A supplement to a late treatise, called An essay for the discovery of some new geometrical problems concerning angular sections, resolving what was there problematically proposed; and with some rectification made in the former essay, showing an easie method truly geometrical, without any conick section, or cubick æquation, to sect any angle or arch of a circle into 3. 5. 7. or any other uneven number of equal parts. By G. K.
         Keith, George, 1639?-1716.
      
       
         
           1697
        
      
       Approx. 13 KB of XML-encoded text transcribed from 4 1-bit group-IV TIFF page images.
       
         Text Creation Partnership,
         Ann Arbor, MI ; Oxford (UK) :
         2006-06 (EEBO-TCP Phase 1).
         A47183
         Wing K216A
         ESTC R216625
         99828350
         99828350
         32777
         
           
            This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of
             Creative Commons 0 1.0 Universal
            . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission.
          
        
      
       
         Early English books online.
      
       
         (EEBO-TCP ; phase 1, no. A47183)
         Transcribed from: (Early English Books Online ; image set 32777)
         Images scanned from microfilm: (Early English books, 1641-1700 ; 1951:14)
      
       
         
           
             A supplement to a late treatise, called An essay for the discovery of some new geometrical problems concerning angular sections, resolving what was there problematically proposed; and with some rectification made in the former essay, showing an easie method truly geometrical, without any conick section, or cubick æquation, to sect any angle or arch of a circle into 3. 5. 7. or any other uneven number of equal parts. By G. K.
             Keith, George, 1639?-1716.
          
           7, [1] p.
           
             printed for the author, and are to be had at the Three Pigeons over against the Exchange, and at his House in Pudding-lane, at the sign of the Golden Ball, where he teacheth the mathematical arts,
             [London :
             [1697?]]
          
           
             G.K. = George Keith.
             Caption title.
             Imprint from colophon; publication date conjectured by Wing.
             Reproduction of the original in the British Library.
          
        
      
    
     
       
         Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford.
         Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. Gap elements of known extent have been transformed into placeholder characters or elements to simplify the filling in of gaps by user contributors.
      
       
         EEBO-TCP is a partnership between the Universities of Michigan and Oxford and the publisher ProQuest to create accurately transcribed and encoded texts based on the image sets published by ProQuest via their Early English Books Online (EEBO) database (http://eebo.chadwyck.com). The general aim of EEBO-TCP is to encode one copy (usually the first edition) of every monographic English-language title published between 1473 and 1700 available in EEBO.
         EEBO-TCP aimed to produce large quantities of textual data within the usual project restraints of time and funding, and therefore chose to create diplomatic transcriptions (as opposed to critical editions) with light-touch, mainly structural encoding based on the Text Encoding Initiative (http://www.tei-c.org).
         The EEBO-TCP project was divided into two phases. The 25,363 texts created during Phase 1 of the project have been released into the public domain as of 1 January 2015. Anyone can now take and use these texts for their own purposes, but we respectfully request that due credit and attribution is given to their original source.
         Users should be aware of the process of creating the TCP texts, and therefore of any assumptions that can be made about the data.
         Text selection was based on the New Cambridge Bibliography of English Literature (NCBEL). If an author (or for an anonymous work, the title) appears in NCBEL, then their works are eligible for inclusion. Selection was intended to range over a wide variety of subject areas, to reflect the true nature of the print record of the period. In general, first editions of a works in English were prioritized, although there are a number of works in other languages, notably Latin and Welsh, included and sometimes a second or later edition of a work was chosen if there was a compelling reason to do so.
         Image sets were sent to external keying companies for transcription and basic encoding. Quality assurance was then carried out by editorial teams in Oxford and Michigan. 5% (or 5 pages, whichever is the greater) of each text was proofread for accuracy and those which did not meet QA standards were returned to the keyers to be redone. After proofreading, the encoding was enhanced and/or corrected and characters marked as illegible were corrected where possible up to a limit of 100 instances per text. Any remaining illegibles were encoded as <gap>s. Understanding these processes should make clear that, while the overall quality of TCP data is very good, some errors will remain and some readable characters will be marked as illegible. Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor.
         The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines.
         Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements).
         
