A treatise of angular sections by John Wallis ...
         Wallis, John, 1616-1703.
      
       
         
           1684
        
      
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         A67419
         Wing W614
         ESTC R186204
         13091627
         ocm 13091627
         97370
         
           
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             A treatise of angular sections by John Wallis ...
             Wallis, John, 1616-1703.
             Wallis, John, 1616-1703. Treatise of algebra.
          
           [2], 68 p., [2] leaves of plates : ill.
           
             Printed by John Playford for Richard Davis ...,
             London :
             1684.
          
           
             Appears at reel 854:14 as part of Wing W613: A treatise of algebra. London, 1685.
             Reproduction of original in Union Theological Seminary Library, New York.
          
        
      
    
     
       
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           Geometry -- Early works to 1800.
        
      
    
     
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           A
           TREATISE
           OF
           Angular
           Sections
           .
        
         
           By
           JOHN
           WALLIS
           ,
           
             D.
             D.
             Professor
             of
          
           Geometry
           
             in
             the
             University
             of
          
           Oxford
           ,
           
             and
             a
             Member
             of
             the
          
           Royal
           Society
           ,
           LONDON
           .
        
         
           printer's or publisher's device
        
         
           LONDON
           :
           Printed
           by
           
             Iohn
             Playford
          
           ,
           for
           
             Richard
             Davis
          
           ,
           Bookseller
           ,
           in
           the
           University
           of
           OXFORD
           ,
           1684.
           
        
      
    
     
       
         
         
         
           A
           TREATISE
           OF
           Angular
           Sections
           .
        
         
           
             
               CHAP.
               1.
            
             
             Of
             the
             Duplication
             and
             Bisection
             of
             an
             ARCH
             or
             ANGLE
             .
          
           
             I.
             LET
             the
             Chord
             (
             or
             Subtense
             )
             of
             an
             Arch
             proposed
             ,
             be
             called
             A
             ,
             (
             or
             E
             ;
             )
             of
             the
             Double
             ,
             B
             ;
             of
             the
             Treble
             ,
             C
             ;
             of
             the
             Quadruple
             ,
             D
             ;
             of
             the
             Quintruple
             ,
             F
             ;
             &c.
             
             The
             Radius
             ,
             R
             ;
             the
             Diameter
             ,
             2R
             .
             (
             But
             sometimes
             we
             shall
             give
             the
             name
             of
             the
             Subtense
             A
             ,
             E
             ,
             &c.
             to
             the
             Arch
             whose
             Subtense
             it
             is
             ;
             yet
             with
             that
             care
             ,
             as
             not
             to
             be
             liable
             to
             a
             mistake
             .
             )
          
           
             II.
             Where
             the
             Subtense
             of
             an
             Arch
             is
             A
             ;
             let
             the
             Versed
             sine
             be
             V
             :
             (
             where
             
             that
             is
             E
             ,
             let
             this
             be
             U.
             )
             Which
             drawn
             into
             (
             or
             Multiplied
             by
             )
             the
             remainder
             of
             the
             Diameter
             (
             2
             R
             −
             V
             )
             makes
             2
             R
             VVq
             ,
             the
             Square
             of
             the
             Right-sine
             :
             (
             this
             Sine
             being
             a
             Mean-proportional
             between
             the
             Segments
             of
             the
             Diameter
             on
             which
             it
             stands
             erect
             ,
             by
             13
             ●
             6.
             )
             That
             is
             ,
             (
             Q
             :
             ½
             B
             :
             )
             the
             Square
             of
             (
             the
             Right-sine
             ,
             or
             )
             half
             the
             Subtense
             of
             the
             double
             Arch
             :
             That
             is
             ,
             2
             R
             V
             −
             Vq
             =
             Q
             :
             ½
             B
             :
             =
             ¼
             Bq.
             
          
           
             III.
             If
             to
             this
             we
             add
             Vq
             (
             the
             Square
             of
             the
             Versed-sine
             ,
             )
             it
             makes
             2
             RV
             =
             (
             ¼
             Bq
             +
             Vq
             =
             )
             Aq.
             (
             And
             ,
             by
             the
             same
             reason
             ,
             2
             R
             U
             =
             Eq.
             )
             That
             is
             ,
          
           
             IV.
             
               The
               Subtense
               of
               an
               Arch
               ,
               is
               a
               Mean
               Proportional
               between
               the
               Diameter
               and
               the
               Versed-sine
               .
            
          
           
             V.
             Again
             ,
             because
             2
             R
             V
             =
             Aq
             ,
             therefore
             (
             dividing
             both
             by
             2
             R
             ,
             )
             
             :
             And
             (
             the
             Square
             thereof
             )
             
             Which
             subtracted
             from
             Aq
             ,
             leaves
             the
             Square
             of
             the
             Right-sine
             ,
             
             (
             And
             ,
             in
             like
             manner
             ,
             
             ,
             and
             
             ,
             and
             
             That
             is
             ,
          
           
           
             VI.
             
               If
               from
               the
               Square
               of
               the
               Subtense
               ,
               we
               take
               its
               Biquadrate
               divided
               by
               the
               Square
               of
               the
               Diameter
               ;
               the
               Remainder
               is
               equal
               to
               the
               Square
               of
               the
               Right-sine
               :
               And
               the
               Square-root
               of
               that
               Remainder
               ,
               to
               the
               Sine
               it self
               :
               And
               ,
               the
               double
               of
               this
               ,
               to
               the
               Subtense
               of
               the
               double
               Arch.
               
            
          
           
             VII
             .
             Accordingly
             ,
             because
             
             therefore
             (
             its
             Quadruple
             )
             
             and
             
             (
             And
             in
             like
             manner
             ,
             
             That
             is
             ,
          
           
             VIII
             .
             
               If
               from
               Four-times
               the
               Square
               of
               a
               Subtense
               ,
               are
               taken
               its
               Biquadrate
               divided
               by
               the
               Square
               of
               the
               Radius
               ;
               the
               Remainder
               is
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               :
               And
               ,
               the
               Quadratick
               Root
               of
               that
               Remainder
               ,
               is
               the
               Subtense
               it self
               .
            
          
           
             IX
             .
             But
             
             That
             is
             ,
          
           
             X.
             
               The
               Rect-angle
               of
               the
               Subtense
               of
               an
               Arch
               ,
               and
               of
               its
               Remainder
               to
               a
               Semicircle
               ,
               divided
               by
               the
               Radius
               ;
               is
               equal
               to
               the
               Subtense
               of
               the
               double
               Arch.
               
            
          
           
             XI
             .
             Because
             
             ;
             therefore
             AE
             =
             RB
             =
             2R
             ×
             ½B
             :
             And
             ,
             R.
             A
             :
             :
             B.
             B
             :
             And
             therefore
             ,
             (
             because
             A
             E
             contain
             a
             Right-angle
             ,
             as
             being
             an
             Angle
             in
             a
             Semicircle
             .
             )
          
           
             XII
             .
             
               In
               a
               Right-angled
               Triangle
               ,
               the
               Rect-angle
               of
               the
               two
               Legs
               containing
               the
               Right-angle
               ,
               is
               equal
               to
               that
               of
               the
               Hypothenuse
               ,
               and
               the
               Perpendicular
               from
               the
               Right-angle
               thereupon
               .
            
             And
             ,
          
           
             XIII
             .
             
               As
               the
               Radius
               ,
               to
               the
               Subtense
               of
               an
               Arch
               ;
               so
               the
               Subtense
               of
               its
               Remainder
               to
               a
               Semicircle
               ,
               is
               to
               that
               of
               the
               double
               Arch.
               
            
          
           
             XIV
             .
             Because
             B
             ,
             the
             Subtense
             of
             a
             double
             Arch
             ,
             doth
             indifferently
             subtend
             the
             two
             Segments
             which
             compleat
             the
             whole
             Circumference
             ;
             and
             ,
             consequently
             ,
             the
             half
             of
             either
             may
             be
             the
             single
             Arch
             of
             this
             double
             :
             It
             is
             therefore
             necessary
             that
             this
             Equation
             have
             two
             (
             Affirmative
             )
             Roots
             ;
             the
             greater
             of
             which
             we
             will
             call
             A
             ;
             and
             the
             lesser
             E
             :
             And
             therefore
             
             That
             is
             ,
          
           
             XV.
             
               Any
               Arch
               ,
               and
               its
               Remainder
               to
               a
               Semicircumference
               ,
               (
               as
               also
               its
               excess
               above
               a
               Semicircumference
               ,
               and
               either
               of
               them
               increased
               by
               one
               or
               more
               Semicircumferences
               ,
               )
               will
               have
               the
               same
               Subtense
               of
               the
               double
               Arch.
            
             For
             in
             all
             these
             Cases
             ,
             the
             Subtense
             of
             the
             single
             Arch
             will
             be
             either
             A
             or
             E.
             
          
           
             XVI
             .
             Because
             
             And
             therefore
             ,
             
             and
             4Aq
             Rq
             −
             Aqq
             (
             =
             Bq
             Rq
             )
             =
             4Eq
             Rq
             −
             Eqq
             :
             Therefore
             (
             by
             Transposition
             )
             4Aq
             Rq
             −
             4Eq
             Rq
             =
             Aqq
             −
             Eqq
             ;
             and
             (
             dividing
             both
             by
             
             That
             is
             ,
          
           
           
             XVII
             .
             
               The
               Square
               of
               the
               Diameter
               ,
               is
               equal
               to
               the
               difference
               of
               the
               Biquadrates
               of
               the
               Subtenses
               of
               two
               Arches
               ,
               (
               which
               together
               complete
               a
               Semicircumference
               )
               divided
               by
               the
               difference
               of
               their
               Squares
               :
               And
               this
               also
               ,
               equal
               to
               the
               sum
               of
               the
               Squares
               of
               those
               Subtenses
               .
            
             That
             is
             ,
             (
             because
             A
             E
             contain
             a
             Right-angle
             .
             )
          
           
             XVIII
             .
             
               In
               a
               Right-angled
               Triangle
               ,
               the
               Square
               of
               the
               Hypothenuse
            
             (
             4Rq
             )
             
               is
               equal
               to
               the
               Squares
               of
               the
               sides
               containing
               the
               Right-angle
               .
            
             (
             Aq+Eq
             .
             )
          
           
             XIX
             .
             Or
             thus
             ,
             Because
             B
             is
             the
             common
             Subtense
             to
             two
             Segments
             ,
             which
             
             together
             complete
             the
             whole
             Circumference
             ;
             and
             therefore
             the
             half
             of
             both
             ,
             complete
             the
             Semicircumference
             :
             If
             therefore
             in
             a
             Circle
             (
             according
             to
             
               Ptolemy's
               Lemma
            
             )
             a
             Trapezium
             be
             inscribed
             ,
             whose
             opposite
             sides
             are
             A
             ,
             A
             ;
             and
             E
             ,
             E
             :
             The
             Diagonals
             will
             be
             Diameters
             ,
             that
             is
             ,
             2R
             :
             And
             ,
             consequently
             ,
             4Rq
             =
             Aq
             +
             Eq
             ;
             as
             before
             .
          
           
             XX.
             Hence
             therefore
             ,
             
               The
               Radius
            
             (
             R
             )
             
               with
               the
               Subtense
               of
               an
               Arch
            
             (
             A
             or
             E
             )
             
               being
               given
               ;
               we
               have
               thence
               the
               Subtense
               of
               the
               double
               Arch
               ,
            
             B
             :
             (
             which
             is
             the
             
               Duplication
               of
               an
               Arch
            
             or
             Angle
             .
             )
             For
             ,
             R
             ,
             A
             ,
             being
             given
             ;
             we
             have
             
             (
             or
             R
             ,
             E
             ,
             being
             given
             ,
             we
             have
             A
             =
             4Rq
             −
             Eq
             :
             )
             And
             ,
             having
             R
             ,
             A
             ,
             E
             ;
             we
             have
             
             ,
             by
             §
             9.
             
          
           
             XXI
             .
             
               The
               Radius
            
             R
             ,
             with
             B
             
               the
               Subtense
               of
               the
               double
               Arch
               ,
               being
               given
               ;
               we
               have
               thence
               the
               Subtense
               of
               the
               single
               Arch
               ,
            
             A
             or
             E.
             (
             which
             is
             the
             
               Bisection
               of
               an
               Arch
            
             or
             Angle
             .
             )
             For
             ,
             by
             §
             14
             ,
             
             :
             And
             therefore
             4Rq
             Aq
             −
             Aqq
             (
             =
             Rq
             Bq
             )
             =
             4Rq
             Eq
             −
             Eqq.
             And
             the
             Roots
             of
             this
             Equation
             ,
             
             ,
             or
             Eq.
             And
             ,
             the
             Quadratick
             Root
             of
             this
             ,
             is
             A
             ,
             or
             E.
             
          
           
             XXII
             .
             Hence
             also
             we
             have
             an
             easie
             
               Method
               ,
               for
               a
               Geometrical
               Construction
               for
            
             
             
               the
               Resolution
               of
               such
               Biquadratick
               Equations
               ;
               or
               Quadratick
               Equations
               of
               a
               Plain
               Root
               ,
               wherein
               the
               Highest
               Power
               is
               Negative
               .
            
             (
             Understand
             it
             in
             Mr.
             Oughtred's
             Language
             :
             Who
             puts
             the
             Absolute
             Quantity
             ,
             Affirmative
             ;
             and
             by
             it self
             ;
             and
             the
             rest
             of
             the
             Equation
             all
             on
             the
             other
             side
             .
             )
             Suppose
             ,
             Rq
             Bq
             =
             4Rq
             Aq
             −
             Aqq
             ,
             or
             (
             putting
             P
             =
             ½
             B
             ,
             )
             4Rq
             Pq
             =
             4Rq
             Aq
             −
             Aqq.
             For
             ,
             dividing
             the
             Absolute
             term
             Rq
             Bq
             ,
             or
             4Rq
             Pq
             ,
             by
             the
             Co-efficient
             of
             the
             middle
             term
             4Rq
             ,
             the
             Result
             is
             ¼Bq
             ,
             or
             Pq
             ;
             and
             its
             Root
             ½
             B
             or
             P.
             Which
             being
             set
             Perpendicular
             on
             a
             Diameter
             equal
             to
             2
             R
             (
             the
             Square
             Root
             of
             that
             Co-efficient
             :
             )
             a
             streight
             Line
             from
             the
             top
             of
             it
             ,
             Parallel
             to
             that
             Diameter
             ,
             will
             (
             if
             the
             Equation
             be
             not
             impossible
             )
             cut
             the
             Circle
             ,
             or
             at
             least
             touch
             it
             :
             From
             which
             Point
             of
             Section
             or
             Contact
             ,
             two
             streight-lines
             drawn
             to
             the
             ends
             of
             the
             Diameter
             ,
             are
             A
             ,
             and
             E
             ,
             the
             two
             Roots
             of
             that
             (
             ambiguous
             )
             Biquadratick
             Equation
             ,
             (
             or
             ,
             if
             we
             call
             it
             a
             Quadratick
             of
             a
             Plain-root
             ,
             the
             Root
             of
             the
             Plain-root
             of
             such
             Quadratick
             Equation
             .
             )
          
           
             XXIII
             .
             And
             this
             Construction
             ,
             is
             the
             same
             with
             the
             Resolution
             of
             this
             Problem
             ;
             
               In
               a
               Right-angled
               Triangle
               ,
               the
               Hypothenuse
               being
               given
               ,
               and
               a
               Perpendicular
               from
               the
               Right-angle
               thereupon
               ,
               to
               find
               the
               other
               sides
            
             ;
             (
             and
             ,
             if
             need
             be
             ,
             the
             Angles
             ,
             the
             Segments
             of
             the
             Hypothenuse
             ,
             and
             the
             Area
             of
             the
             Triangle
             ½
             R
             B
             or
             P
             R.
             )
          
           
             XXIV
             .
             Or
             thus
             :
             Having
             R
             and
             B
             ,
             (
             as
             at
             §
             22.
             )
             with
             the
             Radius
             R
             describe
             a
             Circle
             ;
             and
             therein
             inscribe
             the
             Chord
             B
             ;
             and
             another
             on
             the
             middle
             hereof
             at
             Right-angles
             :
             (
             which
             will
             therefore
             bisect
             that
             ,
             and
             be
             a
             Diameter
             :
             )
             And
             ,
             from
             both
             ends
             of
             this
             ,
             to
             either
             end
             of
             B
             ,
             draw
             the
             Lines
             A
             ,
             E
             ;
             as
             before
             .
             And
             this
             Construction
             is
             better
             than
             the
             former
             ,
             because
             of
             the
             uncertainty
             of
             the
             precise
             Point
             of
             Contact
             or
             Section
             ,
             in
             case
             the
             Section
             be
             somewhat
             Oblique
             .
          
           
           
             XXV
             .
             Now
             if
             it
             be
             desired
             ,
             in
             like
             manner
             ,
             to
             give
             a
             like
             Construction
             ,
             in
             Case
             of
             
               such
               Biquadratick
               Equations
            
             (
             or
             Quadraticks
             of
             a
             Plain-root
             )
             
               where
               the
               highest
               Power
               is
               Affirmative
            
             ;
             (
             though
             that
             be
             here
             a
             Digression
             ,
             as
             in
             all
             the
             rest
             that
             follow
             ,
             to
             §
             35.
             )
             It
             is
             thus
             :
             Suppose
             the
             Equations
             Aqq
             −
             VqAq
             =
             VqEq
             =
             Pqq
             +
             VqPq
             :
             Whose
             (
             Affirmative
             )
             Roots
             are
             Aq
             ,
             and
             Pq
             ;
             (
             and
             therefore
             Vq
             ,
             VqEq
             ,
             and
             consequently
             Eq
             ,
             are
             known
             Quantities
             :
             )
             Therefore
             (
             by
             Transposition
             )
             Aqq
             −
             VqAq
             +
             VqPq
             ;
             and
             (
             dividing
             by
             
             And
             therefore
             Aq
             −
             Vq
             =
             Pq
             ;
             and
             Pq
             +
             Vq
             =
             Aq
             :
             And
             (
             by
             Multiplication
             )
             Aqq
             −
             VqAq
             =
             AqPq
             =
             Pqq
             +
             VqPq
             =
             VqEq
             .
          
