   ,
                   ZP
                   ,
                   and
                   PS
                   ,
                   with
                   the
                   Angle
                   opposite
                   to
                   one
                   of
                   them
                   SZP
                   ,
                   given
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Prick
                 off
                 ZP
                 ,
                 and
                 draw
                 PCT
                 ,
                 and
                 AECE
                 ,
                 by
                 Case
                 1.
                 and
                 the
                 angle
                 SZP
                 by
                 Case
                 2.
                 
                 Lastly
                 ,
                 by
                 Case
                 1.
                 draw
                 the
                 ark
                 FGL
                 ,
                 and
                 mark
                 where
                 it
                 intersects
                 NWZ
                 ,
                 as
                 at
                 S
                 ,
                 then
                 have
                 you
                 the
                 three
                 points
                 TSP
                 ,
                 to
                 draw
                 that
                 ark
                 ,
                 and
                 make
                 up
                 the
                 triangle
                 .
              
            
             
               
               
                 CASE
                 5.
                 
              
               
                 
                   Two
                   Angles
                   SZP
                   ,
                   and
                   ZPS
                   ,
                   with
                   the
                   side
                   opposite
                   to
                   one
                   of
                   them
                   ZS
                   ,
                   given
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                
              
               
                 Draw
                 the
                 ark
                 BDP
                 ,
                 by
                 Case
                 1.
                 and
                 the
                 ark
                 NWZ
                 ,
                 by
                 Case
                 2.
                 at
                 the
                 intersection
                 of
                 these
                 two
                 arks
                 ,
                 set
                 S
                 ,
                 with
                 the
                 tangent
                 of
                 the
                 angle
                 ZPS
                 ,
                 upon
                 the
                 center
                 C.
                 sweep
                 the
                 ark
                 VΔI
                 .
                 Again
                 ,
                 with
                 the
                 secant
                 of
                 the
                 ark
                 ZPS
                 upon
                 the
                 center
                 S
                 ,
                 cross
                 the
                 ark
                 VΔI
                 ,
                 as
                 at
                 the
                 points
                 V
                 and
                 I.
                 Then
                 in
                 case
                 the
                 hypothenuse
                 is
                 less
                 than
                 a
                 quadrant
                 (
                 as
                 here
                 )
                 the
                 point
                 V
                 ,
                 is
                 the
                 center
                 ,
                 and
                 with
                 the
                 extent
                 VS
                 ,
                 draw
                 the
                 ark
                 TSP
                 ,
                 which
                 makes
                 up
                 the
                 triangle
                 .
                 But
                 in
                 case
                 the
                 hypothenuse
                 is
                 equal
                 to
                 a
                 quadrant
                 ,
                 Δ
                 ,
                 is
                 the
                 center
                 ;
                 if
                 more
                 than
                 a
                 quadrant
                 ,
                 I
                 ,
                 is
                 the
                 center
                 ;
                 in
                 which
                 cases
                 the
                 extent
                 from
                 Δ
                 ,
                 or
                 I
                 ,
                 to
                 S
                 ,
                 is
                 the
                 semidiameter
                 of
                 the
                 ark
                 TSP
                 .
              
            
             
               
                 CASE
                 6.
                 
              
               
                 
                   Three
                   Angles
                   ZPS
                   ,
                   and
                   PZS
                   ,
                   and
                   ZSP
                   ,
                   given
                   ,
                   to
                   project
                   the
                   Triangle
                   .
                   See
                   Fig.
                   7.
                   and
                   8.
                   
                
              
               
                 The
                 angles
                 of
                 any
                 spherical
                 triangle
                 may
                 
                 be
                 converted
                 into
                 their
                 opposite
                 sides
                 ,
                 by
                 taking
                 the
                 complement
                 of
                 the
                 greatest
                 angle
                 to
                 a
                 Semicircle
                 for
                 the
                 hypothenuse
                 ,
                 or
                 greatest
                 side
                 .
                 Wherefore
                 by
                 Case
                 1.
                 make
                 the
                 side
                 ZP
                 ,
                 in
                 Fig.
                 4.
                 equal
                 to
                 the
                 angle
                 ZSP
                 ,
                 in
                 Fig.
                 3
                 ,
                 and
                 the
                 side
                 ZS
                 ,
                 in
                 Fig.
                 4.
                 equal
                 to
                 the
                 angle
                 ZPS
                 ,
                 in
                 Fig.
                 3.
                 and
                 the
                 side
                 PS
                 ,
                 in
                 Fig.
                 4.
                 equal
                 to
                 the
                 complement
                 of
                 the
                 angle
                 PZS
                 ,
                 to
                 a
                 Semicircle
                 in
                 Fig.
                 3.
                 
                 Then
                 is
                 your
                 triangle
                 projected
                 where
                 the
                 angle
                 ZPS
                 in
                 Fig.
                 4.
                 is
                 the
                 side
                 ZS
                 ,
                 Fig.
                 3.
                 
                 Again
                 ,
                 the
                 angle
                 ZSP
                 ,
                 Fig.
                 4.
                 is
                 the
                 side
                 ZP
                 ,
                 in
                 Fig.
                 3.
                 
                 Lastly
                 ,
                 the
                 complement
                 of
                 the
                 angle
                 PZS
                 to
                 a
                 Semicircle
                 in
                 Fig.
                 4.
                 is
                 the
                 measure
                 of
                 the
                 hypothenuse
                 ,
                 or
                 side
                 P
                 S
                 ,
                 in
                 Fig.
                 3.
                 
              
               
                 The
                 Triangle
                 being
                 in
                 any
                 of
                 the
                 former
                 Cases
                 projected
                 ,
                 the
                 quantity
                 of
                 any
                 side
                 or
                 angle
                 may
                 be
                 measured
                 by
                 the
                 following
                 rules
                 .
              
               
                 First
                 ,
                 The
                 side
                 Z
                 P
                 ,
                 is
                 found
                 by
                 applying
                 it
                 to
                 a
                 line
                 of
                 chords
                 .
                 Secondly
                 ,
                 CX
                 ,
                 applyed
                 to
                 a
                 line
                 of
                 tangents
                 ,
                 is
                 half
                 the
                 co-tangent
                 of
                 the
                 angle
                 ZPS
                 .
                 Thirdly
                 ,
                 CW
                 applyed
                 to
                 a
                 line
                 of
                 tangents
                 ,
                 is
                 half
                 the
                 co-tangent
                 of
                 the
                 excess
                 of
                 the
                 angle
                 SZP
                 ,
                 above
                 90.
                 
                 Fourthly
                 ,
                 set
                 half
                 the
                 tangent
                 of
                 the
                 angle
                 ZPS
                 ,
                 from
                 C
                 ,
                 to
                 Π
                 ,
                 a
                 ruler
                 laid
                 to
                 ΠS
                 ,
                 cuts
                 the
                 limb
                 at
                 F
                 ;
                 then
                 PF
                 ,
                 applyed
                 
                 to
                 a
                 line
                 of
                 chords
                 ,
                 gives
                 the
                 side
                 PS
                 .
                 Fifthly
                 ,
                 take
                 the
                 complement
                 of
                 the
                 angle
                 PZS
                 ,
                 to
                 a
                 Semicircle
                 ,
                 and
                 set
                 half
                 the
                 tangent
                 of
                 that
                 complement
                 from
                 C
                 ,
                 to
                 λ
                 ,
                 a
                 ruler
                 laid
                 to
                 λS
                 ,
                 cuts
                 the
                 limb
                 at
                 B
                 ,
                 and
                 ZB
                 ,
                 applyed
                 to
                 a
                 line
                 of
                 chords
                 ,
                 gives
                 the
                 side
                 ZS
                 .
                 Sixthly
                 ,
                 a
                 ruler
                 laid
                 to
                 Sλ
                 ,
                 cuts
                 the
                 limb
                 at
                 L.
                 Again
                 ,
                 a
                 ruler
                 laid
                 to
                 SΠ
                 ,
                 cuts
                 the
                 limb
                 at
                 φ
                 ,
                 and
                 L
                 φ
                 ,
                 applyed
                 to
                 a
                 line
                 of
                 chords
                 ,
                 gives
                 the
                 angle
                 ZSP
                 .
              
            
          
           
             The
             end
             of
             the
             first
             Book
             .
          
        
         
           
           
             An
             Appendix
             to
             the
             first
             Book
             .
          
           
             THe
             sights
             which
             are
             necessary
             for
             taking
             any
             Altitude
             .
             Angle
             ,
             or
             distance
             (
             without
             the
             help
             of
             Thread
             or
             Plummet
             )
             are
             only
             three
             ,
             viz.
             one
             turning
             sight
             ,
             and
             two
             other
             sights
             ,
             contrived
             with
             chops
             ,
             so
             that
             they
             may
             slide
             by
             the
             inward
             or
             outward
             graduated
             limbs
             .
             The
             turning
             sight
             hath
             only
             two
             places
             ,
             either
             the
             center
             at
             the
             head
             ,
             or
             the
             center
             at
             the
             beginning
             of
             the
             line
             of
             sines
             on
             the
             fixed
             piece
             ;
             to
             either
             of
             which
             (
             as
             occasion
             requires
             )
             it
             s
             fastened
             with
             a
             sorne
             .
             The
             center
             at
             the
             head
             serving
             for
             the
             graduations
             next
             the
             inward
             limb
             of
             the
             loose
             piece
             .
             And
             the
             center
             at
             the
             beginning
             of
             the
             line
             of
             sines
             serving
             for
             all
             the
             graduations
             next
             the
             outward
             limb
             of
             the
             moveable
             and
             loose
             piece
             both
             .
             Yet
             because
             it
             is
             requisite
             to
             have
             pins
             to
             keep
             the
             loose
             piece
             close
             in
             its
             place
             .
             You
             may
             have
             two
             sights
             more
             to
             supply
             their
             place
             (
             which
             
             sometimes
             you
             may
             make
             use
             of
             )
             and
             so
             the
             number
             of
             sights
             may
             be
             five
             ,
             viz.
             two
             sliding
             sights
             ,
             one
             turning
             sight
             ,
             and
             two
             pin
             sights
             ,
             to
             put
             into
             the
             holes
             at
             the
             end
             of
             the
             fixed
             and
             moveable
             piece
             ,
             to
             hold
             the
             tenons
             of
             the
             loose
             piece
             close
             joynted
             .
             Every
             one
             of
             these
             sights
             hath
             a
             fiducial
             (
             or
             perpendicular
             )
             line
             ,
             drawn
             down
             the
             middle
             of
             them
             ,
             from
             the
             top
             to
             the
             bottom
             ,
             where
             this
             line
             toucheth
             the
             graduations
             on
             the
             limb
             ,
             is
             the
             point
             of
             observation
             .
          
           
             The
             places
             of
             these
             sights
             have
             an
             oval
             proportion
             ,
             about
             the
             middle
             of
             them
             ,
             only
             leaving
             a
             small
             bar
             of
             brass
             ,
             to
             conduct
             the
             fiducial
             line
             down
             the
             oval
             cavity
             ,
             and
             support
             a
             little
             brass
             knot
             (
             with
             a
             sight
             hole
             in
             it
             )
             in
             the
             middle
             of
             that
             bar
             ,
             which
             is
             ever
             the
             point
             to
             be
             looked
             at
             .
             There
             are
             two
             wayes
             of
             observing
             an
             altitude
             with
             help
             of
             these
             sights
             .
             The
             one
             when
             we
             turn
             our
             face
             toward
             the
             object
             .
             This
             is
             called
             a
             forward
             observation
             in
             which
             you
             must
             alwayes
             set
             the
             turning
             sight
             next
             your
             eye
             .
             This
             way
             of
             observation
             will
             not
             exactly
             give
             an
             altitude
             above
             45
             degrees
             .
          
           
             The
             other
             way
             of
             observing
             an
             altitude
             is
             peculiar
             to
             the
             Sun
             in
             a
             bright
             day
             ,
             when
             we
             turn
             our
             back
             toward
             the
             Sun.
             This
             is
             termed
             a
             backward
             observation
             ;
             wherein
             
             You
             must
             have
             one
             of
             the
             sliding
             sights
             next
             Your
             eye
             ,
             and
             the
             turning
             sight
             toward
             the
             Horizon
             .
             This
             serves
             to
             take
             the
             Suns
             altitude
             without
             thread
             ,
             or
             plummet
             ,
             when
             it
             is
             near
             the
             Zenith
             .
          
           
             
               PROBL.
               1.
               
            
             
               
                 To
                 finde
                 the
                 Suns
                 altitude
                 by
                 a
                 forward
                 observation
                 .
              
            
             
               Serve
               the
               turning
               sight
               to
               the
               center
               of
               those
               graduations
               you
               please
               to
               make
               use
               of
               (
               whether
               on
               the
               inward
               or
               outward
               limb
               )
               and
               place
               the
               two
               sliding
               sights
               upon
               the
               respective
               limb
               to
               that
               center
               ;
               this
               done
               ,
               look
               by
               the
               knot
               of
               the
               turning
               sight
               (
               moving
               the
               instrument
               upward
               or
               downward
               )
               until
               you
               see
               the
               knot
               of
               one
               of
               the
               sliding
               sights
               directly
               against
               the
               Sun
               ,
               then
               move
               the
               other
               sliding
               sight
               ,
               until
               the
               knot
               of
               the
               turning
               sight
               ,
               and
               the
               knot
               of
               this
               other
               sliding
               sight
               be
               against
               the
               horizon
               ;
               then
               the
               degrees
               intercepted
               'twixt
               the
               fiducial
               lines
               of
               the
               sliding
               sights
               on
               the
               limb
               ,
               shew
               the
               altitude
               required
               .
            
          
           
             
             
               PROBL.
               2.
               
            
             
               
                 To
                 finde
                 the
                 distance
                 of
                 any
                 two
                 Stars
                 ,
                 &c.
                 by
                 a
                 forward
                 observation
                 .
              