          Keying and markup guidelines are available at the
           Text Creation Partnership web site
          .
        
      
       
         
         
      
    
     
       
         eng
      
       
         
           Geometry -- Early works to 1800.
        
      
    
     
        2005-05 TCP
        Assigned for keying and markup
      
        2005-08 SPi Global
        Keyed and coded from ProQuest page images
      
        2006-01 Judith Siefring
        Sampled and proofread
      
        2006-01 Judith Siefring
        Text and markup reviewed and edited
      
        2006-04 pfs
        Batch review (QC) and XML conversion
      
    
  
   
     
       
         
         
           A
           SUPPLEMENT
           TO
           A
           Late
           TREATISE
           ,
           CALLED
           An
           Essay
           for
           the
           Discovery
           of
           some
           New
           Geometrical
           Problems
           ,
           Concerning
           Angular
           Sections
           ,
           resolving
           what
           was
           there
           Problematically
           proposed
           ;
           and
           with
           some
           Rectification
           made
           in
           the
           former
           Essay
           ,
           showing
           an
           easie
           method
           truly
           Geometrical
           ,
           without
           any
           Conick
           Section
           ,
           or
           Cubick
           Aequation
           ,
           to
           sect
           any
           Angle
           or
           Arch
           of
           a
           Circle
           into
           3.5.7
           .
           or
           any
           other
           Uneven
           Number
           of
           equal
           parts
           .
        
         
           By
           
             G.K.
             
          
        
         
           WHereas
           it
           was
           supposed
           in
           the
           former
           Proposal
           ,
           that
           a
           straight
           Line
           could
           be
           drawn
           through
           the
           extream
           Points
           of
           three
           or
           more
           Concentrick
           Arches
           at
           both
           ends
           (
           the
           Arches
           beginning
           or
           ending
           upon
           a
           straight
           Line
           ●oming
           from
           the
           Center
           of
           those
           Concentrick
           Arches
           )
           having
           equal
           Cords
           ,
           though
           not
           equal
           Arches
           .
           Upon
           further
           consideration
           ,
           it
           is
           found
           ,
           that
           however
           seemingly
           such
           a
           Line
           may
           appear
           to
           be
           straight
           in
           many
           cases
           ,
           as
           when
           the
           Radius
           is
           short
           ,
           or
           the
           Angle
           very
           acute
           ,
           yet
           in
           no
           case
           is
           such
           a
           Line
           mathematically
           straight
           ,
           but
           is
           a
           regular
           Curve
           ,
           and
           can
           be
           as
           regularly
           drawn
           ,
           and
           by
           as
           true
           Geometrical
           Art
           ,
           as
           any
           Parabola
           ,
           or
           other
           Conick
           Section
           can
           ,
           and
           with
           greater
           facility
           and
           readiness
           ,
           and
           which
           any
           Tiro
           who
           understands
           nothing
           of
           Conick
           Sections
           ,
           and
           Cubick
           Equations
           may
           do
           .
           The
           way
           of
           drawing
           the
           said
           Curve
           is
           this
           :
           Let
           a
           short
           cross-Rule
           be
           set
           at
           right
           Angles
           with
           another
           longer
           Rule
           ,
           and
           let
           the
           length
           of
           the
           cross
           Rule
           be
           at
           pleasure
           2
           or
           3
           Inches
           ,
           
           or
           more
           ,
           as
           6
           or
           7
           ,
           as
           ye
           have
           a
           mind
           to
           make
           the
           length
           of
           the
           Cord
           of
           each
           part
           of
           the
           Section
           of
           your
           Angle
           ,
           which
           as
           in
           the
           following
           Figure
           let
           be
           3
           Inches
           ,
           and
           let
           the
           just
           half
           of
           the
           cross-Rule
           be
           on
           the
           left
           side
           of
           the
           long
           Rule
           ,
           and
           let
           a
           small
           Brass
           or
           Steel-Pin
           be
           fixed
           on
           the
           right
           end
           of
           the
           said
           cross
           Rule
           ,
           that
           as
           the
           Rule
           is
           moved
           ,
           may
           make
           an
           Impression
           on
           the
           Paper
           ,
           as
           the
           point
           of
           the
           Compass
           doth
           in
           drawing
           a
           Circle
           .
           The
           length
           of
           the
           longer
           Rule
           is
           to
           be
           as
           occasion
           requireth
           ,
           as
           double
           or
           triple
           the
           length
           of
           the
           other
           .
        