           
           
             XXVI
             .
             The
             Equation
             therefore
             proposed
             (
             dividing
             all
             by
             Vq
             )
             comes
             to
             this
             ,
             
             That
             is
             ,
             
             .
             Whose
             Roots
             are
             
             ,
             and
             
             .
             Namely
             ,
             
             .
             And
             
             .
             And
             these
             Multiplied
             into
             V
             (
             a
             known
             Quantity
             )
             make
             Aq
             ,
             and
             Pq
             :
             Namely
             ,
             
             .
             And
             
             And
             consequently
             ,
             A
             is
             a
             mean
             Proportional
             between
             V
             and
             
             .
             And
             P
             ,
             a
             mean
             Proportional
             between
             V
             and
             
             .
             Therefore
             ,
          
           
             XXVII
             .
             And
             Equation
             being
             proposed
             in
             one
             of
             these
             Forms
             ,
             Aqq
             −
             VqAq
             
             =
             VqEq
             =
             Pqq
             +
             VqPq
             :
             The
             absolute
             term
             (
             VqEq
             )
             being
             divided
             by
             the
             Co-efficient
             of
             the
             middle
             term
             (
             Vq
             ;
             )
             the
             quantity
             resulting
             is
             (
             Eq
             )
             whose
             Square-root
             (
             E
             ,
             )
             set
             Perpendicular
             on
             the
             end
             of
             a
             streight
             Line
             equal
             to
             (
             V
             )
             the
             Square-root
             of
             the
             Co-efficient
             ;
             which
             we
             may
             suppose
             the
             Diameter
             of
             a
             Circle
             ,
             to
             which
             that
             Perpendicular
             is
             a
             Tangent
             :
             On
             the
             same
             Center
             with
             this
             Circle
             (
             and
             on
             the
             same
             Diameter
             continued
             )
             by
             the
             Top
             of
             that
             Perpendicular
             ,
             draw
             a
             second
             Circle
             .
             The
             Diameter
             of
             this
             second
             Circle
             ,
             is
             ,
             by
             that
             Perpendicular
             (
             E
             )
             cut
             into
             two
             Segments
             ,
             which
             are
             the
             Roots
             of
             these
             Equations
             .
             That
             is
             ,
             
             ;
             and
             
          
           
             XXVIII
             .
             Or
             ,
             (
             without
             drawing
             that
             second
             Circle
             ,
             )
             from
             the
             Top
             of
             that
             Perpendicular
             (
             in
             a
             streight
             Line
             through
             the
             Center
             of
             the
             first
             ,
             which
             will
             cut
             the
             Circumference
             in
             two
             Points
             ,
             )
             to
             the
             first
             Section
             ,
             is
             
             ;
             to
             the
             second
             ,
             
          
           
             XXIX
             .
             These
             two
             Roots
             ,
             Multiplied
             one
             into
             the
             other
             ,
             become
             equal
             to
             the
             Absolute
             quantity
             ,
             
             .
             And
             Multiplied
             into
             V
             ,
             become
             Aq
             ,
             Pq
             :
             Or
             ,
             thus
             ,
             P
             is
             a
             mean
             Proportional
             between
             V
             and
             U
             ;
             and
             A
             ,
             between
             V
             and
             V+U
             :
             Or
             thus
             ,
             P
             is
             a
             mean
             Proportional
             between
             V
             and
             
             ;
             and
             (
             because
             by
             §
             25.
             
             Aq
             =
             Pq
             +
             Vq
             ,
             )
             A
             is
             the
             Hypothenuse
             (
             in
             a
             Right-angled
             Triangle
             )
             to
             the
             Legs
             P
             ,
             V.
             And
             this
             is
             no
             contemptible
             Method
             ,
             
               For
               the
               resolving
               Quadratick
               Equations
               of
               a
               Plain-root
               ,
               wherein
               the
               highest
               term
               is
               Affirmative
               .
            
             The
             whole
             Geometrick
             Construction
             ,
             is
             clear
             enough
             from
             the
             Figures
             adjoined
             ;
             where
             yet
             the
             Circles
             ,
             (
             for
             the
             most
             part
             )
             serve
             rather
             for
             the
             Demonstration
             ,
             than
             the
             Construction
             .
          
           
           
             XXX
             .
             Again
             ,
             (
             by
             the
             same
             §
             25.
             )
             
             And
             therefore
             A
             and
             E
             ,
             are
             also
             the
             Legs
             of
             a
             Right-angled
             Triangle
             ,
             whose
             Hypothenuse
             is
             V
             +
             U
             :
             Which
             ,
             by
             P
             (
             a
             Perpendicular
             on
             it
             from
             the
             Right-angle
             )
             is
             cut
             into
             those
             two
             Segments
             .
          
           
             XXXI
             .
             From
             the
             same
             Construction
             therefore
             ,
             we
             have
             also
             the
             Geometrical
             Construction
             of
             this
             Problem
             ;
             
               In
               a
               Right-angled
               Triangle
               ,
               having
               one
               of
               the
               Legs
            
             E
             ,
             
               with
               the
               farther
               Segment
               of
               the
               Hypothenuse
            
             V
             ,
             
               to
               find
               the
               other
               Segment
            
             ;
             (
             and
             so
             the
             whole
             V
             +
             U
             ;
             and
             the
             Perpendicular
             P
             ;
             and
             the
             other
             Leg
             A
             ;
             and
             the
             whose
             Triangle
             .
             )
          
           
             XXXII
             .
             We
             have
             thence
             also
             this
             Analogy
             ;
             
             And
             
             Or
             thus
             ,
             
             And
             
          
           
             XXXIII
             .
             If
             therefore
             we
             make
             V
             the
             Radius
             of
             a
             Circle
             ;
             then
             is
             A
             the
             
             Secant
             ;
             P
             ,
             the
             Tangent
             ;
             E
             a
             Parallel
             to
             the
             Right-sine
             (
             in
             contrary
             position
             )
             from
             the
             end
             of
             the
             Secant
             to
             the
             Diameter
             produced
             .
             If
             we
             make
             A
             the
             Radius
             ;
             then
             is
             P
             the
             Right-sine
             ,
             and
             E
             the
             Tangent
             of
             the
             same
             Arch
             ;
             and
             V
             ,
             the
             Sine
             of
             the
             Complement
             ,
             or
             Difference
             between
             the
             Radius
             and
             Versed
             Sine
             .
             From
             hence
             therefore
             ,
          
           
             XXXIV
             .
             
               The
               Tangent
            
             E
             ,
             
               and
               Sine
               of
               the
               Complement
            
             V
             ,
             
               being
               given
               ;
               we
               have
               the
               Right-sine
            
             P
             ,
             
               and
               the
               Radius
            
             A.
             (
             But
             ,
             §
             25
             ,
             and
             all
             hitherto
             ,
             is
             a
             Digression
             .
             )
          
           
             XXXV
             .
             If
             in
             a
             Semicircle
             on
             the
             Diameter
             2
             R
             ,
             we
             inscribe
             B
             the
             Subtense
             
             of
             a
             double
             Arch
             :
             A
             Perpendicular
             on
             the
             middle
             Point
             hereof
             ,
             will
             cut
             the
             Arch
             of
             that
             Semicircle
             into
             two
             Segments
             ,
             (
             whose
             Subtenses
             are
             A
             ,
             E
             ;
             )
             either
             of
             which
             is
             a
             single
             Arch
             ,
             to
             the
             double
             whereof
             ,
             B
             is
             a
             Subtense
             .
             This
             ,
             as
             to
             E
             ,
             is
             evident
             from
             4
             è
             1
             ,
             and
             28
             è
             3
             :
             And
             ,
             as
             to
             A
             ,
             from
             §
             15
             of
             this
             .
          
           
             XXXVI
             .
             But
             also
             (
             by
             the
             same
             reason
             ,
             )
             the
             Arch
             β
             (
             the
             difference
             of
             the
             Arches
             A
             ,
             E
             ;
             )
             and
             B
             (
             the
             double
             of
             either
             ,
             )
             will
             (
             if
             doubled
             )
             have
             the
             same
             Subtense
             of
             their
             double
             Arch.
             That
             is
             ,
             
               The
               double
               of
               the
               double
            
             (
             of
             either
             )
             
               and
               the
               double
               of
               their
               difference
               ,
               will
               have
               the
               same
               Subtense
               .
            
          
           
             XXXVII
             .
             If
             an
             Arch
             to
             be
             doubled
             ,
             be
             just
             a
             third
             part
             of
             the
             Circumference
             ;
             
             the
             Subtense
             of
             the
             double
             ,
             is
             equal
             to
             that
             of
             the
             single
             Arch.
             (
             For
             the
             same
             Subtense
             ,
             which
             on
             one
             side
             subtends
             two
             Trients
             ,
             doth
             on
             the
             other
             side
             subtend
             but
             one
             .
             )
             That
             is
             ,
             by
             §
             7
             ,
             
             .
             And
             therefore
             (
             by
             Transposition
             )
             
             ,
             and
             3Rq
             =
             Aq.
             That
             is
             ,
          
           
             XXXVIII
             .
             
               The
               Square
               of
               the
               Subtense
               to
               a
               Trient
               of
               the
               Circumference
            
             (
             or
             of
             the
             side
             of
             an
             Equilater
             Triangle
             inscribed
             )
             
               is
               equal
               to
               three
               Squares
               of
               the
               Radius
               .
            
          
           
             XXXIX
             .
             Again
             ,
             the
             same
             being
             the
             Subtense
             of
             the
             double
             Trient
             ,
             and
             of
             the
             double
             Sextant
             ,
             (
             for
             a
             Trient
             and
             a
             Sextant
             compleat
             the
             half
             ,
             ⅓
             +
             ⅙
             =
             ½
             )
             the
             Square
             of
             the
             Subtense
             of
             a
             Sextant
             ,
             (
             Eq
             ,
             )
             is
             the
             difference
             of
             the
             Squares
             of
             that
             of
             the
             Trient
             ,
             and
             (
             the
             Diameter
             or
             )
             that
             of
             the
             Semicircumference
             .
             That
             is
             ,
             4Rq
             −
             Aq
             =
             Eq
             ;
             that
             is
             ,
             (
             by
             §
             preced
             .
             )
             4Rq
             −
             3Rq
             =
             Rq
             =
             Eq
             :
             And
             ,
             E
             =
             R.
             That
             is
             ,
          
           
             XL.
             
               The
               Subtense
               of
               a
               Sextant
            
             (
             or
             side
             of
             the
             inscribed
             Equilater
             Hexagon
             )
             
               is
               equal
               to
               the
               Radius
            
             .
          
        
         
           
           
             
               CHAP.
               II.
            
             Of
             the
             Triplication
             and
             Trisection
             of
             an
             ARCH
             or
             ANGLE
             .
          
           
             I.
             IF
             in
             a
             Circle
             ,
             be
             inscribed
             a
             Quadrilater
             ,
             whose
             three
             sides
             are
             A
             ,
             A
             ,
             A
             ,
             
             (
             Subtenses
             of
             a
             single
             Arch
             )
             and
             the
             fourth
             C
             ,
             (
             the
             Subtense
             of
             the
             Triple
             Arch
             :
             )
             the
             Diagonals
             are
             B
             ,
             B
             ,
             the
             Subtense
             of
             the
             double
             ;
             )
             as
             is
             evident
             .
             But
             it
             is
             evident
             also
             ,
             that
             (
             in
             this
             Case
             )
             A
             is
             less
             than
             a
             Trient
             of
             the
             whole
             Circumference
             .
          
           
             II.
             And
             therefore
             (
             the
             Rect-angle
             of
             the
             Diagonals
             being
             equal
             to
             the
             two
             Rect-angles
             of
             the
             opposite
             sides
             ,
             )
             Bq
             =
             Aq+AC
             ;
             and
             therefore
             Bq
             −
             Aq
             =
             AC
             ,
             and
             
             .
             That
             is
             ,
          
           
             III.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ;
               is
               equal
               to
               the
               Square
               of
               the
               Subtense
               of
               the
               Single
               Arch
               (
               less
               than
               a
               Trient
               of
               the
               Circumference
               )
               and
               the
               Rectangle
               of
               the
               Subtenses
               of
               the
               Sngle
               and
               Treble
               Arch.
            
             And
             therefore
             ,
          
           
             IV.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ,
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               single
               Arch
               ,
               (
               being
               less
               than
               a
               Trient
               ,
               )
               is
               equal
               to
               the
               Rect-angle
               of
               the
               Subtenses
               of
               the
               single
               and
               Treble
               Arch.
            
             And
             consequently
             ,
          
           
             V.
             
               If
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ,
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
               Trient
               ,
               )
               be
               divided
               by
               the
               Subtense
               of
               the
               single
               Arch
               ;
               the
               Result
               is
               the
               Subtense
               of
               the
               Triple
               Arch.
               
            
          
           
             VI.
             Because
             that
             (
             by
             §
             2.
             )
             
             ;
             and
             ,
             that
             B+A
             into
             B
             −
             A
             is
             equal
             to
             Bq
             −
             Aq
             :
             (
             as
             will
             appear
             by
             Multiplication
             :
             )
             .
             Therefore
             ,
             
             That
             is
             ,
          
           
             VII
             .
             
               As
               the
               Subtense
               of
               a
               single
               Arch
               (
               less
               than
               a
               Trient
               )
               to
               the
               sum
               of
               the
               Subtenses
               of
               the
               single
               and
               double
               Arch
               ;
               so
               is
               the
               Excess
               of
               that
               of
               the
               double
               above
               that
               of
               the
               single
               ,
               to
               the
               Subtense
               of
               the
               Triple
               .
            
          
           
             VIII
             .
             Again
             ,
             because
             (
             by
             §
             7
             ,
             
               of
               the
               precedent
               Chapter
            
             ,
             )
             
             :
             Therefore
             ,
             
             :
             And
             therefore
             
             .
             That
             is
             ,
          
           
             IX
             .
             
               The
               Triple
               of
               the
               Subtense
               of
               an
               Arch
               (
               less
               than
               a
               Trient
               ,
               )
               wanting
               the
               Cube
               thereof
               ,
               divided
               by
               the
               Square
               of
               the
               Radius
               ;
               is
               equal
               to
               the
               Subtense
               of
               the
               Triple
               Arch.
               
            
          
           
             X.
             But
             ,
             because
             the
             same
             Subtense
             C
             ,
             subtends
             also
             to
             another
             Segment
             of
             the
             same
             Circle
             ;
             the
             Subtense
             of
             whose
             Trient
             we
             shall
             call
             E
             :
             Therefore
             
             .
          
           
             XI
             .
             And
             because
             the
             three
             Arches
             A
             ,
             A
             ,
             A
             ;
             and
             the
             three
             Arches
             E
             ,
             E
             ,
             E
             ;
             complete
             the
             whole
             Circumference
             :
             (
             as
             is
             evident
             ;
             )
             Therefore
             ,
             once
             A
             ,
             and
             once
             E
             ,
             complete
             a
             Trient
             or
             third
             part
             thereof
             .
             Therefore
             ,
          
           
           
             XII
             .
             
               An
               Arch
               less
               than
               the
               Trient
               of
               a
               Circumference
               ,
               and
               the
               Residue
               of
               that
            
             
             Trient
             ,
             (
             A
             ,
             and
             E
             ,
             )
             
               have
               the
               same
               Subtense
               of
               their
               Triple
               Arch.
               
            
          
           
             XIII
             .
             Again
             ,
             because
             (
             as
             is
             shewed
             already
             )
             
             ;
             and
             therefore
             3RqA
             −
             Ac
             =
             3RqE
             −
             Ec
             ;
             and
             3RqA
             −
             3RqE
             =
             Ac
             −
             Ec
             :
             Therefore
             ,
             (
             dividing
             both
             by
             A
             −
             E
             ,
             )
             
             .
             (
             As
             will
             appear
             upon
             dividing
             Ac
             −
             Ec
             by
             A
             −
             E
             ;
             or
             Multiplying
             A
             −
             E
             into
             Aq+AE+Eq
             .
             )
          
           
             XIV
             .
             But
             (
             by
             §
             37
             ,
             38
             ,
             
               Chap.
               preced
            
             .
             )
             3Rq
             is
             the
             Square
             of
             the
             Subtense
             of
             a
             Trient
             ;
             that
             is
             (
             by
             §
             11
             of
             this
             )
             of
             the
             sum
             of
             the
             Arches
             A
             and
             E.
             Therefore
             ,
          
           
             XV.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               Trient
               of
               the
               Circumference
               of
               a
               Circle
               ,
            
             (
             or
             three
             Squares
             of
             the
             Radius
             ,
             )
             
               is
               equal
               to
               the
               Squares
               of
               the
               Subtenses
               of
               any
               two
               Arches
               completing
               that
               Trient
               ,
               and
               the
               Rect-angle
               of
               them
               .
            
             That
             is
             ,
             (
             putting
             T
             for
             the
             Subtense
             of
             a
             Trient
             )
             Tq
             (
             =
             3Rq
             )
             =
             Aq+AE+Aq
             .
          
           
             XVI
             .
             But
             the
             Angle
             which
             AE
             contain
             ,
             (
             as
             being
             an
             Angle
             in
             the
             Trient
             of
             a
             Circle
             ,
             or
             insisting
             on
             two
             Trients
             ,
             )
             is
             an
             Angle
             of
             120
             Degrees
             .
             And
             therefore
             (
             by
             §
             15.
             )
          
           
             XVII
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               one
               of
               whose
               Angles
               is
            
             120
             
               Degrees
               ;
               the
               Square
               of
               the
               Subtense
               to
               that
               Angle
               ,
               is
               equal
               to
               the
               two
               Squares
               of
               the
               sides
               containing
               it
               ,
               and
               a
               Rect-angle
               of
               those
               sides
               .
            
             (
             For
             ,
             if
             such
             Triangle
             be
             inscribed
             in
             a
             Circle
             ,
             the
             Base
             of
             that
             Triangle
             ,
             will
             be
             the
             subtendent
             of
             a
             Trient
             in
             such
             Circle
             ;
             or
             
             Rq.
             )
          
           
             XVIII
             .
             If
             a
             Quadrilater
             be
             inscribed
             in
             a
             Circle
             ,
             three
             of
             whose
             sides
             are
             
             A
             ,
             E
             ,
             A
             ,
             (
             or
             E
             ,
             A
             ,
             E
             ,
             )
             and
             the
             fourth
             Z
             :
             Each
             of
             the
             Diagonals
             (
             by
             §
             11.
             )
             is
             T
             ,
             the
             Subtense
             of
             a
             Trient
             .
             And
             therefore
             (
             by
             §
             13
             ,
             14
             ,
             15
             ,
             )
             ZE+Aq
             (
             =
             ZA+Eq
             )
             =
             Tq
             =
             3Rq
             =
             Aq+AE+Eq
             .
             And
             ,
             consequently
             ,
             ZE
             =
             AE+Eq
             ,
             and
             ZA
             =
             Aq+AE
             ;
             and
             therefore
             ,
             Z
             =
             A+E
             .
             And
             therefore
             ,
          
           
             XIX
             .
             
               If
               to
               the
               Aggregate
               of
               two
               Arches
            
             A
             ,
             E
             ,
             
               (
               completing
               a
               Trient
               ,
               )
               be
               added
               a
               third
               equal
               to
               either
               of
               them
            
             ;
             Z
             ,
             
               the
               Subtense
               of
               the
               Aggregate
               of
               all
               the
               three
               ,
               is
               equal
               to
               the
               sum
               of
               the
               Subtenses
               of
               those
               two
               .
            
             That
             is
             ,
             Z
             =
             A+E
             .
          
           
             XX.
             But
             the
             same
             Chord
             Z
             ,
             doth
             subtend
             ,
             on
             the
             one
             side
             ,
             to
             a
             Trient
             increased
             by
             the
             Arch
             A
             ;
             and
             ,
             on
             the
             other
             side
             ,
             to
             a
             Trient
             increased
             by
             the
             Arch
             E
             ;
             (
             as
             is
             evident
             ;
             )
             That
             is
             ,
             to
             an
             Arch
             which
             doth
             as
             much
             exceed
             a
             Trient
             (
             or
             want
             of
             two
             Trients
             ,
             )
             as
             the
             Arch
             A
             or
             E
             wants
             of
             a
             Trient
             .
             Therefore
             ,
          
           
             XXI
             .
             