            
             
               Serve
               the
               turning
               sight
               to
               either
               center
               ,
               and
               apply
               the
               two
               sliding
               sights
               to
               the
               respective
               limb
               (
               holding
               the
               instrument
               with
               the
               proportional
               side
               downward
               )
               and
               applying
               the
               turning
               sight
               to
               your
               eye
               ,
               so
               move
               the
               two
               sliding
               sights
               either
               nearer
               together
               ,
               or
               further
               asunder
               ,
               that
               you
               may
               by
               the
               knot
               of
               the
               turning
               sight
               see
               both
               objects
               even
               with
               the
               knots
               of
               their
               respective
               sliding
               sights
               ,
               then
               will
               the
               degrees
               intercepted
               'twixt
               the
               fiducial
               lines
               of
               the
               object
               sights
               on
               the
               limb
               show
               the
               true
               distance
               .
               By
               this
               means
               you
               may
               take
               any
               angle
               for
               surveying
               ,
               &c.
               
            
          
           
             
               PROBL.
               3.
               
            
             
               
                 To
                 finde
                 the
                 Suns
                 Altitude
                 by
                 a
                 backward
                 Observation
                 .
              
            
             
               Serve
               the
               turning
               sight
               to
               the
               center
               at
               the
               beginning
               of
               the
               line
               of
               sines
               ,
               and
               apply
               one
               of
               the
               sliding
               sights
               to
               the
               outward
               
               limb
               of
               the
               loose
               piece
               ,
               and
               the
               other
               to
               the
               outward
               limb
               of
               the
               moveable
               piece
               ;
               and
               turning
               your
               back
               toward
               the
               Sun
               ,
               set
               the
               sliding
               sight
               upon
               the
               moveable
               piece
               next
               your
               eye
               ;
               and
               slide
               it
               upward
               or
               downward
               toward
               the
               end
               ,
               or
               head
               ,
               until
               you
               see
               the
               shadow
               of
               the
               little
               bar
               ,
               or
               edge
               ,
               of
               the
               sight
               on
               the
               loose
               piece
               fall
               directly
               on
               the
               little
               bar
               on
               the
               turning
               sight
               ;
               and
               at
               the
               same
               time
               the
               bar
               of
               the
               sight
               next
               your
               eye
               ,
               and
               the
               bar
               of
               the
               turning
               sight
               to
               be
               in
               a
               direct
               line
               with
               the
               Horizon
               .
               Then
               will
               the
               degrees
               on
               the
               limb
               intercepted
               'twixt
               the
               fiducial
               lines
               of
               the
               sliding
               sights
               (
               if
               you
               took
               the
               shadow
               of
               the
               bar
               )
               or
               'twixt
               the
               fiducial
               line
               of
               the
               sliding
               sight
               next
               your
               eye
               ,
               and
               the
               edge
               of
               the
               other
               sliding
               sight
               (
               when
               you
               took
               the
               shadow
               of
               the
               edge
               )
               be
               the
               true
               altitude
               required
               .
            
          
        
         
           
           
             ΣΚΙΟΓΡΑΦΙΑ
             ,
             OR
             ,
             The
             Art
             of
             Dyalling
             for
             any
             plain
             Superficies
             .
          
           
             LIB
             .
             II.
             
          
           
             
               CHAP.
               I.
               
            
             
               
                 The
                 distinction
                 of
                 Plains
                 ,
                 with
                 Rules
                 for
                 knowing
                 of
                 them
                 .
              
            
             
               ALL
               plain
               Superficies
               are
               either
               horizontal
               ,
               or
               such
               as
               make
               Angles
               with
               the
               Horizon
               .
            
             
               Horizontal
               plains
               are
               those
               ,
               that
               lie
               upon
               an
               exact
               level
               ,
               or
               flat
               .
            
             
               Plains
               ,
               that
               make
               Angles
               with
               the
               Horizon
               are
               of
               three
               sorts
               .
            
             
             
               1.
               
               Such
               as
               make
               right
               angles
               with
               the
               Horizon
               ,
               generally
               known
               by
               the
               name
               of
               erect
               ,
               or
               upright
               plains
               .
            
             
               2.
               
               Such
               as
               make
               acute
               angles
               with
               the
               horizon
               ,
               or
               have
               their
               upper
               edge
               leaning
               toward
               you
               ,
               usually
               termed
               inclining
               plains
               .
            
             
               3.
               
               Such
               as
               make
               obtuse
               angles
               with
               the
               horizon
               ,
               or
               have
               their
               upper
               edge
               falling
               from
               you
               ,
               commonly
               called
               reclining
               plains
               .
            
             
               All
               these
               three
               sorts
               are
               either
               direct
               ,
               viz.
               East
               ,
               West
               ,
               North
               ,
               South
               .
               Or
               else
               
                 
                   Declining
                
                 
                   From
                   South
                   ,
                   toward
                   East
                   ,
                   or
                   West
                   .
                
                 
                   From
                   North
                   ,
                   toward
                   East
                   ,
                   or
                   West
                   .
                
              
            
             
               All
               plain
               Superficies
               whatsoever
               are
               comprized
               under
               one
               of
               these
               terms
               .
               But
               before
               we
               treat
               of
               the
               affections
               ,
               or
               delineation
               of
               Dials
               for
               them
               ;
               it
               will
               be
               requisire
               to
               acquaint
               you
               with
               the
               nature
               of
               any
               plain
               ,
               which
               may
               be
               found
               by
               the
               following
               Problems
               .
            
             
               
                 PROBL.
                 1.
                 
              
               
                 
                   To
                   finde
                   the
                   reclination
                   of
                   any
                   Plain
                   .
                
              
               
                 Apply
                 the
                 outward
                 ledge
                 of
                 the
                 moveable
                 piece
                 to
                 the
                 Plain
                 with
                 the
                 head
                 upward
                 ,
                 
                 and
                 reckoning
                 what
                 number
                 of
                 degrees
                 the
                 thread
                 cuts
                 on
                 the
                 limb
                 (
                 beginning
                 your
                 account
                 at
                 30.
                 on
                 the
                 loose
                 piece
                 ,
                 and
                 continuing
                 it
                 toward
                 60
                 /
                 0
                 on
                 the
                 moveable
                 piece
                 )
                 you
                 have
                 the
                 angle
                 of
                 reclination
                 .
                 If
                 the
                 thread
                 falls
                 directly
                 on
                 60
                 /
                 0
                 upon
                 the
                 moveable
                 piece
                 ,
                 it
                 s
                 an
                 horizontal
                 ;
                 if
                 on
                 30.
                 on
                 the
                 loose
                 piece
                 ,
                 it
                 s
                 an
                 erect
                 plain
                 .
              
            
             
               
                 PROBL.
                 2.
                 
              
               
                 
                   To
                   finde
                   the
                   inclination
                   of
                   any
                   Plain
                   .
                
              
               
                 Apply
                 the
                 outward
                 ledge
                 of
                 the
                 fixed
                 piece
                 to
                 the
                 plain
                 ,
                 with
                 the
                 head
                 upward
                 ,
                 and
                 what
                 number
                 of
                 degrees
                 the
                 thread
                 cuts
                 upon
                 the
                 limb
                 of
                 the
                 loose
                 piece
                 ,
                 is
                 the
                 complement
                 of
                 the
                 plains
                 inclination
                 .
              
            
             
               
                 PROBL.
                 3.
                 
              
               
                 
                   To
                   draw
                   an
                   Horizontal
                   Line
                   upon
                   any
                   Plain
                   .
                
              
               
                 Apply
                 the
                 proportional
                 side
                 of
                 the
                 Instrument
                 to
                 the
                 plain
                 ,
                 and
                 move
                 the
                 ends
                 of
                 the
                 fixed
                 piece
                 upward
                 ,
                 or
                 downward
                 ,
                 until
                 the
                 thread
                 falls
                 directly
                 on
                 60
                 /
                 0.
                 upon
                 the
                 loose
                 piece
                 ;
                 then
                 drawing
                 a
                 line
                 by
                 the
                 outward
                 
                 ledge
                 of
                 the
                 fixed
                 piece
                 ,
                 its
                 horizontal
                 ,
                 or
                 paralel
                 to
                 the
                 horizon
                 .
              
            
             
               
                 PROBL.
                 4.
                 
              
               
                 
                   To
                   draw
                   a
                   perpendicular
                   Line
                   upon
                   any
                   Plain
                   .
                
              
               
                 When
                 the
                 Sun
                 shineth
                 hold
                 up
                 a
                 thread
                 with
                 a
                 plummet
                 against
                 the
                 plain
                 ,
                 and
                 make
                 two
                 points
                 at
                 any
                 distance
                 in
                 the
                 shadow
                 of
                 the
                 thread
                 upon
                 the
                 plain
                 ,
                 lay
                 a
                 ruler
                 to
                 these
                 points
                 ,
                 and
                 the
                 line
                 you
                 draw
                 is
                 a
                 perpendicular
                 .
              
            
             
               
                 PROBL.
                 5.
                 
              
               
                 
                   To
                   finde
                   the
                   declination
                   of
                   any
                   Plain
                   .
                
              
               
                 Apply
                 the
                 outward
                 ledge
                 of
                 the
                 fixed
                 piece
                 to
                 the
                 horizontal
                 line
                 of
                 your
                 plain
                 ,
                 holding
                 your
                 instrument
                 paralel
                 to
                 the
                 horizon
                 .
                 This
                 done
                 ,
                 lift
                 up
                 the
                 thread
                 and
                 plummer
                 ,
                 until
                 the
                 shadow
                 of
                 the
                 thread
                 fall
                 directly
                 upon
                 the
                 pin
                 hole
                 on
                 the
                 fixed
                 piece
                 (
                 where
                 you
                 hang
                 the
                 thred
                 to
                 take
                 altitudes
                 )
                 Then
                 observe
                 how
                 many
                 degrees
                 the
                 shadow
                 of
                 the
                 thread
                 cuts
                 in
                 the
                 limb
                 ,
                 either
                 from
                 the
                 right
                 hand
                 ,
                 or
                 from
                 the
                 left
                 hand
                 0
                 /
                 60.
                 upon
                 the
                 loose
                 piece
                 ;
                 and
                 immediately
                 taking
                 
                 the
                 altitude
                 of
                 the
                 Sun.
                 By
                 lib.
                 1.
                 cap.
                 2.
                 
                 Probl.
                 9
                 ,
                 10
                 ,
                 11.
                 finde
                 the
                 Suns
                 Azimuth
                 from
                 South
                 .
                 And
                 ,
              
               
                 When
                 you
                 make
                 this
                 observation
                 in
                 the
                 morning
                 ,
                 these
                 Cases
                 determine
                 the
                 declination
                 of
                 the
                 plain
                 .
              
               
                 
                   CASE
                   1.
                   
                
                 
                   When
                   the
                   shadow
                   of
                   the
                   thread
                   upon
                   the
                   limb
                   falls
                   on
                   the
                   right
                   hand
                   0
                   /
                   60
                   on
                   the
                   loose
                   piece
                   ,
                   take
                   the
                   difference
                   of
                   the
                   shadow
                   ,
                   and
                   Azimuth
                   (
                   by
                   subtracting
                   the
                   lesse
                   out
                   of
                   the
                   greater
                   )
                   and
                   the
                   residue
                   or
                   remain
                   is
                   the
                   plains
                   declination
                   .
                   From
                   South
                   toward
                   East
                   ,
                   when
                   the
                   Azimuth
                   is
                   greater
                   than
                   the
                   shadow
                   .
                   From
                   South
                   toward
                   West
                   ,
                   when
                   the
                   shadow
                   is
                   greater
                   than
                   the
                   azimuth
                   ,
                   when
                   the
                   shadow
                   and
                   azimuth
                   are
                   equal
                   ,
                   it
                   s
                   a
                   direct
                   South
                   plain
                   .
                   When
                   the
                   difference
                   is
                   just
                   90.
                   its
                   a
                   direct
                   East
                   ,
                   when
                   above
                   90.
                   subtract
                   the
                   difference
                   from
                   180.
                   and
                   the
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   East
                   .
                
              
               
                 
                   CASE
                   2.
                   
                
                 
                   When
                   the
                   shadow
                   falls
                   on
                   the
                   left
                   hand
                   0
                   /
                   60.
                   
                   Adde
                   the
                   azimuth
                   and
                   shadow
                   together
                   ,
                   
                   that
                   sum
                   is
                   the
                   plains
                   declination
                   ;
                   from
                   South
                   toward
                   East
                   ,
                   when
                   under
                   90
                   ;
                   if
                   it
                   be
                   just
                   90
                   ,
                   it
                   s
                   a
                   direct
                   East
                   .
                   If
                   above
                   90.
                   subtract
                   it
                   from
                   180.
                   the
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   East
                   .
                   When
                   the
                   sum
                   is
                   above
                   180.
                   subtract
                   180
                   from
                   it
                   ,
                   and
                   the
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   West
                   .
                
              
               
                 
                   CASE
                   3.
                   
                
                 
                   When
                   the
                   shadow
                   falls
                   upon
                   0
                   /
                   60.
                   the
                   azimuth
                   is
                   the
                   plains
                   declination
                   .
                   When
                   under
                   90
                   ,
                   its
                   South-East
                   ,
                   when
                   equal
                   to
                   90.
                   direct
                   East
                   ,
                   when
                   above
                   90.
                   subtract
                   it
                   from
                   180.
                   the
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   East
                   .
                
                 
                   If
                   you
                   make
                   the
                   observation
                   afternoon
                   ,
                   the
                   following
                   Cases
                   will
                   resolve
                   you
                   .
                
              
               
                 
                   CASE
                   4.
                   
                
                 
                   When
                   the
                   shadow
                   falls
                   on
                   the
                   left
                   hand
                   0
                   /
                   60.
                   the
                   difference
                   'twixt
                   the
                   shadow
                   and
                   azimuth
                   is
                   the
                   declination
                   ;
                   when
                   the
                   shadow
                   is
                   more
                   than
                   the
                   azimuth
                   it
                   declines
                   South-East
                   ,
                   when
                   less
                   ,
                   South-West
                   .
                   When
                   the
                   shadow
                   and
                   azimuth
                   are
                   equal
                   ,
                   it
                   s
                   a
                   direct
                   South
                   plain
                   ,
                   when
                   their
                   difference
                   is
                   equal
                   to
                   90.
                   
                   it
                   s
                   a
                   direct
                   West
                   ;
                   when
                   the
                   difference
                   exceeds
                   90.
                   subtract
                   it
                   from
                   180.
                   the
                   remain
                   is
                   the
                   declination
                   North-West
                   .
                
              
               
                 
                   CASE
                   5.
                   