         
           Having
           thus
           prepared
           your
           two
           Rules
           ,
           the
           one
           cutting
           the
           other
           at
           right
           Angles
           ,
           and
           the
           cross-Rule
           fixed
           to
           it
           ,
           (
           though
           it
           may
           be
           made
           also
           moveable
           on
           it
           )
           suppose
           the
           Angle
           given
           to
           be
           trisected
           is
           BAC
           ,
           measured
           by
           the
           arch
           BMC
           .
           in
           order
           to
           draw
           the
           curve
           Line
           with
           one
           draught
           of
           the
           Hand
           ,
           set
           the
           left
           end
           of
           the
           cross-Rule
           ,
           on
           the
           point
           B
           ,
           and
           from
           B
           let
           it
           run
           or
           slide
           along
           the
           line
           BA
           ,
           and
           as
           it
           runs
           along
           the
           said
           Line
           ,
           let
           the
           left
           side
           of
           the
           long
           Rule
           still
           run
           through
           the
           Center
           ,
           or
           vertical
           point
           A
           ,
           which
           is
           most
           easily
           done
           ;
           and
           let
           it
           run
           or
           slide
           along
           from
           B
           towards
           A
           ,
           until
           the
           other
           end
           of
           the
           cross-Rule
           reach
           at
           least
           to
           the
           line
           AC
           ,
           or
           further
           as
           one
           pleaseth
           ,
           and
           the
           Brass
           Pin
           on
           the
           other
           end
           of
           the
           cross
           Rule
           shall
           describe
           the
           regular
           curve
           FIG
           Having
           thus
           drawn
           the
           curve
           Line
           ,
           at
           the
           distance
           of
           one
           half
           of
           the
           cross
           Rule
           draw
           the
           straight
           Line
           DE
           paralel
           to
           AC
           .
           and
           where
           the
           Curve
           Line
           cuts
           the
           straight
           line
           DE
           as
           at
           I
           ,
           a
           Line
           drawn
           from
           the
           center
           A
           to
           I
           ,
           shall
           by
           true
           Geometry
           ,
           trisect
           the
           given
           angle
           BAC
           .
        
         
           
             The
             Demonstration
             .
          
           
             Seeing
             it
             is
             the
             property
             of
             these
             two
             Rules
             crossing
             each
             other
             at
             right
             Angles
             ,
             where
             ever
             the
             two
             ends
             of
             the
             cross
             Rule
             terminates
             ,
             to
             make
             an
             Isosceles
             triangle
             ,
             making
             always
             two
             right
             angle
             triangles
             ,
             whose
             Bases
             are
             equal
             ,
             and
             the
             Perpendicular
             common
             to
             both
             ,
             therefore
             by
             the
             4.
             1.
             
               el.
               Eucl.
            
             the
             Hypotenusals
             are
             equal
             .
             Therefore
             with
             Radius
             AI
             describing
             the
             arch
             HLIS
             ,
             draw
             the
             Cord
             HI
             .
             and
             from
             I
             let
             fall
             a
             perpendicular
             on
             the
             line
             AC
             .
             as
             IK
             ,
             making
             HI
             =
             LI
             =
             IK
             ,
             therefore
             the
             arches
             of
             those
             equal
             Sines
             are
             equal
             ,
             as
             HL
             =
             LI
             =
             IS
             .
             q.e.d.
             The
             same
             or
             any
             other
             Angle
             obtruse
             or
             acute
             may
             be
             trisected
             into
             3
             equal
             parts
             ,
             without
             the
             curve
             Line
             ,
             or
             any
             part
             of
             it
             ,
             by
             finding
             the
             point
             I
             ,
             which
             can
             be
             found
             without
             the
             curve
             line
             by
             letting
             the
             cross
             Rule
             slide
             or
             run
             along
             the
             line
             AB
             ,
             (
             while
             the
             left
             side
             of
             the
             long
             Rule
             still
             runneth
             through
             the
             center
             A
             )
             either
             upwards
             or
             downwards
             ,
             until
             the
             right
             side
             of
             the
             cross
             Rule
             touch
             the
             