               The
               Aggregate
               of
               the
               Subtenses
               of
               two
               Arches
               ,
               which
               together
               make
               up
               a
               Trient
               ;
               is
               equal
               to
               the
               Subtense
               of
               another
               Arch
               which
               doth
               as
               much
               exceed
               a
               Trient
               ,
               (
               or
               want
               of
               two
               Trients
               ,
               )
               as
               either
               of
               those
               two
               wants
               of
               a
               Trient
               .
            
          
           
             XXII
             .
             The
             same
             will
             in
             like
             manner
             be
             inferred
             ,
             if
             we
             inscribe
             a
             Quadrilater
             
             whose
             opposite
             sides
             are
             A
             ,
             T
             ,
             and
             E
             ,
             T
             ;
             and
             the
             Diagonals
             TZ
             .
             For
             then
             TA+TE
             =
             TZ
             ;
             and
             therefore
             A+E
             =
             Z
             ,
             as
             before
             .
          
           
           
             XXIII
             .
             But
             if
             either
             of
             the
             Arches
             to
             which
             Z
             subtendeth
             (
             greater
             than
             a
             
             Trient
             ,
             and
             less
             than
             two
             Trients
             )
             be
             Tripled
             ;
             the
             Subtense
             of
             this
             Triple
             ,
             is
             the
             same
             with
             that
             of
             the
             Triple
             of
             A
             or
             E.
             For
             the
             Triple
             of
             an
             Arch
             greater
             than
             a
             Trient
             ,
             is
             equal
             to
             one
             whole
             Circumference
             with
             the
             Triple
             of
             that
             Excess
             .
             (
             For
             the
             Triple
             of
             ⅓+A
             ,
             is
             1+3A
             .
             )
             Now
             because
             ,
             when
             we
             have
             once
             gone
             round
             the
             whole
             Circumference
             ,
             we
             are
             just
             there
             where
             at
             first
             we
             began
             ;
             this
             therefore
             (
             as
             to
             this
             Point
             )
             is
             as
             nothing
             ;
             and
             the
             whole
             distance
             to
             be
             acquired
             is
             but
             the
             Triple
             of
             such
             Excess
             ;
             and
             just
             the
             same
             as
             if
             onely
             this
             Excess
             had
             been
             thrice
             taken
             .
          
           
             XXIV
             .
             As
             for
             Example
             :
             If
             the
             Arch
             subtended
             by
             Z
             ,
             be
             β
             γ
             δ
             ,
             (
             that
             is
             ,
             a
             
             Trient
             increased
             by
             the
             Arch
             E
             ;
             )
             and
             to
             this
             we
             add
             a
             second
             equal
             to
             it
             ,
             δ
             ζ
             θ
             ;
             the
             Aggregate
             β
             γ
             δ
             ζ
             θ
             ,
             is
             the
             double
             Arch
             ,
             and
             the
             Subtense
             thereof
             is
             B
             ,
             or
             β
             θ
             ,
             (
             which
             is
             also
             the
             Subtense
             of
             the
             Difference
             of
             the
             Arches
             A
             ,
             E
             :
             )
             and
             if
             to
             these
             two
             ,
             we
             add
             a
             third
             equal
             to
             either
             of
             them
             θ
             γ
             χ
             ;
             then
             is
             β
             γ
             δ
             ζ
             θ
             γ
             χ
             ,
             the
             Triple
             of
             the
             Arch
             first
             proposed
             ;
             and
             the
             Subtense
             hereof
             (
             that
             is
             ,
             the
             Streight-line
             which
             joyns
             the
             beginning
             and
             the
             end
             of
             this
             Triple
             Arch
             )
             is
             β
             χ
             =
             C
             ;
             the
             very
             same
             which
             subtends
             the
             Triple
             of
             E.
             
          
           
             XXV
             .
             And
             just
             the
             same
             would
             come
             to
             pass
             ,
             if
             for
             the
             first
             Arch
             we
             take
             β
             θ
             ζ
             δ
             (
             that
             is
             ,
             a
             Trient
             increased
             by
             A
             ,
             )
             to
             which
             Z
             is
             a
             subtendent
             likewise
             .
             For
             ,
             taking
             a
             second
             equal
             to
             it
             δ
             χ
             γ
             β
             θ
             ;
             the
             Aggregate
             β
             θ
             ζ
             δ
             χ
             γ
             β
             θ
             (
             more
             than
             one
             entire
             Circumference
             )
             is
             the
             double
             Arch
             ,
             and
             the
             Subtense
             thereof
             B
             as
             before
             :
             And
             if
             to
             these
             two
             we
             add
             a
             third
             equal
             to
             either
             ,
             θ
             ζ
             δ
             χ
             ;
             the
             Triple
             Arch
             is
             β
             θ
             ζ
             δ
             χ
             γ
             β
             θ
             ζ
             δ
             χ
             ;
             and
             the
             Subtense
             hereof
             (
             as
             before
             )
             β
             χ
             or
             C
             ;
             the
             same
             with
             the
             subtendent
             of
             the
             Triple
             of
             A.
             And
             therefore
             ,
          
           
             XXVI
             .
             
               The
               Triple
               of
               an
               Arch
               greater
               than
               a
               Trient
               ,
               hath
               the
               same
               Subtense
               with
               the
               Triple
               of
               its
               Excess
               above
               a
               Trient
               .
            
             And
             the
             same
             (
             for
             the
             same
             reason
             )
             holds
             in
             Arches
             greater
             than
             2
             ,
             3
             ,
             or
             more
             Trients
             .
          
           
             XXVII
             .
             But
             note
             here
             ,
             that
             ,
             in
             this
             case
             ;
             That
             is
             ,
             if
             the
             Arch
             to
             be
             Tripled
             be
             greater
             than
             a
             Trient
             ,
             but
             less
             than
             two
             Trients
             ,
             (
             for
             if
             more
             than
             two
             Trients
             ,
             but
             less
             than
             the
             whole
             Circumference
             ,
             it
             is
             the
             same
             as
             if
             it
             were
             less
             than
             a
             Tri●●●●●
             ;
             )
             the
             Subtense
             of
             the
             double
             is
             less
             than
             that
             of
             the
             single
             .
             For
             ,
             in
             such
             case
             ,
             the
             Arch
             will
             differ
             from
             that
             of
             a
             Semicircle
             (
             either
             in
             Excess
             or
             in
             Defect
             )
             by
             less
             than
             ⅙
             of
             the
             whole
             Circumference
             .
             Let
             it
             be
             X.
             If
             therefore
             ½
             ±
             X
             be
             the
             single
             Arch
             ,
             the
             double
             will
             be
             1
             ±
             2X
             ;
             and
             the
             Subtense
             thereof
             (
             whether
             greater
             or
             less
             than
             one
             entire
             revolution
             )
             will
             be
             the
             same
             with
             that
             of
             2X
             :
             And
             therefore
             (
             X
             being
             less
             than
             ⅙
             ,
             )
             2X
             will
             require
             a
             less
             Subtense
             than
             that
             of
             ½
             ±
             X
             ;
             that
             being
             less
             than
             the
             Subtense
             of
             a
             Trient
             ,
             but
             this
             greater
             than
             it
             .
             And
             the
             like
             is
             to
             be
             understood
             in
             other
             cases
             of
             like
             nature
             .
          
           
             XXVIII
             .
             Supposing
             therefore
             ,
             as
             before
             ,
             
             ,
             or
             
             ;
             C
             must
             in
             this
             case
             be
             a
             Negative
             quantity
             :
             Or
             ,
             if
             we
             put
             C
             Affirmative
             ,
             then
             must
             Z
             be
             Negative
             ,
             (
             or
             less
             than
             nothing
             :
             )
             For
             Bq
             −
             Zq
             (
             where
             a
             greater
             quantity
             is
             to
             be
             subtended
             from
             a
             less
             )
             must
             needs
             be
             Negative
             ;
             that
             is
             Bq
             −
             Zq
             =
             Zc
             ;
             where
             Zc
             being
             a
             Negative
             ,
             either
             Z
             or
             C
             must
             be
             so
             too
             ,
             or
             else
             (
             putting
             all
             Affirmative
             )
             Zq
             −
             Bq
             =
             Zc
             ,
             and
             Zq
             =
             Bq+ZC
             .
          
           
             XXIX
             .
             Which
             is
             evident
             also
             from
             the
             Diagram
             ;
             where
             ,
             for
             this
             reason
             ,
             ZZ
             become
             Diagonals
             ;
             and
             both
             BB
             ,
             and
             ZC
             ,
             opposite
             sides
             .
             And
             therefore
             Zq
             −
             Bq
             =
             ZC
             ,
             or
             Zq
             =
             Bq+Zc
             ;
             and
             
             .
             That
             is
             ,
          
           
           
             XXX
             .
             
               The
               Square
               of
               the
               Subtense
               of
               a
               single
               Arch
               ,
               greater
               than
               a
               Trient
               ,
               but
            
             
             
               less
               than
               two
               Trients
               ;
               is
               equal
               to
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ,
               together
               with
               a
               Rect-angle
               of
               the
               Subtenses
               of
               the
               single
               and
               Triple
               Arch.
            
             And
             ,
          
           
             XXXI
             .
             
               The
               Square
               of
               the
               Subtense
               of
               a
               single
               Arch
               (
               greater
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               ,
               )
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ;
               is
               equal
               to
               the
               Rectangle
               of
               the
               Subtenses
               of
               the
               single
               and
               Triple
               Arch.
            
             And
             therefore
             ,
          
           
             XXXII
             .
             
               If
               the
               Square
               of
               the
               Subtense
               of
               a
               single
               Arch
               (
               greater
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               ,
               )
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ,
               be
               divided
               by
               that
               of
               the
               single
               ,
               the
               Result
               is
               the
               Subtense
               of
               the
               Triple
               Arch
               :
            
             (
             Or
             ,
             if
             divided
             by
             that
             of
             the
             Triple
             ,
             the
             Result
             is
             that
             of
             the
             single
             .
             )
             Or
             ,
          
           
             XXXIII
             .
             
               If
               from
               the
               Subtense
               of
               a
               single
               Arch
               (
               greater
               than
               a
               Trient
               but
               less
               than
               two
               Trients
               ,
               )
               we
               Subtract
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               divided
               by
               that
               of
               the
               single
               ;
               the
               Remainder
               is
               equal
               to
               the
               Subtense
               of
               the
               Triple
               .
            
          
           
             XXXIV
             .
             But
             ,
             because
             of
             
             ,
             or
             Z
             )
             Zq
             −
             Bq
             (
             C
             ;
             and
             Zq
             −
             Bq
             =
             Z+B
             into
             Z
             −
             B
             :
             We
             have
             thence
             this
             Analogy
             ,
             
             That
             is
             ,
          
           
             XXXV
             .
             
               As
               the
               Subtense
               of
               a
               single
               Arch
               (
               greater
               than
               a
               Trient
               but
               less
               than
               two
               Trients
               ,
               )
               to
               the
               Aggregate
               of
               the
               Subtenses
               of
               the
               single
               and
               double
               ;
               so
               is
               that
               of
               the
               single
               wanting
               that
               of
               the
               double
               to
               that
               of
               the
               Triple
               .
            
          
           
             XXXVI
             .
             Now
             because
             (
             as
             we
             have
             shewed
             )
             Zq
             −
             Bq
             =
             ZC
             ;
             and
             (
             by
             §
             7
             ,
             8
             ,
             
               Chap.
               preced
            
             .
             )
             
             :
             Therefore
             
             :
             And
             
             .
             That
             is
             ,
          
           
             XXXVII
             .
             
               If
               from
               the
               Cube
               of
               the
               Subtense
               of
               a
               single
               Arch
               (
               greater
               than
               a
               Trient
               but
               less
               than
               two
               Trients
               )
               divided
               by
               the
               Square
               of
               the
               Radius
               ;
               we
               subtract
               the
               Triple
               of
               that
               Subtense
               :
               The
               Remainder
               is
               equal
               to
               the
               Subtense
               of
               the
               Triple
               Arch.
               
            
          
           
             XXXVIII
             .
             If
             the
             Arch
             to
             be
             Tripled
             be
             greater
             than
             two
             Trients
             ,
             it
             is
             the
             same
             as
             if
             it
             were
             less
             than
             one
             Trient
             .
             (
             For
             the
             Residue
             of
             the
             whole
             ,
             to
             which
             it
             also
             subtends
             ,
             is
             then
             less
             than
             a
             Trient
             .
             )
             And
             therefore
             the
             same
             Chord
             (
             suppose
             A
             or
             E
             )
             subtends
             as
             well
             to
             an
             Arch
             greater
             than
             two
             Trients
             ,
             as
             to
             one
             less
             than
             one
             Trient
             .
          
           
             XXXIX
             .
             If
             the
             Arch
             to
             be
             Tripled
             be
             equal
             to
             a
             Trient
             ;
             it
             is
             indifferent
             to
             whether
             of
             the
             two
             cases
             it
             be
             referred
             ,
             (
             that
             of
             the
             greater
             ,
             or
             that
             of
             the
             lesser
             ,
             than
             a
             Trient
             ,
             )
             and
             the
             same
             happens
             if
             it
             be
             supposed
             equal
             to
             two
             or
             more
             Trients
             ,
             or
             to
             one
             or
             more
             intire
             revolutions
             .
          
           
             XL.
             If
             the
             Arch
             to
             be
             Tripled
             be
             greater
             than
             one
             or
             more
             intire
             revolutions
             ;
             its
             Subtense
             is
             the
             same
             with
             that
             of
             its
             Excess
             above
             those
             intire
             revolutions
             ,
             and
             to
             be
             considered
             in
             like
             manner
             ,
             which
             things
             are
             evident
             and
             need
             no
             further
             demonstration
             .
          
           
             XLI
             .
             Now
             what
             hath
             been
             severally
             delivered
             concerning
             the
             Triplication
             of
             an
             Arch
             or
             Angle
             less
             than
             a
             Trient
             ;
             and
             of
             one
             greater
             than
             a
             Trient
             ,
             but
             less
             than
             two
             Trients
             ;
             (
             to
             one
             of
             which
             cases
             every
             Arch
             may
             be
             referred
             ,
             as
             is
             already
             shewed
             ;
             )
             we
             may
             thus
             ,
             jointly
             put
             together
             .
          
           
           
             XLII
             .
             
               The
               Difference
               of
               the
               Squares
               of
               the
               Subtenses
               of
               the
               single
               and
               double
               Arch
            
             
             
               (
               whether
               soever
               of
               them
               be
               the
               greater
               ,
               )
               is
               equal
               to
               the
               Rect-angle
               of
               the
               Subtenses
               of
               the
               single
               and
               Triple
               .
            
             (
             by
             §
             4
             and
             31.
             )
             That
             is
             ,
             Bq
             −
             Aq
             =
             AC
             ,
             and
             Zq
             −
             Bq
             =
             ZC
             .
             And
             therefore
             ,
          
           
             XLIII
             .
             
               If
               the
               Difference
               of
               the
               Squares
               of
               the
               Subtenses
               of
               the
               single
               and
               double
               Arch
               ,
               be
               divided
               by
               the
               Subtense
               of
               the
               single
               ;
               it
               gives
               that
               of
               the
               Triple
               :
            
             If
             ,
             by
             that
             of
             the
             Triple
             ,
             it
             gives
             that
             of
             the
             single
             .
             (
             by
             §
             5
             ,
             32.
             )
             That
             is
             ,
             
             .
             And
             
             .
             And
             
             .
             And
             ,
          
           
             XLIV
             .
             
               As
               the
               Subtense
               of
               the
               single
               Arch
               ,
               to
               the
               Aggregate
               of
               the
               Subtenses
               of
               the
               single
               and
               double
               ;
               so
               is
               the
               Difference
               of
               these
               Subtenses
               ,
               to
               the
               Subtense
               of
               the
               Triple
               .
            
             (
             by
             §
             7
             ,
             35.
             )
             That
             is
             ,
             
             And
             
          
           
             XLV
             .
             Now
             for
             as
             much
             as
             (
             by
             §
             12
             ,
             26.
             )
             the
             Three
             Arches
             A
             ,
             E
             ,
             Z
             ,
             if
             Tripled
             ,
             will
             have
             the
             same
             Subtense
             of
             the
             Triple
             Arch
             C
             :
             'T
             is
             thence
             manifest
             ,
             that
             
               such
               Equation
               as
               this
               (
               which
               concerns
               the
               Trisection
               of
               an
               Arch
               ,
               )
            
             
             
               must
               have
               in
               all
               Three
               Roots
               ,
            
             as
             A
             ,
             E
             ,
             Z
             :
             (
             For
             every
             of
             these
             ,
             upon
             such
             Triplication
             ,
             will
             have
             the
             Subtense
             of
             the
             Triple
             Arch
             ,
             C
             :
             )
             Yet
             so
             ;
             that
             ,
             where
             A
             and
             E
             
               are
               Affirmative
               Roots
            
             ,
             Z
             
               is
               a
               Negative
            
             ;
             and
             contrarywise
             ,
             where
             this
             is
             Affirmative
             ,
             those
             be
             Negative
             .
             That
             is
             ,
             in
             this
             Equation
             ,
             
             ;
             the
             Roots
             are
             +
             A
             ,
             +
             E
             −
             Z.
             But
             ,
             in
             this
             ,
             
             ;
             the
             Roots
             are
             −
             A
             ,
             −
             E
             ,
             +
             Z.
             And
             therefore
             ,
             
             .
             And
             ,
             consequently
             ,
             3ARq
             −
             Ac
             =
             3ERq
             −
             Ec
             =
             CRq
             =
             Zc
             −
             3ZRq
             .
          
           
             XLVI
             .
             Since
             therefore
             3ARq
             −
             Ac
             =
             Zc
             −
             3ZRq
             ;
             and
             consequently
             (
             by
             Transposition
             )
             3ZRq
             +
             3ARq
             =
             Zc
             +
             Ac
             ;
             it
             is
             also
             (
             dividing
             both
             sides
             by
             Z
             +
             A
             ,
             )
             
             .
             (
             For
             Z
             +
             A
             into
             Zq
             −
             ZA
             +
             Aq
             ,
             is
             equal
             to
             Zc
             +
             Ac
             ;
             as
             will
             appear
             by
             Multiplication
             ;
             and
             contrarywise
             ,
             if
             this
             be
             divided
             by
             either
             of
             those
             ,
             the
             Quotient
             will
             be
             the
             other
             of
             them
             ,
             as
             will
             be
             found
             by
             Division
             .
             )
             And
             in
             like
             manner
             ;
             because
             also
             3ERq
             −
             Ec
             =
             Zc
             −
             3ZRq
             ,
             therefore
             3ZRq
             +
             3ERq
             =
             Zc
             +
             Ec
             ;
             and
             
             .
             That
             is
             ,
          
           
             XLVII
             .
             