                
                 
                   When
                   the
                   shadow
                   falls
                   on
                   the
                   right
                   hand
                   0
                   /
                   60.
                   take
                   the
                   sum
                   of
                   the
                   shadow
                   and
                   azimuth
                   ,
                   and
                   that
                   is
                   the
                   declination
                   from
                   South
                   toward
                   West
                   ,
                   when
                   under
                   90.
                   when
                   just
                   90.
                   its
                   a
                   direct
                   West
                   plain
                   ;
                   when
                   more
                   than
                   90.
                   subtract
                   it
                   from
                   180.
                   the
                   remain
                   is
                   the
                   declination
                   North-West
                   ;
                   when
                   the
                   sum
                   is
                   above
                   180.
                   subtract
                   180
                   from
                   it
                   ,
                   and
                   the
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   East
                   .
                
              
               
                 
                   CASE
                   6.
                   
                
                 
                   When
                   the
                   shadow
                   falls
                   upon
                   0
                   /
                   60.
                   the
                   azimuth
                   is
                   the
                   quantity
                   of
                   declination
                   .
                   From
                   South
                   toward
                   West
                   ,
                   when
                   under
                   90.
                   when
                   equal
                   to
                   90.
                   its
                   direct
                   West
                   ;
                   when
                   more
                   than
                   90.
                   subtract
                   it
                   from
                   180.
                   
                   The
                   remain
                   is
                   the
                   declination
                   from
                   North
                   toward
                   West
                   .
                
              
            
             
               
               
                 PROBL.
                 6.
                 
              
               
                 
                   To
                   draw
                   a
                   Meridional
                   Line
                   upon
                   a
                   Horizontal
                   Plain
                   .
                
              
               
                 Draw
                 first
                 a
                 circle
                 upon
                 the
                 plain
                 ,
                 and
                 holding
                 up
                 a
                 thread
                 and
                 plummet
                 (
                 when
                 the
                 Sun
                 shines
                 )
                 so
                 that
                 the
                 shadow
                 of
                 the
                 thread
                 may
                 pass
                 through
                 the
                 center
                 of
                 the
                 circle
                 ,
                 make
                 a
                 point
                 in
                 the
                 circumference
                 where
                 the
                 shadow
                 intersects
                 it
                 .
                 At
                 the
                 same
                 time
                 finding
                 the
                 Suns
                 Azimuth
                 from
                 South
                 ,
                 by
                 a
                 line
                 of
                 chords
                 ,
                 set
                 it
                 upon
                 the
                 limb
                 of
                 the
                 circle
                 from
                 the
                 intersection
                 of
                 the
                 shadow
                 toward
                 the
                 South
                 ,
                 and
                 it
                 gives
                 the
                 true
                 South
                 point
                 .
                 Wherefore
                 laying
                 a
                 ruler
                 to
                 this
                 last
                 point
                 and
                 the
                 center
                 ,
                 the
                 line
                 you
                 draw
                 is
                 a
                 true
                 Meridian
                 .
              
            
          
           
             
             
               CHAP.
               II.
               
            
             
               
                 The
                 affections
                 of
                 all
                 sorts
                 of
                 Plains
                 .
              
            
             
               
                 Sect.
                 1.
                 
              
               
                 
                   The
                   affections
                   of
                   an
                   horizontal
                   Plain
                   .
                
              
               
                 1.
                 
                 THe
                 style
                 ,
                 or
                 cock
                 of
                 every
                 horizontal
                 Dial
                 ,
                 is
                 an
                 angle
                 equal
                 to
                 that
                 latitude
                 for
                 which
                 the
                 Dial
                 is
                 made
                 .
              
               
                 2.
                 
                 The
                 place
                 of
                 the
                 style
                 is
                 directly
                 upon
                 the
                 meridional
                 ,
                 or
                 twelve
                 a
                 clock
                 line
                 ,
                 and
                 the
                 angular
                 point
                 must
                 stand
                 in
                 the
                 center
                 of
                 the
                 hour
                 lines
                 .
              
               
                 3.
                 
                 The
                 rule
                 for
                 drawing
                 the
                 hour
                 lines
                 before
                 six
                 in
                 the
                 morning
                 ,
                 is
                 to
                 draw
                 the
                 respective
                 hour
                 lines
                 afternoon
                 beyond
                 the
                 center
                 ;
                 or
                 for
                 the
                 hour
                 lines
                 after
                 six
                 at
                 evening
                 ,
                 draw
                 the
                 respective
                 hour
                 lines
                 in
                 the
                 morning
                 beyond
                 the
                 center
                 .
              
               
                 4.
                 
                 To
                 place
                 an
                 horizontal
                 Dial
                 upon
                 the
                 plain
                 ;
                 first
                 draw
                 a
                 Meridian
                 line
                 upon
                 the
                 plain
                 by
                 Cap.
                 1.
                 
                 Probl.
                 6.
                 and
                 lay
                 in
                 the
                 line
                 of
                 12.
                 exactly
                 thereon
                 ,
                 with
                 the
                 angular
                 point
                 of
                 the
                 style
                 toward
                 the
                 South
                 ,
                 fasten
                 the
                 Dial
                 upon
                 the
                 plain
                 .
              
            
             
               
               
                 Sect.
                 2.
                 
              
               
                 
                   The
                   affections
                   of
                   erect
                   ,
                   direct
                   South
                   and
                   North
                   Plains
                   .
                
              
               
                 1.
                 
                 The
                 style
                 in
                 both
                 these
                 is
                 an
                 angle
                 equal
                 to
                 the
                 complement
                 of
                 the
                 latitude
                 for
                 which
                 the
                 Dial
                 is
                 made
                 to
                 stand
                 upon
                 the
                 Meridian
                 line
                 ,
                 or
                 perpendicular
                 of
                 your
                 plain
                 ,
                 with
                 the
                 angular
                 point
                 in
                 the
                 center
                 of
                 the
                 hour
                 lines
                 ,
                 and
                 that
                 point
                 in
                 South
                 alwayes
                 upward
                 ,
                 in
                 North
                 alwayes
                 downward
                 .
              
               
                 2.
                 
                 To
                 prick
                 off
                 the
                 Dial
                 from
                 your
                 paper
                 draught
                 upon
                 the
                 plain
                 ,
                 lay
                 the
                 hour
                 line
                 of
                 6.
                 and
                 6.
                 upon
                 the
                 plains
                 horizontal
                 ;
                 and
                 applying
                 a
                 ruler
                 to
                 the
                 center
                 ,
                 and
                 each
                 hour
                 line
                 ,
                 transmit
                 the
                 hour
                 lines
                 from
                 your
                 paper
                 draught
                 to
                 the
                 plain
                 .
              
            
             
               
                 Sect.
                 3.
                 
              
               
                 
                   The
                   affections
                   of
                   erect
                   ,
                   direct
                   East
                   and
                   West
                   Plains
                   .
                
              
               
                 1.
                 
                 In
                 both
                 these
                 the
                 style
                 may
                 be
                 a
                 pin
                 or
                 plate
                 ,
                 equal
                 in
                 length
                 or
                 heighth
                 to
                 the
                 radius
                 of
                 the
                 tangents
                 ,
                 by
                 which
                 you
                 draw
                 the
                 Dial.
                 
              
               
               
                 2.
                 
                 The
                 style
                 in
                 both
                 of
                 them
                 is
                 to
                 stand
                 directly
                 upon
                 the
                 hour
                 of
                 six
                 ,
                 and
                 perpendicular
                 to
                 the
                 plain
                 .
              
            
             
               
                 Sect.
                 4.
                 
              
               
                 
                   The
                   affections
                   of
                   erect
                   declining
                   Plains
                   .
                
              
               
                 1.
                 
                 These
                 are
                 of
                 two
                 sorts
                 ,
                 either
                 such
                 as
                 admit
                 of
                 centers
                 to
                 the
                 hour
                 lines
                 and
                 style
                 ,
                 or
                 such
                 as
                 cannot
                 with
                 conveniency
                 (
                 because
                 of
                 the
                 lowness
                 of
                 the
                 style
                 ,
                 and
                 nearness
                 of
                 the
                 hour
                 lines
                 to
                 each
                 other
                 )
                 be
                 drawn
                 with
                 a
                 center
                 to
                 those
                 lines
                 .
                 Of
                 this
                 latter
                 sort
                 are
                 all
                 such
                 plains
                 ,
                 whose
                 style
                 is
                 an
                 angle
                 less
                 than
                 15
                 degrees
                 .
                 For
                 where
                 the
                 angle
                 of
                 the
                 style
                 is
                 more
                 than
                 15
                 degrees
                 ;
                 those
                 Dials
                 may
                 be
                 drawn
                 with
                 a
                 center
                 to
                 the
                 hour
                 lines
                 .
              
               
                 2.
                 
                 In
                 all
                 erect
                 declining
                 plains
                 with
                 centers
                 the
                 Meridian
                 is
                 the
                 plains
                 perpendicular
                 .
                 In
                 those
                 that
                 admit
                 not
                 of
                 centers
                 in
                 their
                 delineation
                 ,
                 the
                 meridian
                 is
                 parallel
                 to
                 the
                 plains
                 perpendicular
                 .
              
               
                 3.
                 
                 In
                 all
                 declining
                 plains
                 the
                 substile
                 ,
                 or
                 line
                 whereon
                 the
                 style
                 is
                 to
                 stand
                 ,
                 must
                 be
                 placed
                 on
                 that
                 side
                 the
                 meridian
                 ,
                 which
                 is
                 contrary
                 to
                 the
                 coast
                 of
                 declination
                 ;
                 and
                 also
                 in
                 such
                 decliners
                 as
                 admit
                 of
                 centers
                 ,
                 the
                 
                 angle
                 'twixt
                 12.
                 and
                 6.
                 is
                 to
                 be
                 set
                 to
                 the
                 contrary
                 coast
                 to
                 that
                 of
                 declination
                 .
              
               
                 4.
                 
                 In
                 all
                 decliners
                 ,
                 without
                 centers
                 ,
                 the
                 inclination
                 of
                 meridians
                 is
                 to
                 be
                 set
                 from
                 the
                 substyle
                 toward
                 the
                 coast
                 of
                 declination
                 .
              
               
                 5.
                 
                 The
                 proportions
                 in
                 all
                 erect
                 decliners
                 ,
                 for
                 finding
                 the
                 height
                 of
                 the
                 style
                 ,
                 the
                 distance
                 of
                 the
                 substyle
                 from
                 the
                 meridian
                 ,
                 the
                 angle
                 of
                 twelve
                 and
                 six
                 ,
                 with
                 the
                 inclination
                 of
                 meridians
                 (
                 all
                 which
                 may
                 be
                 wrought
                 either
                 by
                 the
                 canon
                 ,
                 or
                 exactly
                 enough
                 for
                 this
                 purpose
                 by
                 the
                 instrument
                 )
                 are
                 as
                 followeth
                 .
              
               
                 
                   To
                   finde
                   the
                   Styles
                   height
                   above
                   the
                   Substile
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   latitude
                   ,
                   so
                   is
                   the
                   cosine
                   of
                   declination
                   ,
                   to
                   the
                   sine
                   of
                   the
                   styles
                   height
                   .
                
              
               
                 
                   To
                   finde
                   the
                   Substyles
                   distance
                   from
                   the
                   Meridian
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   sine
                   of
                   declination
                   ,
                   so
                   is
                   the
                   co-tangent
                   of
                   the
                   latitude
                   to
                   the
                   tangent
                   of
                   the
                   substyle
                   from
                   the
                   Meridian
                   .
                
              
               
                 
                 
                   To
                   finde
                   the
                   angle
                   of
                   twelve
                   and
                   six
                   .
                
                 
                   As
                   the
                   co-tangent
                   of
                   the
                   latitude
                   is
                   to
                   the
                   radius
                   ,
                   so
                   is
                   the
                   sine
                   of
                   declination
                   ,
                   to
                   the
                   co-tangent
                   of
                   six
                   from
                   twelve
                   .
                
              
               
                 
                   To
                   finde
                   the
                   inclination
                   of
                   Meridians
                   .
                
                 
                   As
                   the
                   sine
                   of
                   the
                   latitude
                   is
                   to
                   the
                   radius
                   ,
                   so
                   is
                   the
                   tangent
                   of
                   declination
                   to
                   the
                   tangent
                   of
                   inclination
                   of
                   Meridians
                   .
                
                 
                   6.
                   
                   All
                   North
                   decliners
                   with
                   centers
                   have
                   the
                   angular
                   point
                   of
                   the
                   style
                   downward
                   ,
                   and
                   all
                   South
                   have
                   it
                   upward
                   .
                
                 
                   7.
                   
                   All
                   North
                   decliners
                   without
                   centers
                   have
                   the
                   narrowest
                   end
                   of
                   the
                   style
                   downward
                   ,
                   all
                   South
                   have
                   it
                   upward
                   .
                
                 
                   8.
                   
                   In
                   all
                   decliners
                   without
                   centers
                   take
                   so
                   much
                   of
                   the
                   style
                   as
                   you
                   think
                   convenient
                   ,
                   but
                   make
                   points
                   at
                   its
                   beginning
                   and
                   end
                   upon
                   the
                   substyle
                   of
                   your
                   paper
                   draught
                   ,
                   and
                   transmit
                   those
                   points
                   to
                   the
                   substyle
                   of
                   your
                   plain
                   ,
                   for
                   direction
                   in
                   placing
                   your
                   style
                   thereon
                   .
                
                 
                   9.
                   
                   In
                   all
                   North
                   decliners
                   the
                   Meridian
                   ,
                   or
                   inclination
                   of
                   Meridians
                   is
                   the
                   hour
                   line
                   of
                   twelve
                   at
                   mid-night
                   :
                   in
                   South
                   decliners
                   ,
                   
                   at
                   noon
                   ,
                   or
                   mid-day
                   .
                   This
                   may
                   tell
                   you
                   the
                   true
                   names
                   of
                   the
                   hour
                   lines
                   .
                
                 
                   10.
                   
                   In
                   transmitting
                   these
                   Dials
                   from
                   your
                   paper
                   draught
                   to
                   your
                   plain
                   ,
                   lay
                   the
                   horizontal
                   of
                   your
                   paper
                   draught
                   ,
                   upon
                   the
                   horizontal
                   line
                   of
                   the
                   plain
                   ,
                   and
                   prick
                   off
                   the
                   hours
                   and
                   substyle
                   .
                