             straight
             line
             DE
             which
             shall
             be
             at
             I.
             And
             thus
             without
             any
             need
             of
             noticing
             or
             regarding
             the
             Curve
             Line
             ,
             the
             Point
             1
             is
             found
             ,
             where
             the
             two
             straight
             lines
             HI
             and
             DE
             meet
             together
             :
             And
             as
             thus
             any
             Angle
             may
             be
             trisected
             without
             drawing
             any
             Curve
             Line
             ,
             so
             it
             may
             easily
             and
             truly
             be
             done
             without
             either
             Scale
             or
             Compass
             ,
             other
             than
             what
             the
             two
             cross
             Rules
             are
             ,
             as
             any
             Artist
             may
             easily
             perceive
             .
          
           
           
             If
             any
             object
             against
             this
             Method
             ,
             as
             Mechanical
             ,
             and
             not
             Mathematical
             and
             truly
             Geometrical
             ,
             because
             performed
             by
             an
             Instrument
             ,
             I
             shall
             refer
             them
             to
             two
             great
             Geometricians
             for
             its
             Vindication
             ,
             
             to
             wit
             ,
             
               Des
               Cartes
            
             in
             his
             second
             Book
             of
             Geometry
             ,
             and
             
               Franciscus
               a
               Schoten
            
             in
             his
             Commentary
             on
             him
             ,
             
               argum
               .
               lib.
            
             2.
             both
             which
             do
             prove
             that
             what
             is
             performed
             by
             Instruments
             Geometrically
             made
             ,
             is
             Geometrical
             ,
             otherwise
             the
             plainest
             Geometry
             must
             be
             rejected
             ,
             because
             its
             Figures
             are
             drawn
             by
             Rule
             and
             Compass
             ,
             both
             which
             are
             Instruments
             ,
             and
             not
             only
             Parabolas
             ,
             and
             other
             Conick
             Sections
             which
             are
             Curves
             ,
             but
             divers
             other
             Curves
             ,
             yea
             ,
             all
             such
             that
             can
             be
             drawn
             by
             Art
             ,
             with
             the
             help
             of
             Instruments
             ,
             such
             as
             they
             have
             devised
             ,
             they
             contend
             to
             be
             truly
             Geometrical
             ;
             and
             both
             of
             them
             in
             their
             Geometrical
             Treatises
             ,
             use
             divers
             Instruments
             for
             describing
             Curves
             Geometrically
             much
             more
             difficult
             to
             be
             made
             ,
             and
             with
             more
             difficulty
             to
             be
             used
             ,
             than
             what
             is
             here
             proposed
             of
             two
             simple
             Rules
             ,
             cutting
             one
             another
             at
             right
             Angles
             .
             And
             seeing
             it
             hath
             no
             dependance
             on
             Solids
             ,
             or
             Algebra
             Equations
             ,
             and
             may
             be
             done
             without
             any
             Curve
             Line
             ,
             as
             is
             above
             showed
             ,
             and
             whose
             demonstration
             wholly
             depends
             on
             a
             few
             easie
             Propositions
             of
             the
             first
             Book
             of
             
               Euclid
               ,
            
             I
             see
             not
             why
             it
             may
             not
             be
             called
             Plain
             Geometry
             :
             And
             as
             the
             word
             Mechanical
             is
             used
             to
             signifie
             a
             thing
             not
             Mathematically
             exact
             ,
             but
             coming
             near
             to
             it
             by
             Approximation
             ,
             in
             this
             sense
             it
             is
             not
             Mechanical
             ,
             but
             Mathematical
             ,
             and
             purely
             Geometrical
             ,
             being
             grounded
             on
             as
             good
             demonstration
             ,
             as
             any
             Propositions
             in
             
               Euclid
               ,
            
             and
             being
             but
             a
             Corrolary
             from
             some
             of
             them
             .
          