               Of
               two
               Arches
               ,
               whereof
               the
               one
               exceeds
               the
               other
               by
               a
               Trient
               of
               the
               whole
               Circumference
               ;
               or
               else
               ,
               whereof
               the
               one
               doth
               as
               much
               exceed
               a
               Trient
               as
               the
               other
               wants
               of
               it
               ;
               the
               Squares
               of
               the
               Subtenses
               ,
               wanting
               a
               Rect-angle
               of
               the
               same
               Subtenses
               ,
               are
               equal
               to
               the
               Square
               of
               the
               Subtense
               of
               a
               Trient
               ,
            
             or
             
               Three
               times
               the
               Square
               of
               the
               Radius
               .
            
             That
             is
             ,
             Zq
             −
             ZA
             +
             Aq
             =
             3Rq
             =
             Tq
             =
             Zq
             −
             ZE
             +
             Eq.
             
          
           
             XLVIII
             .
             Now
             the
             Angle
             contained
             by
             the
             Legs
             ZA
             ,
             or
             ZE
             ,
             (
             standing
             on
             the
             Chord
             T
             ,
             )
             is
             an
             Angle
             of
             60
             Degrees
             ;
             (
             as
             being
             an
             Angle
             in
             the
             Circumference
             standing
             on
             an
             Arch
             of
             120
             Degrees
             .
             )
             And
             therefore
             ,
          
           
             XLIX
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               one
               of
               whose
               Angles
               is
               of
            
             60
             
               Degrees
               ;
               the
            
             
             
               Square
               of
               the
               side
               opposite
               to
               this
               Angle
               ,
               is
               equal
               to
               the
               two
               Squares
               of
               the
               sides
               containing
               it
               ,
               wanting
               the
               Rect-angle
               of
               the
               same
               sides
               .
            
             (
             For
             any
             such
             Triangle
             may
             be
             thus
             inscribed
             in
             a
             Circle
             :
             )
             That
             is
             ,
             Zq
             −
             ZA
             +
             Aq
             ,
             (
             or
             Zq
             −
             ZE
             +
             Eq
             ,
             )
             =
             Tq
             =
             3Rq
             .
          
           
           
             L.
             The
             same
             things
             (
             from
             §
             45
             ,
             &c.
             )
             may
             be
             thus
             otherwise
             inferred
             .
             
             Because
             (
             by
             §
             15
             )
             Aq
             +
             AE
             +
             Eq
             =
             3Rq
             ,
             and
             (
             by
             §
             18
             or
             21.
             )
             Z
             =
             A
             +
             E
             ;
             therefore
             Zq
             =
             Aq
             +
             2AE
             +
             Eq
             ,
             and
             ZA
             =
             Aq
             +
             AE
             ,
             (
             and
             ZE
             =
             AE
             +
             Eq
             ,
             )
             and
             therefore
             Zq
             −
             ZA
             =
             AE
             +
             Eq
             ,
             (
             and
             Zq
             −
             ZE
             =
             Aq
             +
             AE
             ;
             )
             and
             consequently
             Zq
             −
             ZA
             +
             Aq
             (
             or
             Zq
             −
             ZE
             +
             Eq
             ,
             )
             =
             Aq
             +
             AE
             +
             Eq
             =
             3Rq
             .
             From
             whence
             the
             rest
             are
             inferred
             as
             before
             .
          
           
             
          
           
             LI.
             Moreover
             ,
             because
             (
             as
             is
             before
             shewed
             )
             
             :
             We
             may
             thence
             infer
             the
             following
             Theorems
             .
          
           
             LII
             .
             
               The
               Difference
               of
               the
               Cubes
               of
               two
               Legs
               containing
               an
               Angle
               of
            
             120
             
               degrees
               ,
               divided
               by
               the
               Difference
               of
               those
               Legs
               ,
               is
               equal
               to
               the
               Square
               of
               the
               Base
               subtended
               to
               it
               .
            
          
           
             LIII
             .
             
               But
               if
               it
               be
               an
               Angle
               of
            
             60
             
               degrees
               ;
               the
               sum
               of
               the
               Cubes
               of
               the
               Legs
               or
               sides
               containing
               it
               ,
               divided
               by
               the
               sum
               of
               those
               sides
               ,
               is
               equal
               to
               the
               Square
               of
               the
               Base
               .
            
             Again
             ,
          
           
             LIV.
             
               The
               Difference
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             120
             
               degrees
               ,
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ,
               is
               equal
               to
               the
               Difference
               of
               the
               Cubes
               of
               those
               Legs
               .
            
          
           
             LV.
             
               But
               if
               it
               be
               an
               Angle
               of
            
             60
             
               degrees
               ;
               the
               sum
               of
               the
               Legs
               Multiplied
               into
               the
               Square
               of
               the
               Base
               is
               equal
               to
               the
               sum
               of
               the
               Cubes
               of
               those
               Legs
               .
            
          
           
             LVI
             .
             Again
             ,
             because
             
             (
             or
             
             and
             
             :
             Therefore
             
             ,
             (
             and
             
             ,
             )
             and
             
             .
             And
             therefore
             ,
          
           
             LVII
             .
             
               The
               Difference
               between
               the
               Triple
               of
               the
               Subtense
               of
               a
               single
               Arch
               ,
               less
               than
               a
               Trient
               ;
               and
               ,
               of
               the
               Subtense
               of
               the
               Triple
               of
               that
               Arch
               ;
               is
               equal
               to
               the
               Cube
               of
               the
               Subtense
               of
               that
               single
               Arch
               ,
               divided
               by
               the
               Square
               of
               the
               Radius
               .
            
             And
             consequently
             ,
             
               That
               Difference
               Multiplied
               into
               the
               Square
               of
               the
               Radius
               ,
               is
               equal
               to
               such
               Cube
               .
            
          
           
             LVIII
             .
             
               The
               sum
               of
               the
               Triple
               of
               the
               Subtense
               of
               a
               single
               Arch
               greater
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               ,
               and
               of
               the
               Subtense
               of
               the
               Triple
               Arch
               ;
               is
               equal
               to
               the
               Cube
               of
               the
               Subtense
               of
               that
               single
               Arch
               divided
               by
               the
               Square
               of
               the
               Radius
               .
            
             And
             consequently
             ,
             
               That
               sum
               Multiplied
               into
               the
               Square
               of
               the
               Radius
               ,
               is
               equal
               to
               such
               Cube
               .
            
          
           
             LIX
             .
             Because
             (
             as
             is
             before
             shewed
             )
             
             :
             Or
             ,
             3ARq
             −
             AC
             =
             3ERq
             −
             Ec
             =
             CRq
             =
             Zc
             −
             3ZRq
             :
             Therefore
             
               the
               Subtense
               of
               an
               Arch
               being
               given
            
             (
             as
             A
             ,
             E
             ,
             or
             Z
             ,
             )
             
               together
               with
               the
               Radius
            
             R
             ;
             
               we
               have
               thence
               the
               Subtense
               of
               the
               Triple
               Arch
            
             C.
             Which
             is
             ,
             
               the
               Triplication
               of
               an
               Arch
               or
               Angle
               .
            
          
           
           
             LX.
             And
             ,
             contrarywise
             ;
             
               The
               Radius
               of
               a
               Circle
            
             R
             ,
             
               and
               the
               Subtense
               of
               the
               Triple
            
             
             Arch
             C
             ,
             
               being
               given
               ;
               we
               have
               thence
               the
               Subtense
               of
               the
               single
               Arch
               ,
            
             (
             A
             ,
             E
             ,
             or
             Z
             ,
             )
             
               by
               resolving
               such
               a
               Cubick
               Equation
               .
            
             Which
             is
             ,
             
               the
               Trisection
               of
               an
               Arch
               or
               Angle
               .
            
             But
             the
             Geometrical
             effection
             thereof
             is
             not
             to
             be
             performed
             by
             Rule
             and
             Compass
             ;
             without
             the
             help
             of
             a
             Conick
             Section
             ,
             or
             some
             Line
             more
             Compounded
             .
          
           
             LXI
             .
             But
             then
             ,
             on
             the
             other
             side
             ;
             
               such
               Cubick
            
             Equations
             may
             be
             resolved
             
               by
               the
               Trisection
               of
               an
               Arch.
            
             For
             ,
             suppose
             a
             Cubick
             Equation
             of
             this
             Form
             ,
             3RqA
             −
             Ac
             ,
             (
             or
             3RqE
             −
             Ec
             ,
             )
             =
             RqC
             ;
             whose
             Root
             A
             (
             or
             E
             )
             is
             sought
             .
             Now
             if
             R
             (
             the
             Square-root
             of
             a
             third
             part
             of
             the
             Co-efficient
             )
             be
             made
             the
             Radius
             of
             a
             Circle
             ,
             (
             that
             is
             ,
             
             ;
             )
             and
             therein
             be
             inscribed
             C
             ,
             which
             is
             the
             Result
             of
             the
             absolute
             term
             divided
             by
             the
             third
             part
             of
             the
             Co-efficient
             ,
             (
             that
             is
             ,
             
             ;
             )
             And
             either
             of
             the
             Arches
             to
             which
             this
             Chord
             subtends
             be
             divided
             into
             three
             equal
             parts
             :
             The
             Chord
             which
             subtendeth
             to
             one
             of
             those
             parts
             is
             an
             (
             Affirmative
             )
             Root
             of
             that
             Equation
             ;
             which
             therefore
             hath
             two
             Affirmative
             Roots
             ;
             suppose
             A
             ,
             and
             E.
             
          
           
             LXII
             .
             But
             it
             hath
             moreover
             a
             Negative
             Root
             ;
             which
             is
             the
             Subtense
             of
             either
             of
             those
             Arches
             (
             whose
             Chord
             is
             A
             ,
             or
             E
             ,
             )
             increased
             by
             a
             Trient
             of
             the
             whole
             Circumference
             ,
             suppose
             Z.
             I
             say
             
               either
               of
               those
               Arches
            
             ;
             for
             the
             same
             Chord
             Z
             ,
             which
             on
             the
             one
             side
             subtends
             a
             Trient
             increased
             by
             the
             Arch
             A
             ,
             subtends
             on
             the
             other
             side
             a
             like
             Trient
             increased
             by
             E.
             
          
           
             LXIII
             .
             But
             if
             the
             Equation
             be
             of
             this
             Form
             ,
             Zc
             −
             3RqZ
             =
             RqC
             :
             The
             process
             is
             just
             the
             same
             in
             all
             Points
             ;
             save
             ,
             that
             then
             ,
             there
             is
             but
             one
             Affirmative
             Root
             ,
             Z
             ;
             and
             two
             Negatives
             ,
             A
             ,
             E.
             
          
           
             LXIV
             .
             But
             ,
             in
             both
             Cases
             ,
             it
             must
             be
             still
             observed
             ,
             
               That
               the
               Chord
            
             C
             
               be
               not
               greater
               than
            
             2R
             .
             (
             For
             when
             this
             happens
             ,
             the
             Chord
             C
             ,
             as
             being
             greater
             than
             the
             Diameter
             ,
             cannot
             be
             inscribed
             in
             such
             Circle
             .
             )
             Or
             ,
             (
             which
             is
             in
             effect
             the
             same
             ;
             )
             
               That
               the
               Square
               of
               Half
               the
               Absolute
               term
               ,
               be
               not
               greater
               than
               the
               Cube
               of
               a
               third
               part
               of
               the
               Co-efficient
               of
               the
               middle
               term
               .
            
             For
             ,
             the
             third
             part
             of
             that
             Co-efficient
             being
             Rq
             ,
             and
             the
             Cube
             thereof
             R
             6
             ;
             and
             half
             the
             Absolute
             quantity
             ½
             RqC
             ,
             and
             the
             Square
             of
             this
             ¼
             RqqCq
             ;
             if
             this
             Square
             be
             greater
             than
             that
             Cube
             ,
             and
             therefore
             (
             dividing
             both
             by
             Rqq
             )
             ¼
             Cq
             greater
             than
             Rq
             ,
             and
             (
             taking
             the
             Roots
             of
             both
             )
             ½
             C
             greater
             than
             R
             ;
             then
             must
             C
             be
             greater
             than
             2R
             the
             Diameter
             ,
             and
             therefore
             cannot
             be
             inscribed
             in
             the
             Circle
             .
             And
             therefore
             ,
             
               when
               this
               happens
               ,
               such
               Equations
               cannot
               be
               thus
               resolved
               by
               the
               Trisection
               of
               an
               Arch.
            
             But
             they
             may
             by
             (
             what
             are
             wont
             to
             be
             called
             )
             
               Cardan's
               Rules
            
             ,
             (
             as
             I
             have
             elsewhere
             shewed
             ,
             )
             the
             consideration
             of
             which
             doth
             not
             belong
             to
             this
             place
             .
          
           
             LXV
             .
             If
             an
             Arch
             to
             be
             Tripled
             be
             a
             Trient
             (
             or
             Two
             ,
             Three
             ,
             Four
             ,
             or
             more
             Trients
             ;
             )
             the
             Triple
             Arch
             will
             therefore
             be
             one
             intire
             Revolution
             ,
             (
             or
             Two
             ,
             Three
             ,
             Four
             ,
             or
             more
             intire
             Revolutions
             ;
             )
             and
             the
             Subtense
             of
             the
             Triple
             will
             be
             nothing
             ;
             (
             the
             beginning
             and
             end
             of
             such
             Triple
             Arch
             being
             the
             same
             Point
             :
             )
             That
             is
             ,
             
             ,
             (
             or
             
             .
             And
             therefore
             
             ,
             
             ,
             
             .
             And
             3Rq
             =
             Aq
             =
             Eq
             =
             Zq.
             That
             is
             ,
             (
             as
             was
             before
             shewed
             )
          
           
             LXVI
             .
             
               The
               Square
               of
               the
               Subtense
               of
               a
               Trient
               ,
            
             (
             or
             of
             the
             side
             of
             an
             Equilater
             inscribed
             Triangle
             )
             
               is
               equal
               to
               three
               Squares
               of
               the
               Radius
               .
            
          
           
           
             LXVII
             .
             If
             an
             Arch
             to
             be
             Tripled
             be
             a
             Quadrant
             ;
             it
             is
             manifest
             that
             the
             Subtense
             
             of
             the
             Triple
             ,
             is
             equal
             to
             that
             of
             the
             single
             .
             (
             For
             the
             same
             Chord
             ,
             subtendeth
             ,
             on
             the
             one
             side
             ,
             to
             Three
             such
             Arches
             ;
             and
             ,
             on
             the
             other
             side
             ,
             but
             to
             one
             .
             )
             That
             is
             ,
             
             ;
             and
             therefore
             
             ;
             and
             2Rq
             =
             Aq.
             That
             is
             ,
          
           
             LXVIII
             .
             
               The
               Square
               of
               the
               Subtense
               of
               a
               Quadrant
            
             (
             or
             of
             the
             side
             of
             an
             inscribed
             Quadrate
             )
             
               is
               equal
               to
               two
               Squares
               of
               the
               Radius
               .
            
          
           
             LXIX
             .
             The
             same
             may
             be
             inferred
             ,
             from
             the
             Bisection
             of
             a
             Semicircumference
             .
             For
             the
             Subtense
             of
             that
             being
             2R
             ;
             and
             therefore
             (
             by
             §
             9
             ,
             
               Chap.
               preced
            
             .
             )
             
             Or
             ,
             (
             putting
             E
             for
             the
             Remainder
             of
             that
             Quadrant
             to
             the
             Semicircle
             )
             
             .
             Or
             ,
             (
             because
             ,
             in
             this
             case
             E
             =
             A
             ,
             )
             
             .
             Therefore
             2Rq
             =
             Aq
             ,
             as
             before
             ;
             and
             
             .
          
           
             LXX
             .
             But
             if
             the
             Arch
             to
             be
             Tripled
             be
             of
             a
             Semicircle
             (
             and
             so
             ,
             greater
             than
             a
             Trient
             ,
             )
             the
             Subtense
             of
             the
             Triple
             will
             be
             the
             same
             with
             that
             of
             the
             single
             ;
             but
             with
             a
             contrary
             sign
             ,
             (
             by
             §
             28
             ;
             )
             and
             therefore
             (
             by
             §
             36.
             )
             
             ;
             that
             is
             ,
             
             ,
             or
             Zc
             =
             4ZRq
             ,
             and
             Zq
             =
             4Rq
             ,
             and
             Z
             =
             2R
             .
             Which
             is
             the
             third
             ,
             or
             Negative
             Root
             ,
             of
             the
             last
             mentioned
             Equation
             ,
             
             ;
             beside
             the
             two
             Affirmatives
             A
             ,
             E
             ,
             the
             Subtenses
             of
             the
             two
             Quadrants
             :
             This
             being
             equal
             to
             the
             Aggregate
             of
             both
             ,
             with
             the
             contrary
             sign
             .
          
           
             LXXI
             .
             Moreover
             ;
             because
             the
             same
             Subtense
             (
             before
             mentioned
             ,
             )
             C
             =
             A
             ,
             subtends
             not
             onely
             to
             the
             Triple
             of
             a
             Quadrant
             on
             the
             one
             side
             ;
             but
             also
             ,
             on
             the
             other
             side
             ,
             to
             the
             single
             Quadrant
             ;
             to
             a
             third
             part
             of
             this
             therefore
             the
             Root
             E
             is
             to
             be
             also
             a
             subtendent
             :
             (
             that
             is
             ,
             to
             an
             Arch
             of
             30
             degrees
             .
             )
             That
             is
             ,
             
             .
          
           
             LXXII
             .
             And
             ,
             because
             (
             by
             §
             15.
             )
             Aq
             +
             AE
             +
             Eq
             =
             3Rq
             ,
             and
             (
             by
             §
             67.
             )
             Aq
             =
             2Rq
             ;
             therefore
             AE
             +
             Eq
             =
             Rq.
             And
             therefore
             (
             by
             resolving
             this
             Equation
             )
             
             .
             And
             therefore
             ,
             
             .
             That
             is
             ,
          
           
             LXXIII
             .
             
               As
               the
               Subtense
               of
               a
               Quadrant
               ,
               to
               the
               Subtense
               of
               a
               Trient
               wanting
               the
               Radius
               ;
               so
               is
               the
               Radius
               ,
               to
               the
               Subtense
               of
               the
               Semi-sextant
            
             ;
             or
             ,
             of
             30
             degrees
             :
             Or
             ,
          
           
             LXXIV
             .
             
               As
               the
               side
               of
               the
               (
               inscribed
               Equilater
               )
               Tetragone
               ,
               to
               the
               Difference
               of
               the
               sides
               of
               the
               Trigone
               and
               Hexagone
               ;
               so
               is
               the
               Radius
               ,
               to
               that
               of
               the
               Dodecagone
               .
            
             (
             Understand
             it
             of
             the
             inscribed
             Equilater
             Figures
             ;
             and
             so
             afterward
             in
             like
             cases
             .
             )
          
           
             LXXV
             .
             And
             because
             (
             as
             before
             )
             
             :
             Therefore
             ,
             
             ,
             or
             ,
             
             into
             Rq.
             And
             so
             ,
             
             Eq.
             That
             is
             ,
          
           
           
             LXXVI
             .
             