              
            
             
               
                 Sect.
                 5.
                 
              
               
                 
                   The
                   affections
                   of
                   direct
                   reclining
                   Plains
                   inclining
                   Plains
                   .
                
              
               
                 
                   For
                   South
                   Recliners
                   ,
                   and
                   North
                   Incliners
                   .
                
                 
                   1.
                   
                   The
                   difference
                   'twixt
                   your
                   co-latitude
                   ,
                   and
                   the
                   reclination
                   inclination
                   is
                   the
                   elevation
                   ,
                   or
                   height
                   of
                   the
                   style
                   .
                
                 
                   2.
                   
                   When
                   the
                   reclination
                   inclination
                   exceeds
                   your
                   co-latitude
                   ,
                   the
                   contrary
                   pole
                   is
                   elevated
                   so
                   much
                   as
                   the
                   excess
                   .
                   Ex.
                   gr
                   .
                   a
                   North
                   recliner
                   ,
                   or
                   South
                   incliner
                   50.
                   d.
                   in
                   lat
                   .
                   52.
                   30.
                   min.
                   the
                   excess
                   of
                   the
                   reclination
                   inclination
                   ,
                   to
                   your
                   co-latitude
                   is
                   12.
                   d.
                   30.
                   min.
                   and
                   so
                   much
                   the
                   North
                   is
                   elevated
                   on
                   the
                   recliner
                   ,
                   and
                   the
                   South
                   pole
                   on
                   the
                   incliner
                   .
                
                 
                 
                   3.
                   
                   When
                   the
                   reclination
                   inclination
                   is
                   equal
                   to
                   the
                   co-latitude
                   ,
                   it
                   s
                   a
                   polar
                   plain
                   .
                
              
               
                 
                   For
                   South
                   incliners
                   ,
                   and
                   North
                   recliners
                   .
                
                 
                   1.
                   
                   The
                   Sum
                   of
                   your
                   co-latitude
                   ,
                   and
                   the
                   reclination
                   inclination
                   is
                   the
                   styles
                   elevation
                   .
                
                 
                   2.
                   
                   When
                   the
                   reclination
                   inclination
                   is
                   equal
                   to
                   your
                   latitude
                   ,
                   it
                   s
                   an
                   equinoctial
                   plain
                   ,
                   and
                   the
                   Dial
                   is
                   no
                   more
                   than
                   a
                   circle
                   divided
                   into
                   24.
                   equal
                   parts
                   ,
                   having
                   a
                   wyer
                   of
                   any
                   convenient
                   length
                   placed
                   in
                   the
                   center
                   perpendicular
                   to
                   the
                   plain
                   for
                   the
                   style
                   .
                
                 
                   3.
                   
                   When
                   the
                   reclination
                   inclination
                   is
                   greater
                   than
                   your
                   latitude
                   ,
                   take
                   the
                   summe
                   of
                   the
                   reclination
                   inclination
                   of
                   your
                   co-latitude
                   from
                   180.
                   and
                   the
                   residue
                   ,
                   or
                   remain
                   is
                   the
                   styles
                   height
                   .
                   But
                   in
                   this
                   case
                   the
                   style
                   must
                   be
                   set
                   upon
                   the
                   plain
                   ,
                   as
                   if
                   the
                   contrary
                   pole
                   was
                   elevated
                   ,
                   viz.
                   These
                   North
                   recliners
                   must
                   have
                   the
                   center
                   of
                   the
                   style
                   upward
                   ,
                   and
                   the
                   South
                   incliners
                   have
                   it
                   downward
                   .
                
                 
                 
                   Note
                   .
                   In
                   all
                   South
                   re-in-cliners
                   North
                   re-in-cliners
                   ,
                   for
                   their
                   delineation
                   the
                   styles
                   height
                   is
                   to
                   be
                   called
                   the
                   co-latitude
                   ,
                   and
                   then
                   you
                   may
                   draw
                   them
                   as
                   erect
                   direct
                   plains
                   ,
                   for
                   South
                   ,
                   or
                   North
                   (
                   as
                   the
                   former
                   rules
                   shall
                   give
                   them
                   )
                   in
                   that
                   latitude
                   ,
                   which
                   is
                   the
                   complement
                   of
                   the
                   styles
                   height
                   .
                
              
               
                 
                   For
                   direct
                   East
                   and
                   West
                   recliners
                   incliners
                   .
                
                 
                   1.
                   
                   In
                   all
                   East
                   and
                   West
                   recliners
                   incliners
                   ,
                   you
                   may
                   refer
                   them
                   to
                   a
                   new
                   latitude
                   ,
                   and
                   new
                   declination
                   ,
                   and
                   then
                   describe
                   them
                   as
                   erect
                   declining
                   plains
                   .
                
                 
                   2.
                   
                   Their
                   new
                   latitude
                   is
                   the
                   complement
                   of
                   that
                   latitude
                   where
                   the
                   plain
                   stands
                   ,
                   and
                   their
                   new
                   declination
                   is
                   the
                   complement
                   of
                   their
                   reclination
                   inclination
                   :
                   But
                   to
                   know
                   which
                   way
                   you
                   are
                   to
                   account
                   this
                   new
                   declination
                   ,
                   remember
                   all
                   East
                   and
                   West
                   recliners
                   are
                   North-East
                   ,
                   and
                   North-West
                   decliners
                   .
                   All
                   East
                   and
                   West
                   incliners
                   are
                   South-East
                   ,
                   and
                   South-West
                   decliners
                   .
                
                 
                   3.
                   
                   Their
                   new
                   latitude
                   and
                   declination
                   known
                   ,
                   you
                   may
                   by
                   Sect
                   4.
                   par
                   .
                   5.
                   finde
                   the
                   
                   substyle
                   from
                   the
                   Meridian
                   ,
                   height
                   of
                   the
                   style
                   ,
                   angle
                   of
                   twelve
                   and
                   six
                   ,
                   and
                   inclination
                   of
                   Meridians
                   ,
                   using
                   in
                   those
                   proportions
                   the
                   new
                   latitude
                   and
                   new
                   declination
                   instead
                   of
                   the
                   old
                   .
                
                 
                   4.
                   
                   In
                   all
                   East
                   and
                   West
                   recliners
                   incliners
                   with
                   centers
                   ,
                   the
                   Meridian
                   lies
                   in
                   the
                   horizontal
                   line
                   of
                   the
                   plain
                   ;
                   in
                   such
                   as
                   have
                   not
                   centers
                   ,
                   its
                   paralel
                   to
                   the
                   horizontal
                   line
                   .
                
              
            
             
               
                 Sect.
                 6.
                 
              
               
                 
                   The
                   affections
                   of
                   declining
                   reclining
                   Plains
                   inclining
                   Plains
                   .
                
              
               
                 The
                 readiest
                 way
                 for
                 these
                 ,
                 is
                 to
                 refer
                 them
                 to
                 a
                 new
                 latitude
                 ,
                 and
                 new
                 declination
                 ,
                 by
                 the
                 subsequent
                 proportions
                 .
              
               
                 
                   1.
                   
                   To
                   finde
                   the
                   new
                   Latitude
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   plains
                   declination
                   ,
                   so
                   is
                   the
                   co-tangent
                   of
                   the
                   reclination
                   inclination
                   to
                   the
                   tangent
                   of
                   a
                   fourth
                   ark
                   .
                
                 
                   In
                   South
                   recliners
                   North
                   incliners
                   get
                   the
                   difference
                   'twixt
                   this
                   fourth
                   ark
                   ,
                   and
                   the
                   latitude
                   of
                   
                   your
                   place
                   ,
                   and
                   the
                   complement
                   of
                   that
                   difference
                   is
                   the
                   new
                   latitude
                   sought
                   .
                   If
                   the
                   fourth
                   ark
                   be
                   less
                   than
                   your
                   old
                   latitude
                   ,
                   the
                   contrary
                   pole
                   is
                   elevated
                   ;
                   if
                   equal
                   to
                   your
                   old
                   latitude
                   ,
                   it
                   s
                   a
                   polar
                   plain
                   .
                
                 
                   In
                   South
                   incliners
                   North
                   recliners
                   the
                   difference
                   'twixt
                   the
                   fourth
                   ark
                   ,
                   and
                   the
                   complement
                   of
                   your
                   old
                   latitude
                   is
                   the
                   new
                   latitude
                   .
                   If
                   the
                   fourth
                   ark
                   be
                   equal
                   to
                   your
                   old
                   co-latitude
                   ,
                   they
                   are
                   equinoctial
                   plains
                   .
                
              
               
                 
                   2.
                   
                   To
                   finde
                   the
                   new
                   declination
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   cosine
                   of
                   the
                   reclination
                   inclination
                   ,
                   so
                   is
                   the
                   sine
                   of
                   the
                   old
                   declination
                   to
                   the
                   sine
                   of
                   the
                   new
                   .
                
              
               
                 
                   3.
                   
                   To
                   finde
                   the
                   angle
                   of
                   the
                   Meridian
                   with
                   the
                   Horizontal
                   Line
                   of
                   the
                   Plain
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   tangent
                   of
                   the
                   old
                   declination
                   ,
                   so
                   is
                   the
                   sine
                   of
                   reclination
                   inclination
                   ,
                   to
                   the
                   co-tangent
                   of
                   the
                   angle
                   of
                   the
                   meridian
                   with
                   the
                   horizontal
                   line
                   of
                   the
                   plain
                   .
                   
                   This
                   gives
                   the
                   angle
                   for
                   its
                   scituation
                   .
                   Observe
                   ,
                   in
                   
                     
                       
                         North
                         incliners
                      
                       
                         less
                         than
                         a
                         polar
                         ,
                         the
                         Meridian
                         lyes
                         .
                      
                       
                         above
                      
                       
                         That
                         end
                         of
                         the
                         Horizontall
                         Line
                         contrary
                         to
                         the
                         Coast
                         of
                         Declination
                         .
                      
                    
                     
                       
                          
                      
                       
                          
                      
                       
                         below
                      
                       
                          
                      
                    
                     
                       
                         South
                         recliners
                      
                       
                         more
                         than
                         a
                         polar
                         ,
                         the
                         Meridian
                         lyes
                         .
                      
                       
                         below
                      
                       
                         That
                         end
                         of
                         the
                         Horizontall
                         next
                         the
                         Coast
                         of
                         Declination
                         .
                      
                    
                     
                       
                          
                      
                       
                          
                      
                       
                         above
                      
                       
                          
                      
                    
                  
                   
                     
                     
                       
                         North
                         recliners
                      
                       
                         less
                         than
                         an
                         equinoctial
                         ,
                         the
                         Meridian
                         lyes
                      
                       
                         above
                      
                       
                         That
                         end
                         of
                         the
                         Horizontalnext
                         the
                         Coast
                         of
                         Declination
                         .
                      
                    
                     
                       
                          
                      
                       
                          
                      
                       
                         below
                      
                       
                         In
                         North
                         recliners
                         this
                         Meridian
                         is
                         12.
                         at
                         midnight
                         .
                      
                    
                     
                       
                          
                      
                       
                         equal
                         to
                         an
                         equinoctial
                         the
                         Meridian
                         descends
                         below
                         the
                         Horizontal
                         at
                         that
                         end
                         contrary
                         to
                         the
                         coast
                         of
                         Declination
                         ,
                         and
                         the
                         substyle
                         lies
                         in
                         the
                         hour
                         line
                         of
                         six
                         .
                      
                       
                          
                      
                       
                          
                      
                    
                     
                       
                         South
                         incliners
                      
                       
                         more
                         than
                         an
                         equinoctial
                         ,
                         the
                         Meridian
                         lyes
                         .
                      
                       
                         below
                      
                       
                         That
                         end
                         of
                         the
                         horizontal
                         contrary
                         to
                         the
                         declination
                         .
                      
                    
                     
                       
                          
                      
                       
                          
                      
                       
                         above
                      
                       
                         In
                         South
                         incliners
                         this
                         Meridian
                         is
                         only
                         useful
                         for
                         drawing
                         the
                         Dial
                         ,
                         and
                         placing
                         the
                         substyle
                         ,
                         for
                         the
                         hour
                         lines
                         must
                         be
                         drawn
                         through
                         the
                         center
                         to
                         the
                         lower
                         side
                         .
                      
                    
                  
                
                 
                 
                   After
                   you
                   have
                   by
                   the
                   former
                   proportions
                   and
                   rules
                   found
                   the
                   new
                   latitude
                   ,
                   new
                   declination
                   ,
                   the
                   angle
                   and
                   scituation
                   of
                   the
                   meridian
                   ,
                   your
                   first
                   business
                   in
                   delineating
                   of
                   the
                   Dial
                   will
                   be
                   (
                   both
                   for
                   such
                   as
                   have
                   centers
                   ,
                   and
                   such
                   as
                   admit
                   not
                   of
                   centers
                   )
                   to
                   set
                   off
                   the
                   meridian
                   in
                   its
                   proper
                   coast
                   and
                   quantity
                   .
                   This
                   done
                   ,
                   by
                   Sect.
                   4.
                   
                   Paragr
                   .
                   5.
                   of
                   this
                   Chapter
                   ,
                   finde
                   the
                   substyles
                   distance
                   from
                   the
                   Meridian
                   ,
                   the
                   height
                   of
                   the
                   style
                   ,
                   angle
                   of
                   twelve
                   and
                   six
                   (
                   and
                   for
                   Dials
                   without
                   centers
                   ,
                   the
                   inclination
                   of
                   Meridians
                   )
                   in
                   all
                   those
                   proportions
                   ,
                   using
                   your
                   new
                   latitude
                   and
                   new
                   declination
                   ,
                   instead
                   of
                   the
                   old
                   ,
                   and
                   setting
                   them
                   off
                   from
                   the
                   Meridian
                   ,
                   according
                   to
                   the
                   directions
                   in
                   Paragr
                   .
                   3.
                   and
                   4.
                   you
                   may
                   draw
                   the
                   Dials
                   by
                   the
                   following
                   rules
                   ,
                   for
                   erect
                   declining
                   plains
                   .
                   In
                   placing
                   of
                   them
                   ,
                   lay
                   the
                   horizontal
                   line
                   of
                   your
                   paper
                   draught
                   upon
                   the
                   horizontal
                   line
                   of
                   your
                   plain
                   ,
                   and
                   prick
                   off
                   the
                   substyle
                   and
                   hour
                   lines
                   .
                