           
             The
             next
             thing
             to
             be
             shown
             is
             the
             Quinquisection
             ,
             where
             to
             make
             one
             Figure
             serve
             to
             both
             ,
             I
             make
             the
             Cross
             Rule
             only
             one
             Inch
             and
             one
             4th
             part
             from
             the
             middle
             line
             AM
             ,
             setting
             off
             on
             both
             sides
             one
             half
             of
             the
             length
             of
             the
             cross
             Rule
             ,
             draw
             the
             paralel
             Lines
             
               ad
            
             and
             
               be
               ,
            
             then
             let
             the
             cross
             Rule
             side
             along
             the
             line
             AB
             ,
             as
             in
             the
             Trisection
             ,
             while
             the
             left
             side
             of
             the
             long
             Rule
             slides
             through
             the
             center
             A
             ,
             the
             other
             end
             of
             the
             cross
             Rule
             shall
             describe
             a
             Curve
             ,
             a
             part
             of
             which
             shall
             be
             
               g
               h
               ,
            
             that
             may
             be
             continued
             at
             pleasure
             .
             Again
             ,
             setting
             the
             right
             end
             of
             the
             cross
             Rule
             one
             the
             Point
             
               d
               ,
            
             let
             it
             slide
             or
             move
             along
             the
             Line
             
               da
               ,
            
             (
             while
             the
             left
             side
             of
             the
             long
             Rule
             runneth
             through
             the
             center
             A
             ,
             )
             the
             left
             end
             of
             the
             cross
             Rule
             shall
             describe
             a
             part
             of
             another
             Curve
             ,
             meeting
             at
             
               h
            
             the
             other
             Curve
             .
             And
             having
             found
             the
             point
             
               h
            
             with
             radius
             A
             
               h
            
             describe
             the
             Arch
             
               v
               h
               z
               x
               y
            
             N
             which
             shall
             give
             
               v
               h
            
             =
             one
             fifth
             of
             the
             whole
             Arch
             ,
             as
             is
             evident
             from
             the
             foregoing
             demonstration
             .
          
           
             The
             Quinquisection
             also
             may
             be
             made
             without
             any
             Curve
             ,
             if
             two
             long
             Rules
             be
             jointed
             together
             like
             a
             Sector
             ,
             and
             each
             have
             a
             moveable
             cross
             Rule
             to
             move
             on
             them
             at
             right
             Angles
             ,
             with
             the
             long
             Rules
             .
             For
             let
             the
             center
             of
             the
             two
             long
             Rules
             be
             fixed
             on
             the
             center
             A
             ,
             and
             let
             the
             2
             cross
             Rules
             be
             moved
             together
             from
             
             B
             and
             
               d
               ,
            
             until
             (
             the
             left
             end
             of
             the
             one
             still
             touching
             the
             right
             end
             of
             the
             other
             )
             the
             right
             end
             of
             the
             Cross
             nearest
             to
             the
             Line
             a
             d
             touch
             upon
             some
             Point
             of
             it
             as
             at
             W
             ,
             the
             Point
             at
             W
             shall
             give
             the
             Quinquisection
             as
             above
             .
          
           
             And
             thus
             a
             true
             Geometrical
             Line
             of
             Cords
             may
             be
             made
             by
             any
             Tiro
             ,
             without
             any
             Conick
             Section
             ,
             or
             Algebra
             Equation
             ,
             and
             without
             any
             Table
             of
             Natural
             Sines
             or
             Arithmetical
             Operation
             ;
             for
             whereas
             