               As
               the
               Radius
               ,
               to
               the
               Excess
               of
               the
               Diameter
               above
               the
               Subtense
               of
            
             
             120
             degrees
             ,
             (
             or
             side
             of
             the
             inscribed
             Trigone
             :
             )
             
               so
               is
               the
               Square
               of
               the
               Radius
               ,
               to
               that
               of
               the
               Subtense
               of
            
             30
             degrees
             ;
             or
             of
             the
             side
             of
             the
             Dodecagone
             .
             And
             therefore
             ,
             (
             that
             of
             Rq
             to
             Eq
             ,
             being
             Duplicate
             to
             that
             of
             R
             to
             E.
             )
          
           
             LXXVII
             .
             
               The
               Proportion
               of
               the
               Radius
               ,
               to
               the
               Difference
               of
               the
               Diameter
               and
               the
               side
               of
               the
               inscribed
               Trigone
               ;
               is
               Duplicate
               to
               that
               of
               the
               Radius
               ,
               to
               the
               side
               of
               the
               inscribed
               Dodecagone
               .
            
             And
             therefore
             ,
          
           
             LXXVIII
             .
             
               The
               (
               Radius
               or
               )
               side
               of
               the
               Hexagone
               ,
               and
               of
               the
               Dodecagone
               ,
               and
               the
               Difference
               of
               the
               Diameter
               from
               that
               of
               the
               Trigone
               ,
               are
               in
               continual
               Proportion
               .
            
          
           
             LXXIX
             .
             And
             (
             because
             
             into
             
             into
             R
             ,
             )
             
               The
               Excess
               of
               the
               Diameter
               above
               the
               Subtense
               of
               the
               Trient
               ,
               Multiplied
               into
               the
               Radius
               ;
               is
               equal
               to
               the
               Square
               of
               the
               Subtense
               of
            
             30
             degrees
             ,
             or
             the
             Semisextant
             .
          
           
             LXXX
             .
             The
             same
             are
             found
             by
             bisecting
             the
             Sextant
             ;
             (
             for
             a
             quarter
             of
             the
             Trient
             ,
             or
             half
             the
             Sextant
             ,
             is
             the
             same
             ;
             )
             in
             this
             manner
             ,
          
           
             LXXXI
             .
             If
             E
             be
             put
             for
             the
             Subtense
             of
             30
             degrees
             ,
             and
             A
             for
             that
             of
             the
             Residue
             to
             a
             Semicircumference
             ,
             or
             of
             150
             degrees
             ;
             then
             because
             the
             Subtense
             of
             a
             Sextant
             ,
             or
             the
             double
             Arch
             of
             E
             ,
             is
             R
             ;
             therefore
             (
             by
             §
             14
             ,
             
               Chap.
               preced
            
             .
             )
             
             :
             And
             Rqq
             =
             4RqAq
             −
             Aqq
             =
             4RqEq
             −
             Eqq
             :
             And
             (
             by
             resolving
             that
             Equation
             )
             
             into
             
             into
             R
             ,
             =
             Aq
             ,
             and
             Eq.
             That
             is
             ,
          
           
             LXXXII
             .
             
               The
               Squares
               of
               the
               Subtenses
               of
            
             150
             ,
             
               and
               of
            
             30
             
               degrees
               ;
               are
               equal
               ,
               that
               to
               the
               Sum
               ,
               this
               to
               the
               Difference
               ,
               (
               of
               the
               Diameter
               and
               side
               of
               the
               inscribed
               Trigone
               )
               Multiplied
               by
               the
               Radius
               .
            
          
           
             LXXXIII
             .
             Again
             ,
             for
             as
             much
             as
             C
             =
             A
             subtends
             as
             well
             the
             Triple
             of
             the
             Arch
             A
             of
             90
             degrees
             ,
             as
             the
             Triple
             of
             the
             Arch
             E
             of
             30
             degrees
             ;
             therefore
             
             is
             the
             Subtense
             of
             a
             Trient
             increased
             by
             the
             Arch
             A
             or
             E
             ;
             that
             is
             ,
             as
             well
             of
             degr
             .
             210
             =
             120
             +
             90
             ,
             as
             of
             150
             =
             120
             +
             30.
             
             Which
             was
             before
             concluded
             at
             §
             81.
             (
             for
             Z
             here
             ,
             is
             the
             same
             with
             A
             there
             )
             and
             Zq
             (
             as
             there
             Aq
             ,
             )
             
             into
             Rq.
             
          
           
             LXXXIV
             .
             And
             the
             same
             is
             yet
             again
             found
             by
             subducting
             
             into
             Rq
             )
             the
             Square
             of
             the
             Subtense
             30
             degrees
             ,
             out
             of
             (
             4Rq
             )
             the
             Square
             of
             the
             Diameter
             :
             (
             because
             150
             +
             30
             =
             180
             degrees
             ,
             complete
             the
             Semicircumference
             :
             )
             For
             if
             from
             4Rq
             we
             subduct
             
             ,
             there
             remains
             
             ,
             or
             ,
             
             into
             Rq
             ,
             the
             Subtense
             of
             150
             degrees
             ;
             and
             therefore
             also
             of
             210
             degrees
             .
          
        
         
           
           
             
               CHAP.
               III.
            
             Of
             the
             Quadruplation
             and
             Quadrisection
             of
             an
             ARCH
             or
             ANGLE
             .
          
           
             I.
             IF
             in
             a
             Circle
             be
             inscribed
             a
             Quadrilater
             ,
             whose
             opposite
             sides
             are
             A
             ,
             A
             ,
             
             (
             the
             subtenses
             of
             a
             single
             Arch
             )
             and
             B
             ,
             D
             ,
             (
             the
             subtenses
             of
             the
             Double
             and
             Quadruple
             )
             the
             Diagonals
             will
             be
             C
             ,
             C
             ,
             (
             the
             subtenses
             of
             the
             Triple
             )
             as
             is
             evident
             .
             (
             But
             it
             is
             evident
             also
             ,
             that
             ,
             in
             this
             Case
             ,
             the
             Arch
             A
             ,
             is
             less
             than
             a
             Quadrant
             of
             the
             whole
             Circumference
             .
             )
          
           
             II.
             And
             therefore
             (
             the
             Rect-angle
             of
             the
             Diagonals
             being
             equal
             to
             the
             two
             Rect-angles
             of
             the
             opposite
             sides
             )
             Cq
             −
             Aq
             =
             BD.
             And
             therefore
             
             ,
             and
             
             That
             is
             ,
          
           
             III.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               Triple
               Arch
               ,
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
               Quadrant
               )
               is
               equal
               to
               the
               Rect-angle
               of
               the
               subtenses
               of
               the
               Double
               and
               Quadruple
               .
            
             And
             ,
             divided
             by
             either
             of
             these
             ,
             it
             gives
             the
             other
             of
             them
             .
          
           
             IV.
             But
             C+A
             into
             C
             −
             A
             is
             equal
             to
             Cq
             −
             Aq.
             And
             therefore
             ,
             
             That
             is
             ,
          
           
             V.
             
               As
               the
               Subtense
               of
               the
               double
               Arch
               ,
               is
               to
               the
               sum
               of
               the
               Subtenses
               of
               the
               Triple
               ,
               and
               of
               the
               single
               (
               this
               being
               less
               than
               a
               Quadrant
               ;
               )
               so
               is
               the
               excess
               of
               the
               Subtense
               of
               the
               Triple
               above
               that
               of
               the
               single
               ,
               to
               that
               of
               the
               Quadruple
               .
            
          
           
             VI.
             And
             because
             (
             by
             §
             8
             ,
             
               Chap.
               preced
            
             .
             )
             
             ;
             and
             therefore
             
             :
             Therefore
             
             .
          
           
             VII
             .
             But
             (
             by
             §
             7
             and
             9
             ,
             Chap.
             29.
             )
             
             .
             And
             therefore
             
             .
             (
             For
             if
             
             be
             divided
             by
             
             ,
             it
             is
             
             :
             And
             this
             again
             divided
             by
             4Rq
             −
             Aq
             ,
             is
             
             :
             But
             this
             last
             Division
             being
             by
             4Rq
             −
             Aq
             ,
             whereas
             (
             according
             to
             the
             value
             of
             B
             )
             it
             should
             be
             divided
             only
             by
             the
             Square
             Root
             hereof
             ,
             therefore
             we
             are
             to
             restore
             a
             Multiplication
             by
             that
             Root
             ;
             which
             makes
             it
             
             .
             )
          
           
             VIII
             .
             And
             then
             ,
             turning
             the
             Equation
             into
             an
             Analogy
             ,
             
             .
             That
             is
             ,
          
           
           
             IX
             .
             
               As
               the
               Cube
               of
               the
               Radius
               ,
               to
               the
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
            
             
             
               Quadrant
               )
               Multiplied
               into
               the
               double
               Square
               of
               the
               Radius
               wanting
               the
               Square
               of
               the
               Subtense
               ,
               so
               is
               the
               Subtense
               of
               what
               this
               Arch
               wants
               of
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             X.
             Or
             thus
             ;
             (
             dividing
             the
             two
             first
             terms
             by
             Rq
             ,
             )
             
             .
             That
             is
             ,
          
           
             XI
             .
             
               As
               the
               Radius
               ,
               to
               the
               double
               of
               the
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
               Quadrant
               )
               wanting
               the
               Cube
               of
               that
               Subtense
               divided
               by
               the
               Square
               of
               the
               Radius
               :
               So
               is
               the
               Subtense
               of
               what
               that
               single
               Arch
               wants
               of
               the
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             XII
             .
             But
             (
             by
             §
             8
             ,
             
               Chap.
               preced
            
             .
             )
             
             .
             And
             therefore
             ,
          
           
             XIII
             .
             
               As
               the
               Radius
               to
               the
               excess
               of
               the
               Triple
               Arch
               above
               that
               of
               the
               single
               (
               less
               than
               a
               Quadrant
               :
               )
               so
               is
               the
               Subtense
               of
               what
               that
               single
               Arch
               wants
               of
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             XIV
             .
             The
             same
             may
             be
             thus
             also
             demonstrated
             :
             Because
             (
             by
             §
             9
             ,
             Chap.
             29.
             )
             
             ,
             is
             the
             Subtense
             of
             the
             double
             Arch
             of
             A
             :
             Therefore
             (
             by
             the
             same
             reason
             )
             
             ,
             the
             Subtense
             of
             the
             double
             Arch
             of
             B
             ;
             that
             is
             ,
             of
             the
             Quadruple
             Arch
             of
             A.
             
          
           
             XV.
             And
             ,
             because
             
             :
             Therefore
             ,
             for
             4Rq
             −
             Bq
             ,
             we
             may
             put
             
             ,
             or
             
             :
             And
             the
             Quadratick
             Root
             hereof
             ,
             
             Multiplied
             into
             
             makes
             
             As
             before
             ,
          
           
             XVI
             .
             We
             may
             also
             ,
             to
             the
             same
             purpose
             ,
             (
             and
             with
             the
             same
             event
             ,
             )
             inscribe
             
             a
             Quadrilater
             ,
             so
             as
             that
             A
             ,
             D
             ,
             and
             A
             ,
             B
             ,
             may
             be
             opposite
             sides
             ,
             and
             B
             ,
             C
             ,
             Diagonals
             .
             For
             then
             BC
             −
             BA
             =
             DA.
             And
             therefore
             
             into
             
             will
             equal
             DA.
             That
             is
             ,
             
             .
             And
             
             ;
             as
             before
             .
             But
             of
             this
             we
             shall
             say
             more
             at
             §
             86.
             &c.
             
          
           
             XVII
             .
             But
             ,
             for
             as
             much
             as
             the
             same
             D
             ,
             Subtends
             not
             only
             the
             Quadruple
             of
             
             the
             Arch
             A
             ,
             but
             also
             the
             Quadruple
             of
             the
             Arch
             E
             ,
             (
             which
             therefore
             ,
             together
             with
             the
             Arch
             A
             ,
             will
             complete
             a
             Quadrant
             of
             the
             whole
             Circumference
             :
             )
             it
             may
             in
             like
             manner
             be
             shewed
             ,
             that
             
             :
             And
             therefore
             ,
          
           
             XVIII
             .
             
               An
               Arch
               less
               than
               a
               Quadrant
               ,
               and
               the
               Arch
               which
               this
               wants
               of
               a
               Quadrant
               ,
               have
               both
               the
               same
               Subtense
               of
               the
               Quadruple
               Arch
               ,
            
             D.
             And
             ,
             accordingly
             ,
             AE
             ,
             are
             two
             Affirmative
             Roots
             of
             that
             Equation
             .
          
           
           
             XIX
             .
             But
             there
             are
             yet
             two
             other
             Roots
             (
             but
             both
             Negative
             ,
             as
             will
             after
             
             appear
             )
             of
             the
             same
             Equation
             ;
             (
             which
             we
             will
             call
             P
             ,
             S
             ;
             )
             whereof
             one
             subtends
             a
             Quadrant
             increased
             by
             the
             Arch
             A
             ;
             the
             other
             ,
             a
             Quadrant
             increased
             by
             the
             Arch
             E.
             For
             it
             is
             manifest
             (
             by
             what
             is
             said
             at
             §
             23
             ,
             
               Chap.
               preced
            
             .
             )
             that
             these
             also
             must
             have
             the
             same
             ●●btense
             of
             the
             Quadruple
             Arch
             ,
             with
             A
             and
             E.
             For
             Four
             times
             ¼+A
             ,
             is
             1+4●
             ,
             and
             will
             therefore
             have
             the
             same
             Subtense
             with
             4A
             .
             And
             the
             like
             of
             Four
             times
             ¼+E
             ,
             which
             is
             1+4E
             ,
             whose
             Subtense
             is
             the
             same
             with
             that
             of
             4E
             .
             (
             And
             the
             like
             will
             follow
             ,
             in
             case
             two
             ,
             three
             ,
             or
             more
             Quadrants
             be
             thus
             increased
             .
             )
             And
             consequently
             ,
          
           
             XX.
             
               An
               Arch
               greater
               than
               a
               Quadrant
               ,
            
             (
             or
             than
             two
             ,
             three
             ,
             or
             more
             Quadrants
             )
             
               will
               require
               the
               same
               Subtense
               of
               its
               Quadruple
               Arch
               ,
               with
               its
               excess
               above
               a
               Quadrant
               ,
            
             (
             or
             above
             these
             two
             ,
             three
             ,
             or
             more
             Quadrants
             .
             )
          
           
             XXI
             .
             But
             the
             same
             P
             ,
             subtends
             as
             well
             to
             a
             Quadrant
             increased
             by
             the
             Arch
             of
             A
             ,
             as
             to
             three
             Quadrants
             wanting
             the
             said
             Arch
             ;
             as
             also
             to
             a
             Semicircumference
             (
             or
             two
             Quadrants
             )
             increased
             by
             the
             Arch
             of
             E
             ,
             or
             wanting
             that
             Arch.
             (
             As
             is
             manifest
             to
             view
             by
             the
             Scheme
             .
             )
             And
             ,
             in
             like
             manner
             ,
             S
             subtends
             as
             well
             to
             a
             Quadrant
             increased
             by
             the
             Arch
             of
             E
             ,
             as
             to
             three
             Quadrants
             wanting
             that
             Arch
             ;
             as
             also
             to
             a
             Semicircumference
             (
             or
             two
             Quadrants
             )
             increased
             by
             the
             Arch
             of
             A
             ,
             or
             wanting
             that
             Arch.
             
          
           
             XXII
             .
             Now
             ,
             that
             P
             ,
             S
             ,
             are
             Negative
             Roots
             ,
             will
             thus
             appear
             .
             For
             ,
             supposing
             (
             for
             instance
             )
             
             ;
             and
             P
             ,
             a
             subtendent
             of
             an
             Arch
             greater
             than
             a
             Quadrant
             :
             (
             But
             ,
             less
             than
             three
             Quadrants
             ;
             otherwise
             it
             is
             the
             same
             as
             if
             it
             were
             less
             than
             one
             Quadrant
             :
             For
             the
             same
             Chord
             which
             subtends
             an
             Arch
             greater
             than
             three
             Quadrants
             ,
             subtends
             also
             to
             less
             than
             one
             :
             )
             P
             will
             in
             this
             case
             be
             greater
             than
             
             the
             Subtense
             of
             a
             Quadrant
             ,
             and
             therefore
             2Rq
             −
             Pq
             a
             Negative
             quantity
             (
             because
             of
             greater
             quantity
             subtracted
             from
             a
             lesser
             ;
             )
             and
             therefore
             also
             P
             must
             be
             Negative
             ,
             that
             so
             2RqP
             −
             Pc
             (
             compounded
             by
             the
             Multiplication
             of
             two
             Negatives
             )
             may
             be
             a
             Positive
             quantity
             ,
             and
             therefore
             the
             whole
             
             Affirmative
             also
             .
             (
             And
             what
             is
             said
             of
             P
             ,
             holds
             in
             like
             manner
             of
             S.
             )
          
           
             XXIII
             .
             But
             if
             we
             chuse
             to
             make
             P
             Affirmative
             ,
             then
             must
             
             ,
             be
             Negative
             :
             And
             therefore
             (
             changing
             the
             signs
             )
             
             ,
             Affirmative
             .
             (
             And
             the
             like
             of
             S.
             )
             But
             ,
             of
             this
             ,
             more
             afterwards
             .
          
           
             XXIV
             .
             But
             for
             what
             reason
             ,
             the
             Equation
             
             ,
             or
             
             ,
             hath
             two
             Affirmative
             Roots
             A
             ,
             E
             ;
             and
             two
             Negatives
             P
             ,
             S
             ;
             for
             the
             same
             reason
             will
             the
             Equation
             
             ,
             or
             
             ,
             have
             two
             Negatives
             A
             ,
             E
             ;
             and
             P
             ,
             S
             ,
             Affirmatives
             .
          
           
             XXV
             .
             If
             now
             we
             consider
             the
             Quadrilater
             ,
             whose
             said
             opposite
             sides
             are
             A
             ,
             A
             ,
             and
             E
             ,
             P
             ;
             then
             (
             because
             the
             Arches
             A
             ,
             E
             ,
             do
             together
             make
             up
             a
             Quadrant
             )
             the
             Diagonals
             Q
             ,
             Q
             ,
             are
             subtenses
             of
             a
             Quadrant
             ,
             (
             or
             sides
             of
             an
             inscribed
             Square
             )
             and
             therefore
             (
             by
             §
             68
             ,
             
               Chap.
               preced
            
             .
             )
             Qq
             =
             2Rq
             ,
             and
             
             .
          
           
           
             XXVI
             .
             And
             therefore
             Qq
             −
             Aq
             =
             2Rq
             −
             Aq
             =
             EP
             ;
             and
             consequently
             
             
             ,
             and
             
             .
          
           
             XXVII
             .
             But
             the
             same
             P
             doth
             also
             subtend
             a
             Semicircumference
             wanting
             the
             Arch
             of
             E
             :
             And
             therefore
             
             And
             
             .
          
           
             XXVIII
             .
             And
             (
             by
             the
             same
             reason
             )
             taking
             a
             Quadrilater
             whose
             opposite
             sides
             are
             E
             ,
             E
             ,
             and
             A
             ,
             S
             ;
             we
             have
             the
             Diagonals
             
             And
             Qq
             −
             Eq
             =
             2Rq
             −
             Eq
             =
             AS
             .
             And
             consequently
             (
             because
             the
             Arches
             A
             and
             S
             do
             complete
             the
             Semicircumference
             )
             
             ;
             and
             
             .
          