                 
                   Only
                   observe
                   .
                   That
                   such
                   South-East
                   ,
                   or
                   South-West
                   recliners
                   ,
                   as
                   have
                   the
                   contrary
                   pole
                   elevated
                   must
                   be
                   described
                   as
                   North-East
                   ,
                   and
                   North-West
                   decliners
                   ,
                   and
                   such
                   North-East
                   ,
                   and
                   North-West
                   incliners
                   as
                   have
                   the
                   contrary
                   pole
                   elevated
                   ,
                   must
                   be
                   described
                   as
                   South-East
                   ,
                   and
                   South-West
                   
                   decliners
                   ,
                   which
                   will
                   direct
                   you
                   which
                   way
                   to
                   set
                   off
                   the
                   substyle
                   ,
                   and
                   hour
                   line
                   of
                   six
                   from
                   the
                   Meridian
                   in
                   those
                   oblique
                   plains
                   ,
                   which
                   admit
                   of
                   centers
                   ,
                   or
                   the
                   substyle
                   from
                   the
                   Meridian
                   ,
                   and
                   the
                   inclination
                   of
                   Meridians
                   from
                   the
                   substyle
                   in
                   such
                   as
                   admit
                   not
                   of
                   Centers
                   .
                
                 
                   4.
                   
                   Declining
                   polar
                   plains
                   must
                   have
                   a
                   peculiar
                   calculation
                   for
                   the
                   substyle
                   and
                   inclination
                   of
                   Meridians
                   ,
                   which
                   is
                   thus
                   .
                
              
               
                 
                   To
                   finde
                   the
                   Angle
                   of
                   the
                   Substile
                   with
                   the
                   Horizontal
                   Line
                   of
                   the
                   Plain
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   sine
                   of
                   the
                   polar
                   plains
                   reclination
                   ,
                   so
                   is
                   the
                   tangent
                   of
                   declination
                   to
                   the
                   co-tangent
                   of
                   the
                   substyles
                   distance
                   from
                   the
                   horizontal
                   line
                   of
                   the
                   plain
                   .
                
              
               
                 
                   To
                   finde
                   the
                   inclination
                   of
                   Meridians
                   .
                
                 
                   As
                   the
                   radius
                   is
                   to
                   the
                   sine
                   of
                   the
                   latitude
                   ,
                   so
                   is
                   the
                   tangent
                   of
                   declination
                   to
                   the
                   tangent
                   of
                   inclination
                   of
                   Meridians
                   .
                
                 
                   5.
                   
                   The
                   reclining
                   declining
                   polar
                   hath
                   the
                   substyle
                   lying
                   below
                   that
                   end
                   of
                   the
                   horizontal
                   line
                   that
                   is
                   contrary
                   to
                   the
                   coast
                   of
                   declination
                   .
                   The
                   inclining
                   declining
                   polar
                   
                   hath
                   the
                   substyle
                   lying
                   above
                   that
                   end
                   of
                   the
                   horizontal
                   line
                   ,
                   contrary
                   to
                   the
                   coast
                   of
                   declination
                   .
                
              
            
          
           
             
               CHAP.
               III.
               
            
             
               
                 The
                 delineation
                 of
                 Dials
                 for
                 any
                 plain
                 Superficies
                 .
              
            
             
               HEre
               it
               will
               be
               necessary
               to
               premise
               the
               explication
               of
               some
               few
               terms
               and
               symbols
               ,
               which
               for
               brevity
               sake
               we
               shall
               hereafter
               make
               use
               of
               .
               Ex.
               gr
               .
            
             
               Rad.
               denotes
               the
               radius
               ,
               or
               sine
               90.
               or
               tangent
               45.
               
            
             
               Tang.
               is
               the
               tangent
               of
               any
               arch
               or
               number
               affixed
               to
               it
               .
            
             
               Cos.
               is
               the
               cosine
               of
               any
               arch
               or
               number
               of
               degrees
               ,
               or
               what
               it
               wants
               of
               90.
               
               Ex.
               gr
               .
               cos
               .
               19.
               is
               what
               19.
               wants
               of
               90.
               that
               is
               71.
               
            
             
               Co-tang
               -
               is
               the
               co-tangent
               of
               any
               ark
               or
               number
               affixed
               to
               it
               ,
               or
               what
               it
               wants
               of
               90.
               
               Ex.
               gr
               .
               co-tangent
               30.
               is
               what
               30.
               wants
               of
               90.
               that
               is
               60.
               
            
             
               =
               This
               is
               a
               note
               of
               equality
               in
               lines
               ,
               numbers
               ,
               or
               degrees
               .
               Ex.
               gr
               .
               AB
               =
               CD
               .
               That
               is
               ,
               the
               line
               AB
               .
               is
               equal
               unto
               ,
               or
               of
               the
               same
               length
               as
               the
               line
               CD
               .
               Again
               ,
               ABC
               =
               EFG
               .
               that
               is
               the
               angle
               ABC
               .
               is
               of
               the
               same
               quantity
               ,
               
               or
               number
               of
               degrees
               ,
               as
               the
               angle
               EFG
               .
               Once
               more
               AB
               =
               CD
               =
               FG
               =
               tang
               .
               15.
               
               That
               is
               AB
               .
               and
               CD
               .
               and
               FG.
               are
               all
               of
               the
               same
               length
               ,
               and
               that
               length
               is
               the
               tangent
               of
               15.
               d.
               
            
             
               ♒
               This
               is
               a
               note
               of
               two
               lines
               being
               paralel
               unto
               ,
               or
               equidistant
               from
               each
               other
               .
               Ex.
               gr
               .
            
             
               FI.
               ♒
               RS.
               that
               is
               the
               line
               FL.
               is
               paralel
               unto
               ,
               or
               equi-distant
               from
               the
               line
               RS.
               
            
             
               
                 Sect.
                 1.
                 
              
               
                 
                   To
                   delineate
                   an
                   horizontal
                   Dial.
                   See
                   Fig.
                   9.
                   
                
              
               
                 First
                 draw
                 the
                 square
                 BCDE
                 .
                 of
                 what
                 quantity
                 the
                 plain
                 will
                 permit
                 .
                 Then
                 make
                 AF
                 =
                 AG
                 =
                 HD
                 =
                 HE
                 =
                 sine
                 of
                 the
                 latitude
                 ,
                 and
                 AH
                 =
                 Radius
                 .
                 Enter
                 HD
                 .
                 in
                 tang
                 .
                 45.
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 set
                 off
                 HI
                 =
                 HK
                 =
                 tang
                 .
                 15.
                 and
                 HO
                 =
                 HL
                 =
                 tang
                 .
                 30.
                 
                 Again
                 ,
                 enter
                 FD.
                 in
                 tang
                 .
                 45.
                 and
                 set
                 off
                 FQ
                 =
                 GN
                 =
                 tang
                 .
                 15.
                 and
                 FP
                 =
                 GM
                 =
                 tang
                 .
                 30.
                 
                 This
                 done
                 ,
              
               
                 Draw
                 AQ
                 .
                 AP.
                 AD.
                 AO
                 .
                 AI.
                 for
                 the
                 hour
                 lines
                 of
                 5.
                 4.
                 3.
                 2.
                 1.
                 afternoon
                 .
                 Again
                 ,
              
               
                 Draw
                 AK
                 .
                 AL.
                 AE
                 .
                 AM.
                 AN.
                 for
                 the
                 hour
                 lines
                 of
                 11.
                 10.
                 9.
                 8.
                 7.
                 before
                 noon
                 .
              
               
                 The
                 line
                 FAG
                 .
                 is
                 for
                 six
                 and
                 six
                 .
                 In
                 the
                 same
                 manner
                 you
                 may
                 prick
                 the
                 quarters
                 of
                 an
                 hour
                 ,
                 reckoning
                 three
                 tangents
                 ,
                 and
                 45.
                 
                 minutes
                 ,
                 for
                 a
                 quarter
                 .
                 How
                 to
                 draw
                 the
                 hour
                 lines
                 before
                 ,
                 and
                 after
                 six
                 ,
                 was
                 mentioned
                 ,
                 Chap.
                 2.
                 
                 Sect.
                 1.
                 
              
            
             
               
                 Sect.
                 2.
                 
              
               
                 
                   To
                   describe
                   an
                   erect
                   ,
                   direct
                   South
                   Dial.
                   See
                   Fig.
                   10.
                   
                
              
               
                 Draw
                 ABCD.
                 a
                 rect-angle
                 parallellogram
                 .
                 Then
                 make
                 AE
                 =
                 EB
                 =
                 CF
                 =
                 FD
                 =
                 cos
                 .
                 of
                 your
                 latitude
                 .
                 And
                 EF
                 =
                 AC
                 =
                 BD
                 =
                 sine
                 of
                 your
                 latitude
                 .
                 Enter
                 CF.
                 in
                 tang
                 .
                 45.
                 and
                 lay
                 down
                 FK
                 =
                 FL
                 =
                 tang
                 .
                 15.
                 and
                 FI
                 =
                 FM
                 =
                 tang
                 .
                 30.
                 
                 Again
                 ,
                 enter
                 AC
                 .
                 in
                 tang
                 .
                 45.
                 and
                 lay
                 down
                 AG
                 =
                 BO
                 =
                 15.
                 and
                 AH
                 =
                 BN
                 =
                 tang
                 .
                 30.
                 with
                 a
                 ruler
                 draw
                 the
                 lines
                 EG
                 .
                 EH
                 .
                 EC
                 .
                 EF.
                 EK
                 .
                 for
                 the
                 hour
                 lines
                 of
                 7.
                 8.
                 9.
                 10.
                 11.
                 in
                 the
                 morning
                 .
                 and
                 EO
                 .
                 EN
                 .
                 ED.
                 EM
                 .
                 EL.
                 for
                 5.
                 4.
                 3.
                 2.
                 1.
                 afternoon
                 ,
                 the
                 line
                 AEB
                 .
                 is
                 for
                 six
                 and
                 six
                 .
                 The
                 line
                 EF.
                 for
                 twelve
                 .
              
               
                 The
                 description
                 of
                 a
                 direct
                 North-Dial
                 differs
                 nothing
                 from
                 this
                 ,
                 only
                 the
                 hour
                 lines
                 from
                 Sun
                 rise
                 to
                 six
                 in
                 the
                 morning
                 ,
                 and
                 from
                 six
                 in
                 the
                 evening
                 ,
                 until
                 Sun
                 set
                 ,
                 must
                 be
                 placed
                 thereon
                 ,
                 by
                 drawing
                 the
                 respective
                 morning
                 and
                 evening
                 hours
                 beyond
                 the
                 center
                 as
                 in
                 the
                 horizontal
                 .
                 See
                 Fig.
                 11.
                 
              
            
             
               
               
                 Sect.
                 3.
                 
              
               
                 
                   To
                   describe
                   an
                   erect
                   ,
                   direct
                   East
                   Dial.
                   See
                   Fig.
                   12.
                   
                
              
               
                 Having
                 drawn
                 ABCD.
                 a
                 rectangle
                 paralellogram
                 ,
                 fix
                 upon
                 any
                 point
                 in
                 the
                 lines
                 AB
                 .
                 and
                 CD
                 .
                 for
                 the
                 line
                 of
                 six
                 ,
                 provided
                 the
                 distance
                 from
                 that
                 point
                 to
                 A.
                 being
                 entred
                 radius
                 in
                 the
                 line
                 of
                 tangents
                 ,
                 the
                 distance
                 from
                 thence
                 to
                 B.
                 may
                 not
                 exceed
                 ,
                 nor
                 much
                 come
                 short
                 of
                 the
                 tangent
                 75.
                 
                 This
                 point
                 being
                 found
                 ,
                 enter
                 the
                 distance
                 from
                 thence
                 to
                 A.
                 (
                 which
                 we
                 shall
                 call
                 6.
                 
                 A.
                 )
                 radius
                 in
                 the
                 line
                 of
                 tangents
                 ,
                 and
                 keeping
                 the
                 sector
                 at
                 that
                 gage
                 ,
                 lay
                 down
                 upon
                 the
                 lines
                 AB
                 .
                 and
                 CD
                 .
                 6.
                 11.
                 
                 =
                 tang
                 .
                 75.
                 6.
                 10
                 
                 =
                 tang
                 .
                 60.
                 and
                 6.
                 9
                 
                 =
                 6.
                 
                 A.
                 =
                 tang
                 .
                 45.
                 and
                 6.
                 8.
                 
                 =
                 6.
                 
                 =
                 4.
                 =
                 tang
                 .
                 30.
                 
                 Lastly
                 ,
                 6.
                 7
                 
                 =
                 6.
                 5
                 =
                 tang
                 .
                 15.
                 draw
                 lines
                 from
                 these
                 points
                 on
                 AB
                 .
                 to
                 the
                 respective
                 points
                 on
                 CD
                 .
                 and
                 you
                 have
                 the
                 hours
                 .
              
               
                 To
                 place
                 it
                 on
                 the
                 plain
                 ,
                 draw
                 the
                 angle
                 DCE
                 .
                 =
                 co-latitude
                 ,
                 and
                 laying
                 ED.
                 on
                 the
                 horizontal
                 line
                 of
                 the
                 plain
                 ,
                 prick
                 off
                 the
                 hours
                 .
              
               
                 The
                 same
                 rules
                 serve
                 for
                 delineation
                 of
                 a
                 West
                 Dial
                 ,
                 only
                 as
                 this
                 hath
                 morning
                 ,
                 that
                 must
                 be
                 marked
                 with
                 afternoon
                 hours
                 .
              
            
             
               
               
                 Sect.
                 4.
                 
              
               
                 
                   To
                   describe
                   an
                   erect
                   declining
                   Dial
                   ,
                   having
                   a
                   Center
                   .
                   See
                   Fig.
                   14.
                   
                
              
               
                 Draw
                 the
                 square
                 BCDE
                 ,
                 and
                 make
                 AC
                 =
                 AK
                 of
                 quantity
                 what
                 you
                 please
                 .
                 Again
                 ,
                 draw
                 AG.
                 12.
                 
                 ♒
                 CE.
                 and
                 KHF.
                 ♒
                 AG.
                 12.
                 