               Euclid
            
             (
             11.4
             .
             )
             hath
             taught
             how
             to
             find
             the
             Cord
             of
             36
             degr
             .
             and
             also
             it
             is
             found
             by
             Quinquisecting
             the
             half
             Circle
             ,
             as
             is
             above
             shewed
             ,
             it
             remains
             only
             to
             trisect
             the
             Arch
             of
             120
             degr
             .
             which
             giveth
             the
             Cord
             of
             40
             ,
             and
             36
             taken
             from
             40
             ,
             leaveth
             4
             degr
             .
             which
             bisected
             gives
             2
             ,
             and
             that
             bisected
             giveth
             1
             ,
             which
             is
             the
             one
             360th
             part
             of
             the
             Circle
             ,
             and
             one
             90th
             of
             the
             Quadrant
             ;
             and
             this
             is
             more
             methodical
             than
             to
             teach
             a
             beginner
             to
             make
             his
             Line
             of
             Cords
             ,
             for
             projecting
             of
             Angles
             ,
             by
             sending
             him
             to
             Conick
             Sections
             ,
             and
             Algebra
             Equations
             or
             the
             Table
             of
             Natural
             Sines
             ,
             which
             he
             is
             not
             capable
             at
             his
             entry
             ,
             nor
             after
             he
             has
             made
             some
             good
             progress
             to
             understand
             (
             it
             being
             to
             teach
             
               ignotum
               per
               ignotius
            
             an
             unknown
             thing
             by
             a
             more
             unknown
             )
             quite
             contrary
             to
             all
             good
             method
             of
             true
             Science
             ,
             such
             as
             Geometry
             is
             .
             The
             method
             of
             the
             Quinquisection
             here
             delivered
             ,
             sufficiently
             showeth
             without
             example
             ,
             any
             other
             Section
             desired
             .
          
           
             The
             Corrolaries
             mentioned
             in
             the
             former
             Treatise
             ,
             with
             the
             Rectification
             here
             made
             ,
             are
             all
             valid
             ,
             some
             of
             the
             chief
             of
             them
             I
             shall
             here
             mention
             .
          
           
           
           
             1.
             
             One
             great
             Use
             is
             to
             teach
             a
             beginner
             how
             to
             make
             a
             true
             Line
             of
             Cords
             ,
             as
             is
             above
             showed
             ,
             and
             how
             to
             divide
             a
             Circle
             into
             any
             parts
             required
             .
          
           
             2.
             
             Another
             great
             use
             
               Descartes
            
             showeth
             in
             his
             third
             Book
             of
             Geometry
             ,
             for
             the
             resolving
             any
             such
             Equation
             in
             Algebra
             as
             z
             3
             =
             +
             p
             zq
             ,
             where
             the
             Root
             z
             is
             an
             unknown
             quantity
             ,
             and
             can
             be
             found
             by
             the
             Trisection
             of
             an
             Angle
             .
          
           
             3.
             
             A
             third
             great
             use
             is
             to
             give
             some
             New
             Promblems
             in
             Practical
             Geometry
             ,
             one
             whereof
             I
             shall
             here
             show
             .
             Let
             a
             straight
             Line
             A
             F
             be
             given
             ,
             (
             see
             the
             
               second
               Figure
            
             )
             and
             it
             is
             required
             on
             the
             point
             A
             to
             erect
             an
             Isosceles
             ABC
             ,
             whose
             side
             BC
             produced
             ,
             shall
             terminate
             on
             a
             limited
             point
             D
             ,
             under
             the
             given
             straight
             line
             AF.
             
          
           
             The
             construction
             is
             thus
             ,
             draw
             a
             straight
             Line
             from
             D
             to
             A
             ,
             as
             D
             A
             ,
             next
             make
             the
             right
             Angle
             FAE
             .
             Divide
             the
             angle
             EAD
             into
             three
             equal
             parts
             ,
             and
             with
             radius
             AD
             describe
             the
             Semicircle
             GFE
             .
             From
             C
             to
             B
             set
             off
             GB
             =
             ED.
             Then
             draw
             the
             line
             AB
             ,
             and
             from
             B
             draw
             the
             line
             BCD
             ,
             which
             shall
             form
             the
             Isosceles
             triangle
             ABC
             ,
             whose
             side
             BC
             being
             produced
             ,
             shall
             terminate
             on
             D.
             q.e.f.
             the
             use
             of
             this
             is
             obvious
             in
             Architecture
             .
          