           
             XXIX
             .
             Now
             because
             (
             as
             at
             §
             27.
             )
             
             ;
             therefore
             
             .
             And
             therefore
             4RqEq
             −
             Eqq
             =
             PqEq
             =
             4Rqq
             −
             4RqAq
             +
             Aqq
             ;
             and
             Aqq
             +
             Eqq
             =
             4AqRq
             +
             4EqRq
             −
             4Rqq
             .
             (
             And
             ,
             in
             like
             manner
             ,
             because
             
             ;
             therefore
             
             ;
             and
             4RqAq
             −
             Aqq
             =
             SqAq
             =
             Rqq
             =
             4Rqq
             −
             4RqEq
             +
             Eqq
             ;
             and
             4AqRq
             +
             4EqRq
             −
             4Rqq
             =
             Aqq
             +
             Eqq.
             )
          
           
             XXX
             .
             Now
             the
             Legs
             A
             ,
             E
             ,
             contain
             a
             Sesquiquadrantal
             Angle
             ,
             or
             of
             135
             Degrees
             :
             (
             As
             being
             an
             Angle
             in
             the
             Peripherie
             ,
             standing
             on
             an
             Arch
             of
             three
             Quadrants
             :
             )
             And
             therefore
             ,
          
           
             XXXI
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Angle
               at
               the
               Top
               is
            
             135
             
               Degrees
               ,
               if
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               containing
               it
               ,
            
             (
             2Aq
             +
             2Eq
             ,
             )
             
               wanting
               the
               Square
               of
               the
               Base
            
             (
             Qq
             =
             2Rq
             ,
             )
             
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
            
             (
             2Rq
             ;
             )
             
               the
               Product
            
             (
             4AqRq
             +
             4EqRq
             −
             4Rqq
             =
             2Aq
             +
             2Eq
             −
             2Rq
             into
             2Rq
             ,
             )
             
               is
               equal
               to
               the
               Biquadrates
               of
               the
               Legs
            
             (
             Aqq
             +
             Eqq.
             )
             So
             that
             ,
          
           
             XXXII
             .
             From
             hence
             appears
             ,
             
               A
               convenient
               Method
               for
               Adding
               of
               Biquadrates
               .
            
          
           
             XXXIII
             .
             
               The
               Subduction
               of
               Biquadrates
            
             ,
             may
             (
             with
             a
             little
             alteration
             )
             be
             performed
             almost
             in
             the
             same
             manner
             .
             But
             it
             is
             more
             conveniently
             done
             by
             
               Multiplying
               the
               Sum
               of
               the
               Squares
               ,
               by
               the
               Difference
               of
               them
               .
            
             (
             For
             Aq
             +
             Eq
             into
             Aq
             −
             Eq
             is
             equal
             to
             Aqq
             −
             Eqq.
             )
             But
             that
             is
             a
             speculation
             not
             of
             this
             place
             .
          
           
             XXXIV
             .
             Again
             ,
             
               In
               such
               Triangle
               (
               whose
               Angle
               at
               the
               Top
               is
               of
            
             135
             
               Degrees
               )
               If
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               ,
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ;
               the
               Product
               is
               equal
               to
               the
               Aggregate
               of
               the
               Biquadrates
               of
               all
               the
               sides
               .
            
             For
             ,
             since
             4AqRq
             +
             4EqRq
             −
             (
             Qqq
             =
             )
             4Rqq
             =
             Aqq
             +
             Eqq
             ;
             therefore
             Aqq
             +
             Eqq
             +
             Qqq
             =
             4AqRq
             +
             4EqRq
             =
             2Aq
             +
             2Eq
             into
             (
             2Rq
             =
             )
             Qq.
             
          
           
           
             XXXV
             .
             Again
             ,
             because
             (
             as
             at
             §
             27
             ,
             28.
             )
             
             
             and
             therefore
             
             ,
             and
             4Rqq
             −
             4RqAq
             +
             Aqq
             =
             PqEq
             =
             4PqRq
             −
             Pqq.
             Therefore
             Pqq
             +
             Aqq
             =
             4PqRq
             +
             4AqRq
             −
             4Rqq
             .
             (
             And
             in
             like
             manner
             ,
             because
             
             ;
             and
             therefore
             
             ,
             and
             4Rqq
             −
             4EqRq
             +
             Eqq
             =
             SqAq
             =
             4SqRq
             −
             Sqq
             :
             Therefore
             ,
             Sqq
             +
             Eqq
             =
             4SqRq
             +
             4EqRq
             −
             4Rqq
             .
             )
          
           
             XXXVI
             .
             But
             both
             A
             ,
             P
             ,
             and
             also
             E
             ,
             S
             ,
             contain
             a
             Semiquadrantal
             Angle
             ,
             or
             of
             45
             Degrees
             :
             (
             As
             being
             an
             Angle
             in
             the
             Periphery
             standing
             on
             a
             Quadrantal
             Arch
             ;
             )
             And
             one
             of
             the
             Angles
             at
             the
             Base
             ,
             Obtuse
             .
             And
             therefore
             ,
          
           
             XXXVII
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Angle
               at
               the
               Top
               is
               of
            
             45
             
               Degrees
               ,
               or
               half
               a
               Right-angle
               (
               one
               of
               the
               other
               being
               Obtuse
               )
               If
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
            
             (
             as
             2Pq
             +
             2Aq
             )
             
               wanting
               the
               Square
               of
               the
               Base
            
             (
             Qq
             =
             2Rq
             )
             
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
            
             (
             2Rq
             )
             
               the
               Product
            
             (
             4PqRq
             +
             4AqRq
             −
             4Rqq
             =
             2Pq
             +
             2Aq
             −
             2Rq
             into
             2Rq
             ,
             )
             
               is
               equal
               to
               the
               Biquadrates
               of
               the
               Legs
               ,
            
             (
             Pqq
             +
             Aqq
             ,
             )
             In
             like
             manner
             ,
             2Sq
             +
             2Eq
             −
             2Rq
             into
             2Rq
             ,
             =
             4SqRq
             +
             4EqRq
             −
             4Rqq
             =
             Sqq
             +
             Eqq.
             So
             that
          
           
             XXXVIII
             .
             Here
             is
             
               another
               Method
               of
               Adding
               Biquadrates
            
             .
          
           
             XXXIX
             .
             And
             likewise
             ;
             
               In
               such
               Triangle
               ,
               whose
               Angle
               at
               the
               Top
               is
               of
            
             45
             
               Degrees
               ,
               (
               and
               one
               of
               the
               other
               Obtuse
               ,
               )
               if
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               ,
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ;
               the
               Product
               is
               equal
               to
               the
               Biquadrates
               of
               all
               the
               sides
               .
            
             For
             ,
             because
             4PqRq
             +
             4AqRq
             −
             4Rqq
             =
             Pqq
             +
             Aqq
             ;
             therefore
             Pqq
             +
             Aqq
             +
             (
             4Rqq
             =
             )
             Qqq
             =
             4PqRq
             +
             4AqRq
             =
             2Pq
             +
             2Aq
             into
             (
             2Rq
             =
             )
             Qq.
             And
             ,
             because
             4SqRq
             +
             4EqRq
             −
             4Rqq
             =
             Sqq
             +
             Eqq
             ,
             therefore
             Sqq
             +
             Eqq
             +
             (
             4Rqq
             =
             )
             Qqq
             =
             4SqRq
             +
             4EqRq
             =
             2Sq
             +
             2Eq
             into
             (
             2Rq
             =
             )
             Qq.
             
          
           
             XL.
             Furthermore
             ;
             If
             in
             a
             Circle
             be
             inscribed
             a
             Quadrilater
             ,
             whose
             opposite
             
             sides
             are
             S
             ,
             A
             ,
             and
             Q
             ,
             Q
             ,
             and
             the
             Diagonals
             P
             ,
             P
             ,
             (
             as
             in
             the
             Scheme
             ;
             )
             Then
             Pq
             −
             (
             Qq
             =
             )
             2Rq
             =
             SA
             .
             And
             therefore
             ,
             
             :
             And
             
             :
             And
             therefore
             ,
             
             ,
             and
             
             .
             And
             consequently
             Pqq
             −
             4PqRq
             +
             4Rqq
             =
             AqSq
             =
             4RqAq
             −
             Aqq
             =
             4RqSq
             −
             Sqq.
             
          
           
             XLI
             .
             And
             ,
             by
             the
             same
             reason
             ,
             If
             a
             Quadrilater
             be
             inscribed
             whose
             opposite
             sides
             are
             E
             ,
             P
             ,
             and
             Q
             ,
             Q
             ;
             and
             the
             Diagonals
             S
             ,
             S
             :
             Then
             Sq
             −
             (
             Qq
             =
             )
             2Rq
             =
             EP
             :
             And
             therefore
             ,
             
             :
             And
             
             :
             And
             therefore
             ,
             
             :
             And
             
             .
             And
             consequently
             ,
             Sqq
             −
             4SqRq
             +
             4Rqq
             =
             PqEq
             =
             4PqRq
             −
             Pqq
             =
             4EqRq
             −
             Eqq.
             
          
           
             XLII
             .
             And
             either
             way
             ,
             we
             may
             conclude
             ,
             Sqq
             +
             Pqq
             =
             4SqRq
             +
             4PqRq
             −
             4Rqq
             .
          
           
           
             XLIII
             .
             But
             P
             ,
             S
             ,
             contain
             half
             a
             Right-angle
             ,
             or
             Angle
             of
             45
             Degrees
             ;
             (
             as
             
             being
             an
             Angle
             in
             the
             Periphery
             standing
             on
             a
             Quadrantal
             Arch
             ;
             )
             and
             both
             the
             other
             Angles
             Acute
             .
             And
             therefore
             ,
          
           
             XLIV
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Angle
               at
               the
               Top
               is
            
             45
             
               Degrees
               ,
               or
               half
               a
               Right
               Angle
               :
               (
               and
               both
               at
               the
               Base
               ,
               acute
               :
               )
               If
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
            
             (
             as
             2Sq
             +
             2Pq
             )
             
               wanting
               the
               Square
               of
               the
               Base
            
             (
             Qq
             =
             2Rq
             )
             
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
            
             (
             2Rq
             )
             
               the
               Product
            
             (
             4SqRq
             +
             4PqRq
             −
             4qq
             =
             2Sq
             +
             2Pq
             −
             2Rq
             into
             2Rq
             )
             
               is
               equal
               to
               the
               Biquadrates
               of
               the
               Legs
               .
            
             (
             Sqq
             +
             Pqq.
             )
          
           
             XLV
             .
             And
             this
             is
             
               a
               third
               Method
               of
               Adding
               Biquadrates
               .
            
          
           
             XLVI
             .
             
               And
               likewise
               ,
               in
               such
               Triangles
               ,
               (
               whose
               Angle
               at
               the
               Top
               is
               of
            
             45
             
               Degrees
               ,
               and
               both
               the
               others
               acute
               ,
               )
               if
               the
               double
               of
               the
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               ,
               be
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ;
               the
               Product
               is
               equal
               to
               the
               Biquadrates
               of
               all
               the
               sides
               .
            
             For
             ,
             since
             Sqq
             +
             Pqq
             =
             4SqRq
             +
             4PqRq
             −
             4Rqq
             ,
             therefore
             Sqq
             +
             Pqq
             +
             (
             4Rqq
             =
             )
             Qqq
             =
             4SqRq
             +
             4PqRq
             =
             2Sq
             +
             2Pq
             into
             2Rq
             .
          
           
             XLVII
             .
             These
             Theorems
             thus
             demonstrated
             severally
             ;
             whether
             the
             Angle
             at
             the
             Top
             be
             of
             135
             Degrees
             ,
             or
             of
             45
             Degrees
             ;
             and
             this
             whether
             the
             Triangle
             be
             Acute-angled
             ,
             or
             Obtuse-angled
             ,
             (
             to
             either
             of
             which
             we
             may
             refer
             the
             Rectangled
             ;
             )
             may
             be
             thus
             reduced
             to
             these
             Generals
             .
          
           
             XLVIII
             .
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Angle
               at
               the
               Vertex
               is
               either
               of
            
             135
             
               Degrees
               ,
               or
               of
            
             4●
             
               Degrees
               ;
               the
               double
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               containing
               it
               ,
               wanting
               the
               Square
               of
               the
               Base
               ,
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ,
               is
               equal
               to
               the
               Biquadrates
               of
               the
               two
               Legs
               .
            
             (
             Which
             is
             the
             Addition
             of
             Biquadrates
             .
             )
             By
             §
             31
             ,
             37
             ,
             44.
             
          
           
             XLIX
             .
             
               And
               that
               double
               Aggregate
               of
               the
               Squares
               of
               the
               Legs
               ,
               Multiplied
               into
               the
               Square
               of
               the
               Base
               ,
               is
               equal
               to
               the
               Biquadrates
               of
               all
               the
               three
               sides
               .
            
             By
             §
             34
             ,
             39
             ,
             46.
             
          
           
             L.
             Now
             ,
             the
             Equation
             ,
             (
             at
             §
             29.
             )
             4AqRq
             +
             4EqRq
             −
             4Rqq
             =
             Aqq
             +
             Eqq
             (
             and
             the
             other
             like
             to
             it
             at
             §
             35
             ,
             42.
             )
             is
             a
             Quadratick
             Equation
             of
             a
             Plain
             Root
             :
             Whereof
             the
             Root
             is
             2Rq
             ;
             the
             Co-efficient
             of
             the
             middle
             Term
             ,
             2Aq
             +
             2Eq
             ;
             which
             is
             therefore
             equal
             to
             the
             sum
             of
             two
             Quantities
             ,
             whose
             Rect-angle
             is
             equal
             to
             the
             Absolute
             Quantity
             Aqq
             +
             Eqq.
             
          
           
             LI.
             If
             we
             therefore
             order
             this
             according
             to
             the
             Rule
             of
             other
             Equations
             of
             the
             same
             form
             ;
             and
             ,
             accordingly
             ,
             from
             Aqq
             +
             2AqEq
             +
             Eqq
             (
             the
             Square
             of
             half
             the
             Co-efficient
             Aq
             +
             Eq
             )
             we
             subtract
             (
             the
             Absolute
             Quantity
             )
             Aqq
             +
             Eqq
             ;
             the
             Remainder
             is
             2AqEq
             :
             And
             the
             Square
             Root
             of
             this
             (
             
             )
             Added
             to
             ,
             or
             Subducted
             from
             ,
             half
             the
             Co-efficient
             Aq
             +
             Eq
             ,
             gives
             the
             Root
             of
             that
             Equation
             
             .
          
           
             LII
             .
             But
             ,
             of
             this
             Ambiguous
             Equation
             ,
             't
             is
             evident
             that
             we
             are
             to
             make
             choise
             of
             the
             greater
             Root
             ,
             in
             the
             case
             of
             §
             29
             :
             Because
             the
             Angle
             at
             the
             Vertex
             (
             135
             Degrees
             )
             is
             greater
             than
             a
             Right-angle
             ;
             and
             therefore
             the
             Square
             of
             the
             Base
             (
             Qq
             )
             is
             to
             be
             greater
             than
             (
             Aq
             +
             Eq
             )
             the
             Squares
             of
             the
             two
             sides
             containing
             it
             .
             And
             therefore
             
             .
             That
             is
             ,
          
           
             LIII
             .
             
               If
               to
               the
               Squares
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             135
             
               Degrees
               ,
               (
               or
               three
               halfs
               of
               a
               Right-angle
               ,
               )
               we
               add
               the
               Rect-angle
               of
               those
               Legs
               Multiplied
               by
               
               ;
               the
               Aggregate
               is
               equal
               to
               the
               Square
               of
               the
               Base
               .
            
          
           
           
             LIV.
             in
             the
             same
             manner
             may
             be
             shewed
             ,
             that
             the
             Equations
             of
             §
             35.
             
             Pqq
             
             +
             Aqq
             =
             4PqRq
             +
             4AqRq
             −
             4Rqq
             ,
             (
             or
             Sqq
             +
             Eqq
             =
             4SqRq
             +
             4EqRq
             −
             4Rqq
             ,
             )
             and
             of
             §
             42.
             
             Sqq
             +
             Pqq
             =
             4SqRq
             +
             4PqRq
             −
             4Rqq
             ;
             are
             Quadratic
             Equations
             of
             a
             Plain
             Root
             2Rq
             .
             But
             ,
             in
             all
             these
             (
             't
             is
             manifest
             )
             the
             lesser
             Root
             is
             to
             be
             chosen
             ,
             because
             the
             Angle
             at
             the
             Vertex
             (
             being
             of
             45
             Degrees
             )
             is
             less
             than
             a
             Right
             Angle
             ;
             and
             therefore
             the
             Square
             of
             the
             Base
             less
             than
             the
             two
             Squares
             of
             the
             Legs
             .
             And
             therefore
             ,
             the
             Root
             ,
             
             ;
             and
             
             ;
             and
             
             .
             That
             is
             ,
          
           
             LV.
             
               If
               from
               the
               Squares
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             45
             Degrees
             (
             or
             half
             a
             Right-angle
             ,
             )
             
               we
               subtract
               the
               Rect-angle
               of
               these
               Legs
               Multiplied
               by
               
               :
               the
               Remainder
               is
               equal
               to
               the
               Square
               of
               the
               Base
               .
            
          
           
             LVI
             .
             Or
             we
             may
             put
             both
             together
             ,
             thus
             :
             
               If
               to
               the
               Squares
               of
               the
               Legs
               ,
               bè
               Added
               ,
               if
               they
               contain
               an
               Angle
               of
            
             135
             
               Degrees
               ;
               or
               subtracted
               thence
               ,
               if
               they
               contain
               an
               Angle
               of
            
             45
             
               Degrees
               ;
               a
               Rect-angle
               of
               those
               Legs
               Multiplied
               into
               
               :
               The
               Result
               is
               equal
               to
               the
               Square
               of
               the
               Base
               .
            
             By
             §
             53
             ,
             55.
             
          
           
             LVII
             .
             We
             are
             next
             to
             Note
             ,
             That
             the
             subtenses
             E
             and
             P
             ,
             as
             also
             A
             and
             S
             ,
             
             (
             whose
             two
             Arches
             do
             together
             make
             up
             a
             Semicircumference
             ,
             )
             do
             (
             by
             §
             9
             ,
             Chap.
             29.
             )
             require
             the
             same
             Subtense
             of
             the
             double
             Arch
             :
             And
             therefore
             much
             more
             ,
             the
             same
             Subtense
             of
             the
             Quadruple
             .
             That
             is
             ,
             
             is
             the
             Subtense
             of
             the
             double
             Arch
             both
             of
             E
             ,
             and
             of
             P
             :
             And
             
             ,
             of
             the
             double
             Arch
             of
             A
             ,
             and
             of
             S.
             