                 By
                 a
                 line
                 of
                 chords
                 ,
                 set
                 off
                 the
                 angles
                 of
                 the
                 substyle
                 ,
                 style
                 ,
                 and
                 hour
                 of
                 six
                 from
                 twelve
                 ;
                 having
                 first
                 found
                 these
                 angles
                 by
                 Chap.
                 2.
                 
                 Sect.
                 4.
                 
                 Paragr
                 .
                 5.
                 
                 This
                 done
                 ,
                 make
                 a
                 mark
                 in
                 A
                 6.
                 where
                 it
                 intersects
                 KF
                 -
                 as
                 at
                 H.
                 Then
                 enter
                 AK
                 .
                 in
                 the
                 secant
                 of
                 the
                 plains
                 declination
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 take
                 out
                 the
                 secant
                 of
                 the
                 latitude
                 ,
                 which
                 place
                 from
                 A
                 to
                 G.
                 upon
                 the
                 the
                 line
                 A.
                 12.
                 and
                 again
                 ,
                 from
                 H.
                 (
                 which
                 is
                 the
                 intersection
                 of
                 the
                 paralel
                 FK
                 .
                 with
                 the
                 line
                 of
                 six
                 )
                 unto
                 F.
                 This
                 done
                 ,
                 lay
                 a
                 ruler
                 to
                 the
                 points
                 F.
                 and
                 G.
                 and
                 draw
                 a
                 line
                 until
                 it
                 intersects
                 CE.
                 as
                 FG.
                 3.
                 
                 Lastly
                 ,
              
               
                 Enter
                 GF
                 .
                 in
                 tang
                 .
                 45.
                 and
                 set
                 off
                 GL
                 =
                 GN
                 =
                 tang
                 .
                 15.
                 and
                 GM
                 =
                 GO
                 =
                 tang
                 .
                 30.
                 
                 Again
                 ,
                 enter
                 HF.
                 in
                 tang
                 .
                 45.
                 and
                 set
                 off
                 HR
                 =
                 tang
                 .
                 15.
                 and
                 HP
                 =
                 tang
                 .
                 30.
                 
                 A
                 ruler
                 laid
                 to
                 these
                 points
                 ,
                 and
                 the
                 center
                 ,
                 you
                 may
                 draw
                 the
                 hours
                 lines
                 from
                 six
                 in
                 the
                 morning
                 unto
                 
                 three
                 afternoon
                 .
                 For
                 the
                 other
                 hour
                 lines
                 ,
                 do
                 thus
                 ,
                 Produce
                 the
                 line
                 EC
                 .
                 and
                 likewise
                 HA.
                 beyond
                 the
                 center
                 ,
                 until
                 they
                 intersect
                 each
                 other
                 as
                 at
                 S.
                 Then
                 setting
                 off
                 ST
                 =
                 HR
                 .
                 and
                 S.
                 4.
                 
                 =
                 HP
                 .
                 you
                 have
                 the
                 hour
                 points
                 after
                 three
                 in
                 the
                 afternoon
                 ,
                 until
                 six
                 ,
                 although
                 none
                 are
                 proper
                 beyond
                 the
                 hour
                 line
                 of
                 four
                 ;
                 only
                 by
                 drawing
                 them
                 on
                 the
                 other
                 side
                 the
                 center
                 ,
                 they
                 help
                 you
                 to
                 the
                 hour
                 lines
                 before
                 six
                 in
                 the
                 morning
                 .
              
            
             
               
                 Sect.
                 5.
                 
              
               
                 
                   To
                   describe
                   an
                   erect
                   declining
                   Plain
                   without
                   a
                   Center
                   .
                   See
                   Fig.
                   12.
                   
                
              
               
                 The
                 delineation
                 of
                 these
                 Dials
                 is
                 the
                 most
                 difficult
                 of
                 any
                 ,
                 and
                 therefore
                 I
                 shall
                 be
                 the
                 larger
                 in
                 their
                 description
                 .
              
               
                 1.
                 
                 By
                 Chap.
                 2.
                 
                 S.
                 4.
                 
                 Paragr
                 .
                 5.
                 finde
                 the
                 angles
                 of
                 the
                 style
                 ,
                 substyle
                 ,
                 and
                 inclination
                 of
                 Meridians
                 .
              
               
                 2.
                 
                 Having
                 found
                 the
                 inclination
                 of
                 Meridians
                 ,
                 make
                 the
                 following
                 Table
                 for
                 the
                 distance
                 of
                 every
                 hour
                 line
                 and
                 quarter
                 from
                 the
                 substyle
                 .
                 Where
                 I
                 take
                 for
                 example
                 a
                 North
                 plain
                 declining
                 East
                 72.
                 d.
                 45.
                 min.
                 in
                 lat
                 .
                 52.
                 30.
                 min.
                 
              
               
               
                 
                   
                     
                       Imcl
                       .
                       mer
                       .
                    
                     
                       hou
                       .
                    
                     
                       quart
                       .
                    
                  
                   
                     
                       76
                       =
                       30.
                       
                    
                     
                       12
                    
                     
                       00
                    
                  
                   
                     
                       20
                       =
                       15
                    
                     
                       3
                    
                     
                       ...
                    
                  
                   
                     
                       16
                       =
                       30
                    
                     
                       4
                    
                     
                       00
                    
                  
                   
                     
                       12
                       =
                       45
                    
                     
                       4
                    
                     
                       .
                    
                  
                   
                     
                       09
                       =
                       00
                    
                     
                       4
                    
                     
                       ..
                    
                  
                   
                     
                       05
                       =
                       15
                    
                     
                       4
                    
                     
                       ...
                    
                  
                   
                     
                       01
                       =
                       30
                    
                     
                       5
                    
                     
                       00
                    
                  
                   
                     
                       02
                       =
                       15
                    
                     
                       5
                    
                     
                       .
                    
                  
                   
                     
                       06
                       =
                       00
                    
                     
                       5
                    
                     
                       ..
                    
                  
                   
                     
                       09
                       =
                       45
                    
                     
                       5
                    
                     
                       ...
                    
                  
                   
                     
                       13
                       =
                       30
                    
                     
                       6
                    
                     
                       00
                    
                  
                   
                     
                       17
                       =
                       15
                    
                     
                       6
                    
                     
                       .
                    
                  
                   
                     
                       21
                       =
                       00
                    
                     
                       6
                    
                     
                       ..
                    
                  
                   
                     
                       24
                       =
                       45
                    
                     
                       6
                    
                     
                       ...
                    
                  
                   
                     
                       28
                       =
                       30
                    
                     
                       7
                    
                     
                       00
                    
                  
                   
                     
                       32
                       =
                       15
                    
                     
                       7
                    
                     
                       .
                    
                  
                   
                     
                       36
                       =
                       00
                    
                     
                       7
                    
                     
                       ..
                    
                  
                   
                     
                       39
                       =
                       45
                    
                     
                       7
                    
                     
                       ...
                    
                  
                   
                     
                       43
                       =
                       30
                    
                     
                       8
                    
                     
                       00
                    
                  
                   
                     
                       47
                       =
                       15
                    
                     
                       8
                    
                     
                       .
                    
                  
                   
                     
                       51
                       =
                       00
                    
                     
                       8
                    
                     
                       ..
                    
                  
                   
                     
                       54
                       =
                       45
                    
                     
                       8
                    
                     
                       ...
                    
                  
                   
                     
                       58
                       =
                       30
                    
                     
                       9
                    
                     
                       00
                    
                  
                   
                     
                       62
                       =
                       15
                    
                     
                       9
                    
                     
                       .
                    
                  
                   
                     
                       66
                       =
                       00
                    
                     
                       9
                    
                     
                       ..
                    
                  
                   
                     
                       69
                       =
                       45
                    
                     
                       9
                    
                     
                       ...
                    
                  
                   
                     
                       73
                       =
                       30
                    
                     
                       10
                    
                     
                       00
                    
                  
                
              
               
                 The
                 manner
                 of
                 drawing
                 this
                 Table
                 is
                 thus
                 .
                 The
                 inclination
                 of
                 Meridians
                 (
                 which
                 because
                 its
                 a
                 North
                 decliner
                 ,
                 is
                 twelve
                 at
                 midnight
                 .
                 )
                 I
                 finde
                 76.
                 d.
                 30.
                 min.
                 
                 Now
                 considering
                 the
                 Sun
                 never
                 riseth
                 till
                 more
                 than
                 half
                 an
                 hour
                 after
                 three
                 in
                 this
                 latitude
                 ,
                 I
                 know
                 that
                 one
                 quarter
                 before
                 four
                 is
                 the
                 first
                 line
                 proper
                 for
                 this
                 plain
                 :
                 Therefore
                 reckoning
                 15.
                 d.
                 for
                 an
                 hour
                 ,
                 or
                 3.
                 d.
                 45.
                 m.
                 for
                 a
                 quarter
                 of
                 an
                 hour
                 ,
                 I
                 finde
                 three
                 hours
                 ,
                 three
                 quarters
                 (
                 the
                 distance
                 of
                 a
                 quarter
                 before
                 four
                 from
                 midnight
                 )
                 to
                 answer
                 56.
                 d.
                 15.
                 min.
                 which
                 being
                 subtracted
                 from
                 76.
                 d.
                 30.
                 min.
                 the
                 inclination
                 of
                 Meridians
                 there
                 remains
                 
                 20.
                 d.
                 15.
                 m.
                 for
                 the
                 distance
                 of
                 one
                 quarter
                 before
                 four
                 from
                 the
                 substyle
                 .
                 Again
                 ,
                 from
                 20.
                 d.
                 15.
                 min.
                 subtract
                 3.
                 d.
                 45.
                 min.
                 (
                 the
                 quantity
                 of
                 degrees
                 for
                 one
                 quarter
                 of
                 an
                 hour
                 )
                 and
                 there
                 remains
                 16.
                 d.
                 30.
                 min.
                 for
                 the
                 distance
                 of
                 the
                 next
                 line
                 from
                 the
                 substyle
                 ,
                 which
                 is
                 the
                 hour
                 of
                 four
                 in
                 the
                 morning
                 .
                 Thus
                 for
                 every
                 quarter
                 of
                 an
                 hour
                 continue
                 subtracting
                 3.
                 d.
                 45.
                 min.
                 until
                 your
                 residue
                 or
                 remain
                 be
                 less
                 than
                 3.
                 d.
                 45.
                 min.
                 and
                 then
                 first
                 subtracting
                 that
                 residue
                 out
                 of
                 3.
                 d.
                 45.
                 min.
                 
                 This
                 new
                 residue
                 gives
                 the
                 quantity
                 of
                 degrees
                 for
                 that
                 line
                 on
                 the
                 other
                 side
                 the
                 substyle
                 .
                 Now
                 when
                 you
                 are
                 passed
                 to
                 the
                 other
                 side
                 of
                 the
                 substyle
                 ,
                 continue
                 adding
                 3.
                 d.
                 45.
                 min.
                 to
                 this
                 last
                 remain
                 for
                 every
                 quarter
                 of
                 an
                 hour
                 ,
                 and
                 so
                 make
                 up
                 the
                 table
                 for
                 what
                 hours
                 are
                 proper
                 to
                 the
                 plain
                 .
              
               
                 3.
                 
                 Draw
                 the
                 square
                 ABCD.
                 of
                 what
                 quantity
                 the
                 plain
                 will
                 admit
                 ,
                 and
                 make
                 the
                 angle
                 CAG
                 equal
                 to
                 the
                 angle
                 of
                 the
                 substyle
                 with
                 twelve
                 .
                 Again
                 ,
                 cross
                 the
                 line
                 AG.
                 in
                 any
                 two
                 convenient
                 points
                 ,
                 as
                 E.
                 and
                 F.
                 at
                 right
                 angles
                 by
                 the
                 lines
                 KL
                 .
                 and
                 CM
                 .
              
               
                 4.
                 
                 Take
                 the
                 distance
                 from
                 the
                 center
                 unto
                 45.
                 the
                 radius
                 to
                 the
                 lesser
                 lines
                 of
                 tangents
                 ,
                 which
                 is
                 continued
                 to
                 76.
                 on
                 the
                 Sector
                 side
                 ,
                 enter
                 this
                 distance
                 in
                 45.
                 on
                 the
                 larger
                 lines
                 of
                 tangents
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 
                 gage
                 ,
                 take
                 out
                 the
                 tang
                 .
                 20.
                 d.
                 15.
                 min.
                 (
                 which
                 is
                 the
                 distance
                 of
                 the
                 first
                 line
                 from
                 the
                 substyle
                 )
                 set
                 this
                 from
                 73.
                 d.
                 30.
                 min.
                 (
                 the
                 distance
                 of
                 your
                 last
                 hour
                 line
                 from
                 the
                 substyle
                 ,
                 as
                 you
                 see
                 by
                 the
                 Table
                 )
                 toward
                 the
                 end
                 upon
                 the
                 lesser
                 line
                 of
                 tangents
                 ,
                 and
                 where
                 it
                 toucheth
                 as
                 here
                 at
                 75.
                 05.
                 call
                 that
                 the
                 gage
                 tangent
                 .
              
               
                 5.
                 
                 Enter
                 the
                 whole
                 line
                 KL
                 .
                 in
                 the
                 gage
                 tangent
                 ,
                 which
                 in
                 this
                 example
                 is
                 75.
                 d.
                 05.
                 min.
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 take
                 out
                 the
                 tangent
                 73.
                 30.
                 min.
                 which
                 is
                 your
                 last
                 hour
                 ,
                 and
                 set
                 from
                 L.
                 on
                 the
                 line
                 KL
                 .
                 unto
                 V.
                 Again
                 ,
                 take
                 out
                 the
                 tangent
                 20.
                 d.
                 15.
                 min.
                 which
                 is
                 your
                 first
                 line
                 ,
                 and
                 set
                 it
                 from
                 K.
                 towards
                 V.
                 and
                 if
                 it
                 meet
                 in
                 V.
                 it
                 proves
                 the
                 truth
                 of
                 your
                 work
                 ,
                 and
                 a
                 line
                 drawn
                 through
                 V.
                 paralel
                 unto
                 AG.
                 is
                 the
                 true
                 substyle
                 line
                 .
                 Then
                 keeping
                 the
                 Sector
                 at
                 its
                 former
                 gage
                 ,
                 set
                 off
                 the
                 tangents
                 of
                 the
                 hours
                 ,
                 and
                 quarters
                 (
                 as
                 you
                 finde
                 them
                 in
                 the
                 Table
                 )
                 from
                 V.
                 towards
                 K.
                 and
                 from
                 V.
                 towards
                 L.
                 making
                 points
                 for
                 them
                 in
                 the
                 line
                 KL
                 .
                 Lastly
                 ,
                 enter
                 the
                 radius
                 of
                 your
                 tangents
                 to
                 these
                 hour
                 points
                 in
                 the
                 radius
                 of
                 secants
                 ,
                 and
                 set
                 off
                 the
                 secant
                 of
                 the
                 styles
                 height
                 from
                 V.
                 to
                 T.
                 Thus
                 have
                 you
                 the
                 hour
                 points
                 and
                 style
                 on
                 one
                 line
                 of
                 contingence
                 .
                 To
                 mark
                 them
                 out
                 upon
                 the
                 other
                 line
                 do
                 thus
                 .
              