           
             4.
             
             A
             fourth
             great
             use
             is
             to
             give
             us
             some
             New
             Problems
             in
             Geography
             and
             Navigation
             .
          
           
             
               Example
               .
            
          
           
             There
             are
             four
             places
             A.
             B
             C.
             D
             so
             situated
             .
             A
             is
             distant
             from
             D
             100
             Leagues
             ,
             and
             beareth
             South-Easterly
             from
             it
             70
             degr
             .
             B
             is
             distant
             from
             A
             100
             Leagues
             South-Westerly
             .
             C
             is
             distant
             from
             B
             100
             Leagues
             North-Westerly
             C
             and
             A
             are
             in
             the
             same
             Latitude
             and
             so
             that
             these
             three
             places
             B.
             C.
             D
             lie
             in
             a
             straight
             Line
             one
             from
             another
             .
          
           
             Q.
             What
             is
             the
             distance
             betwixt
             these
             two
             places
             ,
             B
             and
             D
             ,
             and
             the
             Course
             on
             the
             Rhumb
             Line
             betwixt
             A
             and
             B
             ,
             and
             the
             distance
             betwixt
             A
             and
             C.
             
          
           
             The
             Resolution
             .
             Divide
             the
             angle
             EAD
             into
             three
             equal
             parts
             ,
             and
             make
             CAB
             =
             one
             third
             part
             of
             EAD
             ,
             and
             draw
             the
             line
             BAD
             .
             Thus
             the
             four
             places
             A.
             B.
             C.
             D
             shall
             be
             duly
             situated
             ,
             and
             an
             Isosceles
             Triangle
             shall
             be
             formed
             ABD
             ,
             whose
             side
             AB
             =
             AD
             =
             100
             Leagues
             ,
             and
             the
             angle
             BAD
             =
             86
             degr
             .
             40.
             consequently
             by
             plain
             
             Trigonometry
             ,
             the
             angle
             of
             the
             course
             GAB
             being
             found
             ,
             which
             is
             23
             degr
             .
             30
             min.
             the
             angle
             ABD
             ,
             its
             double
             is
             46
             degr
             .
             40
             =
             ADB
             ,
             by
             the
             Rule
             of
             Opposits
             .
          
           
             As
             the
             Sine
             of
             46
             d.
             40
             to
             AD
             100
             Leagues
             ,
             so
             the
             Sine
             of
             86.
             40
             
               
                 Log.
                 
              
               
                 9.861757
              
               
                 2.000000
              
               
                 9.999265
              
               
                 11.999265
              
               
                 9.861757
              
               
                 to
                 BD
                 137.
                 
                 ●
                 .
                 fere
                 2.137508
              
            
          
           
             From
             which
             substracting
             BC
             100
             Leagues
             ,
             there
             remaineth
             CD
             37.
             
             ●
             .
             as
             was
             required
             .
          
           
             A
             Fifth
             great
             use
             of
             the
             Trisection
             ,
             and
             other
             Sections
             is
             ,
             having
             the
             Ratio
             of
             any
             2
             Angles
             given
             in
             any
             plain
             .
             Triangle
             ,
             to
             find
             the
             quantity
             of
             them
             ,
             if
             the
             quantity
             of
             the
             3d
             Angle
             be
             given
             ,
             without
             any
             regard
             to
             their
             sides
             .
             What
             other
             uses
             these
             Angular
             Sections
             may
             have
             ,
             is
             left
             to
             the
             search
             of
             Industrious
             Artists
             .
          
        
      
    
     
       
         
           
             London
             ,
          
           Printed
           for
           the
           Author
           ,
           and
           are
           to
           be
           had
           at
           the
           
             Three
             Pigeons
          
           over
           against
           the
           
             Exchange
             ,
          
           and
           at
           his
           House
           in
           
             Pudding-lane
             ,
          
           at
           the
           Sign
           of
           the
           
             Golden
             Ball
             ,
          
           where
           he
           Teacheth
           the
           Mathematical
           Arts.