          
           
             LVIII
             .
             The
             Subtense
             therefore
             of
             the
             Triple
             Arch
             of
             E
             ,
             (
             less
             than
             a
             Quadrant
             ,
             and
             therefore
             ,
             much
             more
             ,
             less
             than
             a
             Trient
             ,
             )
             is
             
             ,
             (
             by
             §
             2
             ,
             5
             ,
             
               Chap.
               preced
            
             .
             )
             as
             being
             the
             Square
             of
             the
             Subtense
             of
             the
             double
             Arch
             
             ,
             wanting
             the
             Square
             of
             the
             Subtense
             of
             the
             single
             Arch
             Eq
             ,
             divided
             by
             the
             Subtense
             of
             the
             single
             Arch
             E.
             
          
           
             LIX
             .
             But
             the
             same
             Subtense
             of
             that
             Triple
             Arch
             ,
             (
             by
             §
             8
             ,
             9
             ,
             
               Chap.
               preced
            
             .
             )
             is
             
             .
          
           
             LX.
             Therefore
             
             And
             
             ,
             the
             Aggregate
             of
             the
             subtenses
             of
             the
             Triple
             and
             single
             .
          
           
             LXI
             .
             Which
             may
             also
             be
             thus
             proved
             :
             Because
             Pq
             +
             Eq
             =
             4Rq
             (
             as
             being
             in
             a
             Semicircle
             ,
             )
             and
             therefore
             Pq
             =
             4Rq
             −
             Eq
             ,
             and
             PqE
             =
             4RqE
             −
             Ec
             ,
             or
             PqE
             −
             RqE
             =
             3RqE
             −
             Ec.
             Therefore
             is
             
             the
             Subtense
             of
             the
             Trible
             ;
             and
             
             the
             Aggregate
             of
             the
             subtenses
             of
             the
             Triple
             and
             single
             .
          
           
             LXII
             .
             And
             ,
             by
             just
             the
             same
             reason
             ,
             
             is
             the
             Subtense
             of
             the
             Triple
             Arch
             of
             A
             :
             And
             
             the
             Aggregate
             of
             the
             subtenses
             of
             the
             Triple
             and
             single
             .
          
           
           
             LXIII
             .
             Now
             the
             Arch
             of
             P
             ,
             (
             a
             Quadrant
             increased
             by
             A
             ,
             it
             s
             greater
             Segment
             )
             
             being
             greater
             than
             a
             Trient
             ,
             but
             less
             than
             two
             Trients
             ;
             the
             Subtense
             of
             its
             Triple
             Arch
             is
             
             (
             by
             §
             32.
             
             
               Chap.
               preced
            
             .
             )
             And
             
             .
             (
             By
             §
             37.
             
             
               Chap.
               preced
            
             .
             )
          
           
             LXIV
             .
             And
             therefore
             ,
             
             the
             Subtense
             of
             the
             Triple
             Arch
             ;
             and
             
             ,
             the
             Difference
             of
             the
             Subtenses
             of
             the
             single
             and
             Triple
             ;
             that
             is
             ,
             the
             Excess
             of
             the
             Subtense
             of
             the
             single
             above
             that
             of
             the
             Triple
             .
          
           
             LXV
             .
             But
             the
             Arch
             S
             (
             a
             Quadrant
             increased
             by
             its
             lesser
             Segment
             E
             )
             because
             it
             may
             be
             either
             lesser
             or
             greater
             than
             a
             Trient
             ,
             according
             as
             the
             Arch
             E
             is
             less
             or
             greater
             than
             30
             Degrees
             ;
             the
             Subtense
             of
             the
             Triple
             Arch
             will
             be
             either
             
             ,
             if
             the
             Arch
             S
             be
             less
             than
             a
             Trient
             ;
             or
             ,
             if
             greater
             ,
             
             .
             And
             ,
             accordingly
             ,
             
             ,
             will
             be
             either
             the
             Sum
             or
             Difference
             of
             the
             subtenses
             of
             the
             single
             and
             Triple
             Arch
             ,
             according
             as
             S
             is
             less
             or
             greater
             than
             a
             Trient
             .
          
           
             LXVI
             .
             Moreover
             ;
             having
             shewed
             (
             at
             §
             2.
             )
             that
             in
             a
             Quadruplication
             of
             an
             Arch
             less
             than
             a
             Quadrant
             ,
             Cq
             −
             Aq
             =
             BD
             ,
             (
             as
             wherein
             the
             Subtense
             of
             the
             Triple
             is
             greater
             than
             that
             of
             the
             single
             ;
             and
             therefore
             C
             ,
             C
             ,
             Diagonals
             ,
             and
             A
             ,
             A
             ,
             opposite
             sides
             :
             )
             Now
             ,
             if
             the
             Arch
             to
             be
             Quadrupled
             be
             greater
             than
             a
             Quadrant
             ,
             (
             but
             less
             than
             three
             Quadrants
             )
             as
             that
             of
             P
             or
             S
             ;
             the
             Subtense
             of
             the
             single
             will
             be
             greater
             than
             that
             of
             the
             Triple
             .
             For
             ,
             supposing
             the
             single
             Arch
             to
             be
             ½
             ∓
             A
             (
             and
             A
             less
             than
             ¼
             )
             the
             Triple
             will
             be
             ½
             ∓
             3A
             ;
             the
             Subtense
             of
             which
             will
             be
             the
             same
             with
             that
             of
             ½
             ∓
             3A
             (
             for
             one
             whole
             revolution
             is
             ,
             in
             this
             case
             ,
             Equivalent
             to
             nothing
             ;
             )
             and
             this
             (
             so
             long
             as
             A
             remains
             less
             than
             ¼
             )
             will
             be
             farther
             (
             either
             in
             excess
             or
             defect
             )
             from
             a
             Semicircumference
             (
             and
             therefore
             require
             a
             less
             Chord
             ,
             )
             than
             ½
             ∓
             A.
             
          
           
             LXVII
             .
             And
             therefore
             ,
             in
             this
             case
             ,
             P
             ,
             P
             ,
             (
             or
             S
             ,
             S
             ,
             )
             become
             Diagonals
             ,
             and
             C
             ,
             C
             ,
             opposite
             sides
             .
             And
             ,
             consequently
             ,
             Pq
             −
             Cq
             =
             BD
             (
             and
             Sq
             −
             Cq
             =
             bD
             ;
             )
             And
             
             .
             That
             is
             ,
          
           
             LXVIII
             .
             
               The
               Square
               of
               the
               Subtense
               of
               an
               Arch
               greater
               than
               a
               Quadrant
               (
               but
               less
               than
               three
               Quadrants
               )
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               Triple
               Arch
               ;
               is
               equal
               to
               the
               Rect-angle
               of
               the
               subtenses
               of
               the
               Double
               and
               Quadruple
               .
            
             And
             therefore
             ,
             divided
             by
             one
             of
             these
             ,
             it
             gives
             the
             other
             .
          
           
             LXIX
             .
             But
             P+C
             into
             P
             −
             C
             ,
             is
             equal
             to
             Pq
             −
             Cq.
             And
             therefore
             ,
             
             .
             (
             And
             ,
             in
             like
             manner
             ,
             
             )
             That
             is
             ,
          
           
             LXX
             .
             
               As
               the
               Subtense
               of
               the
               double
               Arch
               ,
               to
               the
               Aggregate
               of
               the
               subtenses
               of
               the
               Triple
               ,
               and
               of
               the
               single
               (
               greater
               than
               a
               Quadrant
               ,
               but
               less
               than
               three
               Quadrants
               ;
               )
               So
               is
               the
               excess
               of
               the
               Subtense
               of
               the
               single
               Arch
               above
               that
               of
               the
               Triple
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               .
            
          
           
           
             LXXI
             .
             Since
             therefore
             the
             Subtense
             of
             the
             Triple
             Arch
             P
             =
             ⅓+A
             (
             being
             
             greater
             than
             a
             Trient
             )
             is
             
             ;
             whose
             Square
             is
             
             :
             If
             this
             be
             taken
             from
             Pq
             ,
             and
             the
             Remainder
             (
             
             )
             divided
             by
             
             (
             as
             at
             §
             7
             ;
             )
             the
             Result
             is
             
             .
             That
             is
             ,
             dividing
             it
             first
             by
             
             ,
             and
             the
             Result
             by
             4Rq
             −
             Pq
             ,
             or
             by
             −
             Pq+4Rq
             ,
             and
             then
             restoring
             a
             Multiplication
             by
             
             )
          
           
             LXXII
             .
             And
             therefore
             (
             changing
             the
             equality
             in
             an
             Anology
             )
             
             .
          
           
             LXXIII
             .
             The
             same
             will
             happen
             if
             we
             take
             the
             Arch
             S
             =
             ⅓+E
             .
             For
             though
             this
             may
             be
             either
             greater
             or
             less
             than
             a
             Trient
             ,
             according
             as
             E
             is
             greater
             or
             less
             than
             30
             Degrees
             ;
             and
             (
             accordingly
             )
             the
             Triple
             thereof
             ,
             either
             
             ,
             or
             
             :
             Yet
             this
             doth
             not
             alter
             the
             case
             at
             all
             ;
             for
             ,
             either
             way
             ,
             the
             Square
             of
             it
             is
             the
             same
             .
             And
             therefore
             (
             making
             the
             Subduction
             and
             Division
             ,
             as
             §
             71.
             )
             
             .
             And
             
             .
             That
             is
             ,
          
           
             LXXIV
             .
             
               As
               the
               Cube
               of
               the
               Radius
               ,
               to
               the
               Subtense
               of
               an
               Arch
               greater
               than
               a
               Quadrant
               (
               but
               less
               than
               three
               Quadrants
               ,
               )
               Multiplied
               into
               the
               Square
               of
               the
               Subtense
               ,
               wanting
               two
               Squares
               of
               the
               Radius
               ;
               so
               is
               the
               Subtense
               of
               its
               Difference
               from
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             LXXV
             .
             Or
             thus
             ;
             
             Or
             ,
             
             .
             That
             is
             ,
          
           
             LXXVI
             .
             
               As
               the
               Radius
               ,
               to
               the
               Cube
               of
               the
               Subtense
               of
               an
               Arch
               (
               greater
               than
               a
               Quadrant
               ,
               but
               less
               than
               three
               Quadrants
               )
               divided
               by
               the
               Square
               of
               the
               Radius
               ,
               wanting
               the
               double
               of
               that
               Subtense
               ;
               so
               is
               the
               Subtense
               of
               the
               Difference
               from
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             LXXVII
             .
             Or
             thus
             ,
             because
             
             ;
             And
             also
             (
             in
             case
             the
             Arch
             S
             be
             also
             greater
             than
             a
             Trient
             )
             
             .
             Therefore
             ,
             
             (
             And
             
             )
             That
             is
             ,
          
           
             LXXVIII
             .
             
               As
               the
               Radius
               ,
               to
               the
               Aggregate
               of
               the
               Subtenses
               of
               the
               Triple
               Arch
               and
               of
               the
               single
               (
               this
               being
               greater
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               ,
               )
               so
               is
               the
               Subtense
               of
               its
               Difference
               from
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             LXXIX
             .
             But
             if
             the
             Arch
             S
             (
             though
             greater
             than
             a
             Quadrant
             )
             be
             less
             than
             a
             Trient
             ;
             or
             greater
             than
             two
             Trients
             ,
             but
             less
             than
             three
             Quadrants
             :
             That
             is
             ,
             
             .
             And
             therefore
             ,
             
             That
             is
             ,
          
           
             LXXX
             .
             
               As
               the
               Radius
               ,
               to
               the
               Subtense
               of
               an
               Arch
               greater
               than
               a
               Quadrant
               ,
               but
               less
               than
               a
               Trient
               (
               or
               greater
               than
               two
               Trients
               ,
               but
               less
               than
               three
               Quadrants
               )
               wanting
               the
               Subtense
               of
               the
               Triple
               Arch
               ;
               so
               is
               the
               Subtense
               of
               its
               Difference
               from
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
           
             LXXXI
             .
             All
             which
             are
             evident
             from
             the
             Scheme
             ;
             where
             the
             Chord
             D
             subtends
             
             the
             Quadruple
             of
             the
             Arches
             of
             A
             ,
             E
             ,
             P
             ,
             and
             S
             :
             And
             B
             subtends
             the
             double
             of
             the
             Arches
             E
             and
             P
             ,
             and
             b
             the
             double
             of
             the
             Arches
             of
             A
             and
             S.
             
          
           
             LXXXII
             .
             And
             ,
             in
             the
             Quadrilater
             whose
             sides
             B
             ,
             D
             ,
             be
             opposite
             and
             Parallel
             ;
             and
             C
             ,
             C
             ,
             opposite
             sides
             ;
             and
             P
             ,
             P
             ,
             Diagonals
             ;
             Pq
             −
             Cq
             =
             BD
             ,
             and
             
             .
             And
             likewise
             ,
             in
             the
             Quadrilater
             wherein
             b
             D
             are
             opposite
             and
             Parallel
             ;
             c
             c
             opposite
             sides
             ;
             and
             S
             S
             Diagonals
             ;
             Sq
             −
             cq
             =
             bD
             ,
             and
             
             .
          
           
             LXXXIII
             .
             And
             ,
             in
             the
             same
             Figure
             ,
             where
             not
             only
             the
             Arch
             of
             P
             ,
             but
             of
             S
             also
             ,
             are
             supposed
             greater
             than
             a
             Trient
             ;
             two
             of
             the
             Chords
             S
             ,
             S
             ,
             (
             as
             well
             as
             P
             ,
             P
             ,
             )
             cut
             the
             Chord
             D.
             
          
           
             LXXXIV
             .
             But
             in
             the
             other
             Figure
             ,
             where
             the
             Arch
             of
             S
             is
             supposed
             (
             greater
             
             than
             a
             Quadrant
             ,
             but
             )
             less
             than
             a
             Trient
             ;
             the
             case
             is
             somewhat
             different
             .
             For
             here
             b
             (
             the
             Subtense
             of
             the
             double
             Arch
             of
             S
             )
             falling
             on
             the
             other
             side
             of
             D
             (
             the
             Subtense
             of
             the
             Quadruple
             ,
             )
             the
             Chord
             D
             is
             not
             cut
             by
             any
             of
             the
             Chords
             S.
             
          
           
             LXXXV
             .
             But
             it
             comes
             to
             the
             same
             pass
             ,
             for
             these
             two
             Chords
             S
             S
             (
             whether
             they
             cut
             or
             not
             cut
             the
             Chord
             D
             ,
             )
             being
             no
             ingredients
             of
             the
             inscribed
             Quadrilater
             ,
             (
             but
             serve
             only
             to
             shew
             that
             b
             is
             the
             Subtense
             of
             the
             double
             Arch
             ;
             )
             it
             is
             however
             ,
             Sq
             −
             cq
             =
             bD
             .
          
           
             LXXXVI
             .
             The
             same
             things
             as
             before
             ,
             may
             be
             yet
             otherwise
             demonstrated
             
             (
             and
             more
             commodiously
             )
             in
             this
             manner
             ;
             Namely
             ,
             if
             instead
             of
             the
             Quadrilater
             whose
             four
             sides
             and
             two
             Diogonals
             are
             A
             ,
             A
             ,
             C
             ,
             C
             ,
             B
             ,
             D
             ;
             we
             take
             A
             ,
             A
             ,
             B
             ,
             B
             ,
             C
             ,
             D
             ;
             (
             taking
             the
             subtences
             of
             the
             single
             and
             double
             ,
             twice
             ;
             but
             ,
             of
             the
             Triple
             ,
             and
             Quadruple
             ,
             once
             :
             )
             with
             almost
             the
             same
             variety
             of
             cases
             ,
             as
             before
             .
             For
             ,
          
           
             LXXXVII
             .
             If
             the
             Subtense
             of
             the
             single
             Arch
             be
             A
             (
             or
             E
             ,
             )
             less
             than
             a
             Quadrant
             ;
             then
             A
             ,
             B
             ,
             and
             A
             ,
             D
             ,
             will
             be
             opposite
             sides
             ;
             and
             B
             ,
             C
             ,
             Diagonals
             .
             And
             therefore
             ,
             CB
             −
             AB
             =
             AD.
             And
             consequently
             
             into
             
             :
             equal
             to
             AD.
             That
             is
             ,
             
             .
             And
             
             ,
             as
             before
             .
             And
             for
             the
             same
             reason
             ,
             
             .
             And
             
             .
          
           
             LXXXVIII
             .
             If
             the
             Subtense
             of
             the
             single
             Arch
             be
             P
             (
             or
             S
             )
             greater
             than
             a
             
             Quadrant
             ,
             and
             even
             greater
             than
             a
             Trient
             :
             (
             but
             less
             than
             two
             Trients
             :
             )
             Then
             B
             ,
             C
             ,
             and
             B
             ,
             P
             ,
             (
             or
             B
             ,
             S
             ,
             )
             will
             be
             opposite
             sides
             ;
             and
             D
             ,
             P
             ,
             (
             or
             D
             ,
             S
             ,
             )
             Diagonals
             .
             And
             therefore
             BC+BP
             =
             PD
             ,
             (
             or
             BC+BS
             =
             SD
             .
             )
             And
             consequently
             ,
             
             into
             
             equal
             to
             PD
             .
             That
             is
             ,
             
             .
             And
             
             As
             before
             .
             And
             ,
             by
             the
             same
             reason
             ,
             BC+BS
             =
             SD
             (
             if
             S
             also
             be
             greater
             than
             a
             Trient
             )
             and
             
             .
          
           
           
             LXXXIX
             .
             But
             if
             the
             single
             Arch
             be
             that
             of
             S
             (
             greater
             than
             a
             Quadrant
             ,
             but
             )
             
             less
             than
             a
             Trient
             ;
             (
             or
             P
             greater
             than
             two
             Trients
             ,
             but
             less
             than
             three
             Quadrants
             ;
             )
             then
             B
             ,
             C
             ,
             and
             D
             ,
             S
             ,
             are
             opposite
             sides
             ;
             and
             B
             ,
             S
             ,
             Diagonals
             .
             And
             therefore
             ,
             BS
             −
             BC
             =
             DS.
             And
             consequently
             ,
             
             into
             
             .
             That
             is
             ,
             
             .
             And
             
             :
             As
             before
             .
             And
             in
             like
             manner
             ,
             BP
             −
             BC
             =
             PD
             (
             if
             the
             Arch
             of
             P
             be
             greater
             than
             two
             Trients
             ,
             which
             is
             the
             same
             as
             if
             less
             than
             one
             ;
             )
             and
             
             .
          
           
             XC
             .
             From
             all
             which
             ariseth
             this
             General
             Theorem
             :
             
               The
               Rect-angle
               of
               the
               Subtenses
               of
               the
               single
               and
               of
               the
               Quadruple
               Arch
               ,
               is
               equal
               to
               the
               Subtense
               of
               the
               double
               Multiplied
               into
               the
               Excess
               of
               the
               Subtense
               of
               the
               Triple
               above
               that
               of
               the
               single
               ,
               in
               case
               this
               be
               less
               than
               a
               Quadrant
               (
               or
               more
               than
               three
               Quadrants
               ;
               )
               or
               ,
               into
               the
               Excess
               of
               the
               Subtense
               of
               the
               single
               above
               that
               of
               the
               Triple
               ,
               in
               case
               the
               single
               be
               more
               than
               a
               Quadrant
               but
               less
               than
               a
               Trient
               (
               or
               more
               than
               two
               Trients
               ,
               but
               less
               than
               three
               Quadrants
               ;
               )
            
             or
             ,
             lastly
             ,
             into
             the
             Sum
             of
             the
             Subtenses
             of
             the
             Triple
             and
             
               single
               ,
               in
               case
               this
               be
               more
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               .
            