               
               
                 Set
                 the
                 radius
                 to
                 the
                 hour
                 points
                 upon
                 the
                 former
                 line
                 of
                 contingence
                 ,
                 from
                 h.
                 to
                 p.
                 on
                 the
                 line
                 ChM.
                 and
                 entring
                 hV.
                 as
                 Radius
                 in
                 the
                 line
                 of
                 tangents
                 ,
                 take
                 out
                 the
                 tangent
                 of
                 the
                 styles
                 height
                 ,
                 and
                 set
                 from
                 p.
                 to
                 r.
                 Again
                 ,
                 enter
                 hr.
                 radius
                 in
                 your
                 line
                 of
                 tangents
                 ,
                 and
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 take
                 out
                 the
                 tangents
                 for
                 each
                 hour
                 ,
                 and
                 quarter
                 ,
                 according
                 to
                 the
                 table
                 ,
                 and
                 lay
                 them
                 down
                 from
                 h.
                 to
                 the
                 proper
                 side
                 of
                 the
                 substyle
                 toward
                 C.
                 or
                 M.
                 and
                 applying
                 a
                 ruler
                 to
                 the
                 respective
                 points
                 on
                 KL
                 .
                 and
                 CM
                 .
                 draw
                 the
                 lines
                 for
                 the
                 hours
                 and
                 quarters
                 .
                 Lastly
                 ,
                 enter
                 hr.
                 radius
                 on
                 the
                 lines
                 of
                 secants
                 ,
                 and
                 taking
                 out
                 the
                 secant
                 of
                 the
                 styles
                 height
                 ,
                 set
                 it
                 from
                 h.
                 to
                 S.
                 and
                 draw
                 the
                 line
                 ST
                 .
                 for
                 the
                 style
                 .
              
            
             
               
                 Sect.
                 6.
                 
              
               
                 
                   To
                   describe
                   a
                   direct
                   polar
                   Dial.
                   See
                   Fig.
                   15.
                   
                
              
               
                 Draw
                 BCDE
                 .
                 a
                 rectangle
                 paralellogram
                 ,
                 from
                 the
                 middle
                 of
                 BC.
                 to
                 the
                 middle
                 of
                 DE.
                 draw
                 the
                 line
                 12.
                 or
                 substyle
                 ,
                 appoint
                 what
                 place
                 you
                 please
                 in
                 BC.
                 or
                 CD
                 .
                 for
                 the
                 hour
                 point
                 of
                 7.
                 in
                 the
                 morning
                 ,
                 and
                 5.
                 afternoon
                 .
                 Then
                 ,
              
               
                 Enter
                 12.
                 7
                 
                 =
                 12.
                 5.
                 
                 In
                 the
                 tangent
                 75.
                 and
                 set
                 off
                 12.
                 1.
                 
                 =
                 12.
                 11.
                 
                 =
                 tang
                 .
                 15.
                 and
                 12.
                 2.
                 
                 =
                 12.
                 10
                 =
                 tang
                 .
                 30.
                 and
                 12.
                 3.
                 
                 =
                 12.
                 9.
                 
                 =
                 tang
                 .
                 
                 45.
                 
                 Lastly
                 ,
                 12.
                 4.
                 
                 =
                 12.
                 8.
                 
                 =
                 tang
                 .
                 60.
                 
                 From
                 these
                 points
                 draw
                 the
                 hour
                 lines
                 of
                 7.
                 8.
                 9.
                 10.
                 11.
                 12.
                 1.
                 2.
                 3.
                 4.
                 5.
                 which
                 are
                 all
                 the
                 hours
                 proper
                 for
                 these
                 plains
                 .
              
            
             
               
                 Sect.
                 7.
                 
              
               
                 
                   To
                   draw
                   a
                   declining
                   Polar
                   .
                   See
                   Fig.
                   16.
                   
                
              
               
                 1.
                 
                 By
                 Chap.
                 2.
                 
                 Sect.
                 6.
                 
                 Par.
                 4.
                 finde
                 the
                 inclination
                 of
                 Meridians
                 ,
                 and
                 distance
                 of
                 the
                 substyle
                 from
                 the
                 horizontal
                 .
              
               
                 2.
                 
                 By
                 Chap.
                 3.
                 
                 Sect.
                 5.
                 
                 Par.
                 2.
                 make
                 a
                 Table
                 for
                 the
                 distance
                 of
                 the
                 hour
                 points
                 from
                 the
                 substyle
                 .
              
               
                 3.
                 
                 Draw
                 the
                 square
                 BCDE
                 .
                 Set
                 off
                 the
                 angle
                 CAG
                 .
                 for
                 the
                 substyle
                 ,
                 and
                 cross
                 that
                 substyle
                 line
                 at
                 right
                 angles
                 in
                 any
                 two
                 convenient
                 places
                 ,
                 as
                 at
                 H.
                 and
                 K.
                 with
                 the
                 lines
                 PHS.
                 and
                 RKT.
                 for
                 contingent
                 lines
                 .
              
               
                 4.
                 
                 Take
                 any
                 convenient
                 length
                 for
                 your
                 styles
                 height
                 ,
                 and
                 enter
                 it
                 radius
                 in
                 your
                 line
                 of
                 tangents
                 ,
                 keeping
                 the
                 Sector
                 at
                 that
                 gage
                 ,
                 prick
                 off
                 the
                 hours
                 from
                 the
                 substyle
                 (
                 by
                 your
                 table
                 )
                 upon
                 both
                 the
                 contingent
                 lines
                 .
                 Draw
                 lines
                 by
                 the
                 points
                 in
                 both
                 contingents
                 ,
                 and
                 you
                 have
                 the
                 hours
                 :
              
               
                 For
                 all
                 other
                 declining
                 reclining
                 inclining
                 plains
                 ,
                 it
                 would
                 be
                 needless
                 (
                 I
                 presume
                 )
                 to
                 insist
                 upon
                 the
                 description
                 of
                 them
                 :
                 Sith
                 so
                 much
                 
                 hath
                 been
                 already
                 mentioned
                 ,
                 Chap.
                 2.
                 
                 S.
                 6.
                 that
                 there
                 can
                 scarcely
                 be
                 any
                 mistake
                 ,
                 unless
                 through
                 meer
                 wilfulness
                 ,
                 or
                 grandnegligence
                 .
              
            
          
           
             
               CHAP.
               IV.
               
            
             
               
                 To
                 determine
                 what
                 hour
                 lines
                 are
                 proper
                 for
                 any
                 plain
                 Superficles
                 .
                 By
                 projection
                 of
                 the
                 Sphere
                 .
                 See
                 Fig.
                 17.
                 
              
            
             
               DRaw
               the
               Circle
               NESW
               .
               representing
               the
               Horizon
               ,
               and
               crossing
               it
               into
               quadrants
               N.
               is
               North.
               S.
               South
               ,
               E.
               East
               ,
               W.
               West
               ,
               NS
               .
               the
               Meridian
               (
               which
               let
               be
               infinitely
               produced
               )
               Z.
               the
               center
               represents
               the
               Zenith
               .
               To
               finde
               the
               pole
               set
               half
               the
               co-tangent
               of
               the
               latitude
               from
               Z.
               toward
               N.
               it
               gives
               the
               point
               P.
               for
               the
               pole
               or
               the
               point
               through
               which
               all
               the
               hour
               lines
               must
               pass
               .
            
             
               The
               Suns
               declination
               in
               Cancer
               subtracted
               from
               the
               latitude
               ,
               and
               the
               tangent
               of
               half
               the
               remain
               set
               from
               Z.
               to
               Π.
               gives
               the
               intersection
               of
               Cancer
               with
               the
               Meridian
               .
               Again
               ,
               adde
               the
               complement
               of
               the
               Suns
               delineation
               in
               Cancer
               ,
               unto
               the
               complement
               of
               your
               latitude
               ,
               and
               the
               tangent
               of
               half
               that
               sum
               set
               from
               Z.
               to
               ψ
               ,
               gives
               the
               diameter
               of
               that
               tropick
               ,
               half
               ψ
               ,
               is
               the
               radius
               to
               describe
               it
               .
               Half
               the
               tangent
               of
               your
               latitude
               set
               from
               Z.
               
               to
               Q.
               gives
               that
               point
               for
               the
               intersection
               of
               the
               equator
               with
               the
               Meridian
               ;
               and
               the
               co-secant
               of
               the
               latitude
               set
               from
               oe
               .
               toward
               N.
               gives
               the
               point
               ζ.
               the
               center
               of
               the
               equator
               .
            
             
               Adde
               the
               Suns
               greatest
               declination
               (
               or
               his
               declination
               in
               Capricorn
               )
               to
               the
               latitude
               ,
               and
               the
               tangent
               of
               half
               that
               sum
               set
               from
               Z.
               toward
               S.
               gives
               the
               point
               φ
               ,
               where
               Capricorn
               intersects
               the
               Meridian
               .
               Subtract
               the
               Suns
               declination
               in
               Capricorn
               from
               your
               latitude
               ,
               and
               that
               remain
               subtract
               from
               180.
               the
               tangent
               of
               half
               this
               last
               remain
               ,
               set
               from
               Z.
               toward
               N.
               gives
               the
               point
               T.
               the
               diameter
               of
               Capricorn
               ,
               and
               half
               the
               distance
               T
               φ.
               is
               the
               radius
               to
               describe
               it
               .
            
             
               Set
               the
               secant
               of
               the
               latitude
               from
               P.
               towards
               S.
               it
               gives
               the
               point
               H.
               the
               center
               of
               the
               hour
               line
               of
               six
               cross
               the
               line
               ZSH.
               at
               right
               angles
               in
               the
               point
               H.
               Then
               entring
               PH.
               Radius
               on
               the
               lines
               of
               tangents
               ,
               set
               off
               the
               hour
               centers
               both
               wayes
               from
               H.
               reckoning
               15.
               d.
               for
               an
               hour
               .
               Lastly
               ,
               setting
               one
               point
               of
               your
               Compasses
               in
               these
               center
               points
               ,
               extend
               the
               other
               to
               P.
               and
               with
               that
               radius
               describe
               the
               hour
               lines
               .
               Thus
               have
               you
               the
               sphere
               projected
               ,
               the
               following
               Sections
               will
               determine
               the
               hours
               for
               all
               plains
               .
            
             
               
               
                 Sect.
                 1.
                 
              
               
                 
                   To
                   determine
                   the
                   hour
                   lines
                   for
                   erect
                   direct
                   plains
                   .
                   Fig.
                   17.
                   
                
              
               
                 The
                 line
                 NS
                 .
                 represents
                 an
                 erect
                 East
                 ,
                 and
                 West
                 plain
                 .
                 That
                 side
                 next
                 W.
                 is
                 West
                 ,
                 the
                 other
                 side
                 next
                 E.
                 is
                 East
                 ,
                 where
                 you
                 may
                 see
                 that
                 the
                 Sun
                 shines
                 upon
                 the
                 East
                 until
                 twelve
                 ,
                 or
                 noon
                 ,
                 and
                 at
                 that
                 time
                 comes
                 upon
                 the
                 West
                 .
                 The
                 fine
                 WE
                 .
                 represents
                 a
                 direct
                 North
                 and
                 South
                 plain
                 ,
                 the
                 side
                 next
                 N.
                 is
                 North
                 ,
                 the
                 other
                 next
                 S.
                 is
                 South
                 ,
                 where
                 the
                 North
                 cuts
                 the
                 tropick
                 of
                 Cancer
                 (
                 which
                 in
                 the
                 hour
                 lines
                 you
                 finde
                 'twixt
                 7.
                 and
                 8.
                 in
                 the
                 morning
                 ;
                 and
                 again
                 'twixt
                 4.
                 and
                 5.
                 afternoon
                 )
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 ,
                 and
                 coming
                 on
                 that
                 plain
                 .
                 Where
                 the
                 South
                 cuts
                 the
                 equator
                 ,
                 which
                 is
                 in
                 the
                 points
                 of
                 six
                 ,
                 and
                 six
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 ,
                 and
                 coming
                 on
                 that
                 plain
                 .
              
            
             
               
                 Sect.
                 2.
                 
              
               
                 
                   To
                   determin
                   the
                   hour
                   lines
                   for
                   direct
                   reclining
                   inclining
                   Plains
                   .
                   Fig.
                   17.
                   
                
              
               
                 NBS
                 .
                 on
                 the
                 convex
                 side
                 is
                 a
                 West
                 incliner
                 ,
                 where
                 it
                 cuts
                 Capricorn
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 that
                 plain
                 ,
                 afternoon
                 .
                 On
                 the
                 concave
                 side
                 its
                 an
                 East
                 recliner
                 ,
                 where
                 it
                 
                 cuts
                 Cancer
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 that
                 plain
                 ,
                 afternoon
                 .
              
               
                 NCS.
                 On
                 the
                 convex
                 side
                 is
                 an
                 East
                 incliner
                 ,
                 where
                 it
                 cus
                 Capricorn
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 .
                 On
                 the
                 concave
                 side
                 it
                 is
                 a
                 West
                 recliner
                 ,
                 where
                 it
                 cuts
                 Cancer
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 in
                 the
                 morning
                 .
              
               
                 WDE.
                 On
                 the
                 convex
                 side
                 is
                 a
                 South
                 incliner
                 ,
                 where
                 until
                 D.
                 reach
                 below
                 oe
                 .
                 it
                 hath
                 all
                 hours
                 from
                 six
                 to
                 six
                 ,
                 and
                 until
                 D.
                 reacheth
                 below
                 φ.
                 it
                 may
                 have
                 the
                 twelve
                 a
                 clock
                 line
                 .
                 But
                 when
                 D.
                 reacheth
                 below
                 oe
                 .
                 draw
                 a
                 paralel
                 of
                 declination
                 to
                 pass
                 through
                 the
                 point
                 D.
                 and
                 the
                 intersection
                 of
                 that
                 paralel
                 with
                 the
                 limb
                 of
                 the
                 circle
                 NE
                 SW
                 .
                 doth
                 among
                 the
                 hour
                 lines
                 ,
                 shew
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 upon
                 that
                 plain
                 in
                 the
                 morning
                 ,
                 and
                 going
                 off
                 again
                 afternoon
                 ,
                 when
                 D.
                 reacheth
                 below
                 φ.
                 the
                 intersection
                 of
                 the
                 ark
                 WDE.
                 with
                 the
                 tropick
                 of
                 Capricorn
                 ,
                 shews
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 that
                 plain
                 before
                 noon
                 ,
                 and
                 coming
                 on
                 again
                 ,
                 afternoon
                 .
                 And
                 the
                 intersection
                 of
                 the
                 tropick
                 of
                 Capricorn
                 with
                 the
                 limb
                 shews
                 the
                 first
                 hour
                 in
                 the
                 morning
                 the
                 Sun
                 comes
                 on
                 ,
                 and
                 the
                 last
                 hour
                 afternoon
                 ,
                 that
                 it
                 staves
                 upon
                 that
                 plain
                 .
                 The
                 convex
                 side
                 of
                 WDE.
                 is
                 a
                 north
                 recliner
                 ,
                 where
                 it
                 cuts
                 Cancer
                 ,
                 is
                 the
                 
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 ,
                 and
                 coming
                 on
                 again
                 afternoon
                 .
              
               
                 WFE.
                 On
                 the
                 convex
                 side
                 is
                 a
                 North
                 incliner
                 ,
                 where
                 it
                 cuts
                 Cancer
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 ,
                 and
                 coming
                 on
                 afternoon
                 .
                 On
                 the
                 concave
                 side
                 is
                 a
                 South
                 recliner
                 ,
                 where
                 until
                 F.
                 reach
                 beyond
                 P.
                 it
                 enjoyes
                 the
                 Sun
                 only
                 from
                 six
                 to
                 six
                 .
                 When
                 F.
                 reacheth
                 beyond
                 P.
                 where
                 the
                 ark
                 cuts
                 Cancer
                 ,
                 you
                 finde
                 how
                 much
                 before
                 six
                 in
                 the
                 morning
                 the
                 Sun
                 comes
                 on
                 ,
                 or
                 after
                 six
                 at
                 evening
                 it
                 goes
                 off
                 .
              
               
                 To
                 draw
                 any
                 of
                 these
                 arks
                 ,
                 Ex.
                 gr
                 .
                 the
                 ark
                 NBS
                 .
                 do
                 thus
                 .
                 Set
                 the
                 tangent
                 of
                 half
                 the
                 reclination
                 inclination
                 from
                 Z.
                 on
                 the
                 line
                 ZW
                 .
                 and
                 it
                 gives
                 the
                 point
                 B.
                 produce
                 ZE.
                 and
                 set
                 the
                 co-secant
                 of
                 the
                 reclination
                 inclination
                 from
                 B.
                 towards
                 E.
                 which
                 reacheth
                 to
                 G.
                 then
                 G.
                 is
                 the
                 center
                 .
                 &
                 GB
                 .
                 the
                 radius
                 to
                 draw
                 that
                 ark
                 .
              
               
                 Note
                 .
                 The
                 Semidiameter
                 of
                 the
                 circle
                 SENW
                 .
                 is
                 radius
                 to
                 all
                 the
                 tangents
                 ,
                 and
                 secants
                 ,
                 which
                 you
                 make
                 use
                 of
                 for
                 placing
                 any
                 oblique
                 plain
                 upon
                 the
                 Scheme
                 .
              
            
             
               
                 Sect.
                 3.
                 
              
               
                 
                   To
                   determin
                   the
                   hour
                   lines
                   for
                   erect
                   declining
                   plains
                   .
                   Fig.
                   18.
                   
                
              
               
                 For
                 South-East
                 ,
                 or
                 North-West
                 plains
                 .
                 
                 By
                 a
                 line
                 of
                 chords
                 set
                 the
                 angle
                 of
                 declination
                 on
                 the
                 limb
                 from
                 W.
                 toward
                 S.
                 as
                 H.
                 lay
                 a
                 ruler
                 to
                 HZ
                 .
                 and
                 draw
                 HZK.
                 which
                 on
                 that
                 side
                 next
                 NW
                 .
                 represents
                 the
                 North-West
                 ,
                 and
                 where
                 the
                 line
                 cuts
                 Cancer
                 ,
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 afternoon
                 ,
                 and
                 staying
                 until
                 Sun
                 set
                 .
                 But
                 if
                 it
                 cut
                 Cancer
                 twice
                 ,
                 then
                 in
                 the
                 morning
                 hours
                 it
                 shews
                 what
                 time
                 the
                 Sun
                 goes
                 off
                 this
                 plain
                 ,
                 having
                 all
                 hours
                 from
                 Sun
                 rise
                 to
                 that
                 time
                 ,
                 and
                 in
                 the
                 evening
                 hours
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 again
                 ,
                 and
                 staying
                 till
                 sun
                 set
                 .
                 That
                 side
                 HZK.
                 next
                 SE.
                 represents
                 a
                 South-East
                 .
                 Where
                 the
                 line
                 cuts
                 the
                 equator
                 in
                 the
                 evening
                 hours
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 ,
                 where
                 it
                 cuts
                 Cancer
                 ,
                 in
                 the
                 morning
                 hours
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 that
                 plain
                 .
                 For
                 North-East
                 ,
                 or
                 South-West
                 ,
                 set
                 the
                 declination
                 by
                 a
                 line
                 of
                 chords
                 from
                 E.
                 towards
                 S.
                 as
                 L.
                 lay
                 a
                 ruler
                 to
                 ZL
                 .
                 and
                 draw
                 the
                 line
                 LZR.
                 which
                 on
                 that
                 side
                 next
                 NE.
                 is
                 North-East
                 ,
                 where
                 it
                 cuts
                 Cancer
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 .
                 If
                 it
                 cuts
                 Cancer
                 twice
                 ,
                 you
                 have
                 in
                 the
                 evening
                 hours
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 again
                 ,
                 and
                 staying
                 until
                 Sun-set
                 .
              
               
                 On
                 that
                 side
                 next
                 SW
                 .
                 is
                 the
                 South-west
                 .
                 Where
                 it
                 cuts
                 the
                 equator
                 ,
                 or
                 Capricorn
                 in
                 the
                 morning
                 hours
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 
                 on
                 ,
                 where
                 it
                 cuts
                 Cancer
                 in
                 the
                 evening
                 hours
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 .
              
            
             
               
                 Sect.
                 4.
                 
              
               
                 
                   To
                   determin
                   the
                   hours
                   of
                   declining
                   reclining
                   Plains
                   inclining
                   Plains
                   .
                   Fig.
                   18.
                   
                
              
               
                 First
                 ,
                 set
                 in
                 the
                 plain
                 according
                 to
                 its
                 declination
                 .
                 By
                 Sect.
                 3.
                 
                 Ex.
                 gr
                 .
                 LZR.
                 a
                 North-East
                 ,
                 or
                 South-West
                 declining
                 50.
                 d.
                 00.
                 min.
                 
                 This
                 done
                 Cross
                 the
                 line
                 LZR.
                 representing
                 the
                 declination
                 of
                 the
                 plain
                 ,
                 at
                 right
                 angles
                 in
                 the
                 point
                 Z.
                 as
                 CZBG
                 .
                 Then
                 for
                 North-East
                 incliners
                 ,
                 or
                 South-West
                 recliners
                 ,
                 set
                 half
                 the
                 tangent
                 of
                 the
                 reclination
                 inclination
                 from
                 Z.
                 toward
                 C.
                 〈◊〉
                 T.
                 and
                 set
                 the
                 co-secant
                 of
                 the
                 reclination
                 inclination
                 from
                 T.
                 toward
                 B.
                 as
                 TG
                 .
                 Then
                 G.
                 is
                 the
                 center
                 ,
                 and
                 GT
                 .
                 the
                 radius
                 to
                 describe
                 the
                 ark
                 RTL.
                 Whose
                 convex
                 side
                 represents
                 a
                 North-East
                 incliner
                 ,
                 where
                 it
                 cuts
                 Libra
                 or
                 Capricorn
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 ;
                 if
                 it
                 cuts
                 Cancer
                 twice
                 ,
                 the
                 intersection
                 of
                 Cancer
                 with
                 the
                 evening
                 hours
                 shews
                 what
                 time
                 the
                 Sun
                 comes
                 again
                 upon
                 such
                 a
                 plain
                 afternoon
                 ,
                 and
                 continues
                 till
                 Sun
                 setting
                 .
                 The
                 concave
                 side
                 is
                 a
                 South-West
                 recliner
                 ,
                 where
                 it
                 cuts
                 
                 Cancer
                 in
                 the
                 morning
                 hours
                 ,
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 ,
                 in
                 case
                 it
                 intersects
                 Cancer
                 twice
                 in
                 the
                 evening
                 hours
                 ,
                 you
                 have
                 the
                 time
                 that
                 the
                 Sun
                 goes
                 off
                 .
              
               
                 For
                 a
                 North-East
                 recliner
                 ,
                 or
                 South-West
                 incliner
                 ,
                 set
                 the
                 point
                 T.
                 from
                 Z.
                 toward
                 B.
                 and
                 the
                 point
                 G.
                 from
                 T.
                 (
                 so
                 placed
                 )
                 toward
                 C.
                 and
                 draw
                 the
                 ark
                 on
                 that
                 side
                 RZL.
                 toward
                 B.
                 whose
                 convex
                 side
                 will
                 represent
                 a
                 South-West
                 incliner
                 ,
                 and
                 where
                 the
                 ark
                 cuts
                 the
                 equator
                 or
                 Capricorn
                 ,
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 that
                 plain
                 .
                 The
                 concave
                 side
                 is
                 a
                 North-East
                 recliner
                 ,
                 where
                 the
                 ark
                 cuts
                 Cancer
                 ,
                 is
                 the
                 time
                 for
                 the
                 Suns
                 going
                 off
                 that
                 plain
                 .
                 When
                 the
                 ark
                 cuts
                 Cancer
                 twice
                 ,
                 the
                 Sun
                 comes
                 on
                 again
                 before
                 it
                 sets
                 .
              
               
                 For
                 a
                 North-West
                 recliner
                 ,
                 or
                 South-East
                 incliner
                 .
                 Enter
                 the
                 declination
                 by
                 Sect.
                 3.
                 as
                 HZK.
                 Cross
                 it
                 in
                 the
                 point
                 Z.
                 at
                 right
                 angles
                 ,
                 as
                 OZD.
                 set
                 half
                 the
                 tangent
                 of
                 the
                 reclination
                 inclination
                 from
                 Z.
                 toward
                 O.
                 as
                 V.
                 and
                 the
                 co-secant
                 of
                 the
                 reclination
                 inclination
                 from
                 V.
                 toward
                 D.
                 as
                 F.
                 then
                 is
                 F.
                 the
                 center
                 ,
                 and
                 FV.
                 the
                 radius
                 to
                 draw
                 the
                 ark
                 HVK.
                 Where
                 it
                 cuts
                 Cancer
                 the
                 hour
                 lines
                 ,
                 tell
                 you
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 ,
                 and
                 entring
                 again
                 afternoon
                 ,
                 upon
                 the
                 North-West
                 
                 recliner
                 .
                 Where
                 it
                 cuts
                 the
                 equator
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 the
                 South-East
                 ,
                 where
                 it
                 cuts
                 Cancer
                 in
                 the
                 morning
                 hours
                 is
                 the
                 time
                 of
                 the
                 Suns
                 coming
                 on
                 that
                 plain
                 .
              
               
                 For
                 a
                 North-West
                 incliner
                 ,
                 or
                 South-East
                 recliner
                 ,
                 set
                 the
                 point
                 V.
                 from
                 Z.
                 toward
                 D.
                 and
                 the
                 point
                 F.
                 set
                 from
                 V.
                 (
                 so
                 placed
                 )
                 toward
                 O.
                 and
                 draw
                 the
                 ark
                 on
                 that
                 side
                 Z.
                 next
                 D.
                 Then
                 where
                 the
                 convex
                 side
                 cuts
                 Cancer
                 ,
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 in
                 the
                 morning
                 ;
                 and
                 coming
                 on
                 again
                 afternoon
                 upon
                 the
                 North-West
                 incliner
                 .
                 Where
                 the
                 concave
                 side
                 cuts
                 the
                 equator
                 ,
                 you
                 have
                 the
                 time
                 of
                 the
                 Suns
                 going
                 off
                 the
                 South-East
                 recliner
                 ;
                 where
                 it
                 intersects
                 Cancer
                 ,
                 is
                 the
                 time
                 of
                 his
                 coming
                 on
                 that
                 plain
                 in
                 the
                 morning
                 .
              
               
                 Note
                 .
                 All
                 the
                 precedent
                 rules
                 about
                 plains
                 are
                 appropriated
                 to
                 us
                 that
                 live
                 in
                 Northern
                 Hemisphere
                 ,
                 In
                 case
                 any
                 one
                 would
                 apply
                 them
                 to
                 the
                 South
                 Hemisphere
                 :
                 What
                 is
                 here
                 called
                 North
                 ,
                 there
                 name
                 South
                 ,
                 and
                 what
                 we
                 here
                 term
                 South
                 ,
                 there
                 call
                 North
                 ,
                 and
                 the
                 rules
                 are
                 the
                 same
                 .
              
               
                 —
                 Si
                 quid
                 novisti
                 plenius
                 istis
                 ,
                 promptius
                 istis
                 ,
                 rectius
                 istis
                 ,
                 Candidus
                 imperti
                 :
                 Sinon
                 ,
                 His
                 u●ere
                 mecum
                 .
              
            
          
        
         
           FINIS
           .