             That
             is
             ,
             AD
             :
             =
             B
             into
          
           
             
               C
               −
               A
               ;
               if
               the
               Arch
               of
               A
               be
               less
               than
               a
               Quadrant
               ,
               or
               greater
               than
               three
               Quadrants
               .
            
             
               A
               −
               C
               ;
               if
               it
               be
               greater
               than
               a
               Quadrant
               ,
               but
               less
               than
               a
               Trient
               ;
               or
               greater
               than
               two
               Trients
               ,
               but
               less
               than
               three
               Quadrants
               .
            
             
               A+C
               ;
               if
               it
               be
               greater
               than
               a
               Trient
               ,
               but
               less
               than
               two
               Trients
               .
            
          
           
             XCI
             .
             And
             ,
             universally
             ,
             
             .
             That
             is
             ,
             
               if
               the
               Difference
               of
               2RqA
               and
               A
               c
            
             (
             whereof
             that
             is
             the
             greater
             if
             the
             single
             Arch
             be
             less
             than
             a
             Quadrant
             ,
             or
             greater
             than
             three
             Quadrants
             ;
             but
             this
             if
             contrarywise
             ;
             )
             
               divided
               by
               Rc
               ,
               be
               Multiplied
               into
            
             
             
               Product
               is
               equal
               to
               D.
            
             
          
           
             XCII
             .
             And
             therefore
             ,
             
             .
             That
             is
             ,
          
           
             XCIII
             .
             
               As
               the
               Cube
               of
               the
               Radius
               ,
               to
               the
               Solid
               of
               the
               Subtense
               of
               the
               single
               Arch
               into
               the
               Difference
               of
               the
               Square
               of
               it self
               ,
               and
               of
               the
               double
               Square
               of
               the
               Radius
               :
               So
               is
               the
               Subtense
               of
               the
               Difference
               of
               that
               single
               Arch
               from
               a
               Semicircumference
               ,
               to
               the
               Subtense
               of
               the
               Quadruple
               Arch.
               
            
          
           
             XCIV
             .
             Now
             what
             was
             before
             said
             :
             (
             at
             §
             15
             ,
             Chap.
             29.
             )
             That
             the
             Subtense
             
             of
             an
             Arch
             ,
             with
             that
             of
             its
             Remainder
             to
             a
             Semicircumference
             (
             or
             of
             its
             Excess
             above
             a
             Semicircumference
             )
             will
             require
             the
             same
             Subtense
             of
             the
             double
             Arch
             ;
             is
             the
             same
             as
             to
             say
             ,
             that
             ,
             
               From
               any
               Point
               of
               Circumference
               ,
               two
               Subtenses
               drawn
               to
               the
               two
               ends
               of
               any
               inscribed
               Diameter
               ,
            
             (
             as
             A
             ,
             E
             ,
             )
             
               will
               require
               the
               same
               Subtense
            
             (
             B
             )
             
               of
               the
               double
               Arch.
            
             
          
           
             XCV
             .
             And
             what
             is
             said
             :
             (
             at
             §
             12
             ,
             26
             ,
             
               Chap.
               preced
            
             .
             )
             That
             the
             Subtense
             
             of
             an
             Arch
             less
             than
             a
             Trient
             ,
             and
             of
             its
             Residue
             to
             a
             Trient
             (
             as
             A
             ,
             E
             ,
             )
             and
             of
             a
             Trient
             increased
             by
             either
             of
             those
             ,
             (
             as
             Z
             ,
             )
             will
             have
             the
             same
             Subtense
             of
             the
             Triple
             Arch
             ;
             is
             the
             same
             in
             effect
             with
             this
             ,
             that
             ,
             
               From
               any
               Point
               of
               the
               Circumference
               ,
               three
               subtenses
               drawn
               to
               the
               three
               Angles
               of
               any
               inscribed
               (
               Regular
               )
               Trigone
            
             (
             as
             A
             ,
             E
             ,
             Z
             ,
             )
             
               will
               have
               the
               same
               Subtense
            
             (
             C
             )
             
               of
               the
               Triple
               Arch.
            
             
          
           
           
             XCVI
             .
             And
             what
             is
             said
             here
             :
             (
             at
             §
             18
             ,
             20.
             )
             That
             the
             Subtense
             of
             an
             
             Arch
             less
             than
             a
             Quadrant
             ,
             and
             of
             its
             Residue
             to
             a
             Quadrant
             ,
             (
             as
             A
             ,
             E
             ,
             )
             and
             of
             a
             Quadrant
             increased
             by
             either
             of
             these
             ,
             (
             as
             P
             ,
             S
             ,
             )
             will
             have
             the
             same
             Subtense
             of
             the
             Quadruple
             Arch
             :
             Is
             the
             same
             with
             this
             ,
             that
             ,
             
               From
               any
               Point
               of
               the
               Circumference
               ,
               Four
               Subtenses
               drawn
               to
               the
               four
               Angles
               of
               any
               inscribed
               (
               Regular
               )
               Tetragone
               ,
            
             (
             as
             A
             ,
             E
             ,
             P
             ,
             S
             ,
             )
             
               will
               have
               the
               same
               Subtense
            
             (
             D
             )
             
               of
               the
               Quadruple
               Arch.
            
             
          
           
             XCVII
             .
             But
             the
             same
             holds
             ,
             respectively
             ,
             in
             other
             Multiplications
             of
             Arches
             ;
             as
             five
             Subtenses
             from
             the
             same
             Point
             ,
             to
             the
             five
             Angles
             of
             an
             inscribed
             (
             Regular
             )
             Pentagon
             ;
             and
             six
             ,
             to
             the
             six
             Angles
             of
             an
             Hexagon
             ;
             &c.
             
             Will
             have
             the
             same
             Subtense
             of
             the
             Arches
             Quintuple
             ,
             Sextuple
             ,
             &c.
             
             For
             they
             all
             depend
             on
             the
             same
             common
             Principle
             ,
             That
             a
             Semicircumference
             Doubled
             ,
             a
             Trient
             Tripled
             ,
             a
             Quadrant
             Quadrupled
             ,
             a
             Quintant
             Quintupled
             ,
             a
             Sextant
             Sextupled
             ,
             &c.
             
             Make
             one
             entire
             Revolution
             ;
             which
             as
             to
             this
             business
             ,
             is
             the
             same
             as
             nothing
             .
             And
             therefore
             ,
             universally
             ,
          
           
             XCVIII
             .
             
               From
               any
               Point
               of
               the
               Circumference
               ,
               two
               ,
               three
               ,
               four
               ,
               five
               ,
               six
               ,
               or
               more
               subtenses
               ,
               drawn
               to
               so
               many
               (
               ends
               of
               the
               Diameter
               ,
               or
               )
               Angles
               of
               a
               (
               Regular
               )
               Polygone
               of
               so
               many
               Angles
               ,
               however
               inscribed
               ,
               will
               have
               the
               same
               Subtense
               of
               the
               Arch
               Multiplied
               by
               the
               number
               of
               such
               ends
               or
               Angles
               .
            
             And
             therefore
             ,
          
           
             CXIX
             .
             
               An
               Equation
               belonging
               to
               such
               Multiplication
               or
               Section
               of
               an
               Arch
               or
               Angle
               ,
               must
               have
               so
               many
               Roots
            
             (
             Affirmative
             or
             Negative
             )
             
               as
               is
               the
               Exponent
               of
               such
               Multiplication
               or
               Section
               .
            
             As
             two
             for
             the
             Bisection
             ,
             three
             for
             the
             Trisection
             ,
             four
             for
             the
             Quadrisection
             ,
             five
             for
             the
             Quinquisection
             :
             And
             so
             forth
             .
          
           
             C.
             And
             consequently
             ,
             
               Such
               Equations
               may
               accordingly
               be
               resolved
               ,
               by
               such
               Section
               of
               an
               Angle
               .
            
             As
             was
             before
             noted
             (
             at
             §
             61
             ,
             
               Chap.
               preced
            
             .
             )
             of
             the
             Trisection
             of
             an
             Angle
             .
          
        
         
           
           
             
               CHAP.
               IV.
            
             Of
             the
             Quintuplation
             and
             Quinquisection
             of
             an
             ARCH
             or
             ANGLE
             .
          
           
             I.
             IF
             in
             a
             Circle
             be
             inscribed
             a
             Quadrilater
             ,
             whose
             sides
             A
             ,
             F
             ,
             (
             the
             Subtenses
             
             of
             the
             single
             Arch
             and
             the
             Quintuple
             ,
             )
             be
             Parallel
             ;
             B
             ,
             B
             ,
             (
             subtenses
             of
             the
             double
             )
             opposite
             :
             The
             Diagonals
             will
             be
             C
             ,
             C
             ,
             (
             the
             subtenses
             of
             the
             Triple
             ,
             )
             as
             is
             evident
             from
             the
             Figure
             .
             But
             it
             is
             evident
             also
             ,
             that
             ,
             in
             this
             case
             ,
             the
             single
             Arch
             must
             be
             less
             than
             a
             Quintant
             (
             or
             fifth
             part
             )
             of
             the
             whole
             Circumference
             .
          
           
             II.
             And
             therefore
             (
             the
             Rect-angle
             of
             the
             Diagonals
             being
             equal
             to
             the
             two
             Rect-angles
             of
             the
             opposite
             sides
             ,
             )
             Cq
             −
             Bq
             =
             AF.
             (
             And
             by
             the
             same
             reasons
             cq
             −
             bq
             =
             EF.
             )
             That
             is
             ,
          
           
             III.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               Triple
               Arch
               ,
               wanting
               the
               Square
               of
               the
               Subtense
               of
               the
               double
               Arch
               ,
               is
               equal
               to
               the
               Rect-angle
               of
               the
               Subtenses
               of
               the
               single
               and
               of
               the
               Quintuple
               ;
               the
               single
               Arch
               being
               less
               than
               a
               fifth
               part
               of
               the
               whole
               Circumference
               .
            
          
           
             IV.
             And
             therefore
             ,
             if
             it
             be
             divided
             by
             one
             of
             them
             ;
             it
             gives
             the
             other
             .
             That
             is
             ,
             
             ;
             and
             
             .
             (
             And
             ,
             in
             like
             manner
             
             ;
             and
             
             .
          
           
             V.
             But
             C+B
             into
             C
             −
             B
             is
             equal
             to
             Cq
             −
             Bq.
             And
             therefore
             ,
             
             That
             is
             ,
          
           
             VI.
             
               As
               the
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
               fifth
               part
               of
               the
               whole
               Circumference
               )
               to
               the
               Aggregate
               of
               the
               subtenses
               of
               the
               Triple
               and
               double
               ;
               so
               is
               the
               Excess
               of
               the
               Subtense
               of
               the
               Triple
               above
               that
               of
               the
               double
               ,
               to
               that
               of
               the
               Quintuple
               .
            
          
           
             VII
             .
             And
             because
             (
             by
             §
             8.
             
             Chap.
             30.
             )
             
             ;
             and
             therefore
             
             .
             And
             (
             by
             §
             7.
             
             Chap.
             29.
             )
             
             Therefore
             ,
             
             And
             
             .
             That
             is
             ,
          
           
             VIII
             .
             
               If
               ,
               to
               the
               Quintuple
               of
               the
               Subtense
               of
               an
               Arch
               less
               than
               a
               Quintant
               ,
               wanting
               the
               Quintuple
               of
               the
               Cube
               of
               the
               same
               Subtense
               divided
               by
               the
               Square
               of
               the
               Radius
               ,
               be
               added
               the
               Quadricube
            
             (
             or
             fifth
             Power
             )
             
               of
               the
               same
               Subtense
               divided
               by
               the
               Biquadrate
               of
               the
               Radius
               ;
               the
               Result
               is
               the
               Subtense
               of
               the
               Quintuple
               Arch.
               
            
          
           
             IX
             .
             The
             same
             may
             be
             otherwise
             thus
             evinced
             ;
             taking
             a
             Quadrilater
             whose
             
             opposite
             sides
             are
             A
             ,
             A
             ,
             and
             F
             ,
             C
             ;
             and
             the
             Diagonals
             D
             ,
             D.
             And
             therefore
             ,
             Dq
             −
             Aq
             =
             CF.
             (
             And
             ,
             in
             like
             manner
             ,
             dq
             −
             Eq
             =
             cF.
             )
             That
             is
             ,
          
           
           
             X.
             
               The
               Square
               of
               the
               Subtense
               of
               the
               Quadruple
               Arch
               ,
               wanting
               the
               Square
               of
               the
            
             
             
               Subtense
               of
               the
               single
               Arch
               (
               less
               than
               a
               Quintant
               ,
               )
               is
               equal
               to
               the
               Rect-angle
               of
               the
               subtenses
               of
               the
               Triple
               and
               Quintuple
               .
            
             And
             being
             divided
             by
             either
             of
             these
             ,
             it
             gives
             the
             other
             of
             them
             .
          
           
             XI
             .
             And
             (
             because
             D+A
             into
             D
             −
             A
             is
             equal
             to
             Dq
             −
             Aq
             ,
             )
             
             (
             And
             
             )
             That
             is
             ,
          
           
             XII
             .
             
               As
               the
               Subtense
               of
               the
               Triple
               Arch
               ,
               to
               the
               sum
               of
               the
               subtenses
               of
               the
               Quadruple
               and
               of
               the
               single
               (
               this
               being
               less
               than
               a
               Quintant
               ,
               )
               so
               is
               the
               Difference
               of
               these
               ,
               to
               the
               Subtense
               of
               the
               Quintuple
               .
            
          
           
             XIII
             .
             But
             ,
             (
             by
             §
             7
             ,
             
               Chap.
               preced
               .
            
             )
             
             .
             And
             therefore
             ,
             
             Which
             abated
             by
             Aq
             ,
             leaves
             
             .
             And
             this
             divided
             by
             
             ;
             gives
             
             .
             As
             before
             ,
          
           
             XIV
             .
             The
             same
             ,
             is
             a
             third
             way
             ,
             thus
             evinced
             ;
             Inscribing
             a
             Quadrilater
             ,
             
             whose
             opposite
             sides
             are
             A
             ,
             C
             ,
             and
             A
             ,
             F
             ;
             and
             the
             Diagonals
             B
             ,
             D.
             And
             therefore
             AC+AF
             =
             BD
             ;
             and
             BD
             −
             AC
             =
             AF.
             (
             And
             in
             like
             manner
             ,
             bd
             −
             cE
             =
             EF.
             )
             That
             is
             ,
          
           
             15.
             
             
               The
               Rect-angle
               of
               the
               subtenses
               of
               the
               double
               and
               Quadruple
               Arch
               ,
               wanting
               that
               of
               the
               subtenses
               of
               the
               single
               (
               being
               less
               than
               a
               Quintant
               )
               and
               of
               the
               Triple
               ;
               is
               equal
               to
               the
               Rect-angle
               of
               the
               subtenses
               of
               the
               single
               and
               Quintuple
               .
            
             And
             ,
             being
             divided
             by
             either
             of
             these
             ,
             gives
             the
             other
             of
             them
             .
          
           
             XVI
             .
             And
             therefore
             ,
             
             (
             And
             
             )
             That
             is
             ,
          
           
             XVII
             .
             
               As
               the
               Subtense
               of
               a
               single
               Arch
               (
               less
               than
               a
               Quintant
               ,
               )
               to
               that
               of
               the
               double
               ;
               so
               is
               that
               of
               the
               Quadruple
               ,
               to
               the
               Aggregate
               of
               the
               subtenses
               of
               the
               Triple
               and
               Quintuple
               .
            
          
           
             XVIII
             .
             But
             
             .
             And
             
             Therefore
             ,
             
             into
             
             .
             Likewise
             ,
             
             ;
             and
             therefore
             ,
             
             .
             And
             therefore
             ,
             
             .
             And
             
             .
             As
             before
             ,
          
           
             XIX
             .
             Or
             ,
             we
             may
             thus
             compute
             it
             :
             Because
             
             (
             as
             before
             ;
             )
             therefore
             
             .
             And
             therefore
             ,
             (
             subtracting
             
             ,
             )
             
             .
             As
             before
             ,
          
           
           
             XX.
             The
             same
             way
             ,
             a
             fourth
             way
             ,
             be
             thus
             evinced
             ;
             Inscribing
             a
             Quadrilater
             
             whose
             opposite
             sides
             are
             B
             ,
             F
             ,
             and
             B
             ,
             A
             ;
             and
             the
             Diagonals
             C
             ,
             D.
             And
             therefore
             BA+BF
             =
             CD
             ,
             and
             CD
             −
             BA
             =
             BF
             .
             (
             And
             ,
             in
             like
             manner
             ,
             cd
             −
             bE
             =
             BF
             .
             )
             And
             
             .
             That
             is
             ,
          
           
             XXI
             .
             
               The
               Rect-angle
               of
               the
               subtenses
               of
               the
               Treble
               and
               Quadruple
               Arches
               ,
               wanting
               that
               of
               the
               subtenses
               of
               the
               double
               and
               single
               (
               this
               being
               less
               than
               a
               Quintant
               ,
               )
               is
               equal
               to
               that
               of
               the
               subtenses
               of
               the
               double
               and
               Quintuple
               .
            
             And
             being
             divided
             by
             the
             one
             ,
             it
             gives
             the
             other
             .
          
           
             XXII
             .
             And
             therefore
             ,
             
             (
             And
             
             )
             That
             is
             ,
          
           
             XXIII
             .
             
               As
               the
               subtense
               of
               the
               double
               Arch
               ,
               to
               that
               of
               the
               Triple
               ,
               so
               is
               that
               of
               the
               Quadruple
               ,
               to
               the
               Aggregate
               of
               the
               subtenses
               of
               the
               single
               (
               being
               less
               than
               a
               Quintant
               )
               and
               of
               the
               Quintuple
               .
            
          
           
             XXIV
             .
             But
             
             ;
             and
             
             .
             Therefore
             ,
             
             .
             Likewise
             ,
             
             :
             And
             therefore
             ,
             
             .
             And
             consequently
             ,
             
             .
             And
             (
             dividing
             by
             
             :
             )
             
             .
             As
             before
             ,
          
           
             XXV
             .
             Or
             ,
             we
             may
             thus
             compute
             it
             :
             Because
             
             :
             Therefore
             ,
             (
             dividing
             by
             
             ;
             )
             
             .
             And
             
             .
             As
             before
             ,
          
           
             XXVI
             .
             Or
             thus
             ;
             because
             BA+BF
             =
             CD
             ;
             and
             therefore
             ,
             
             :
             
             And
             also
             ,
             
             :
             And
             
             :
             Therefore
             ,
             
             ;
             and
             this
             Multiplied
             by
             
             ,
             makes
             
             :
             And
             
             .
             As
             before
             .
          
           
             XXVII
             .
             But
             if
             the
             Arch
             to
             be
             Quintupled
             be
             just
             the
             fifth
             part
             of
             the
             whole
             Circumference
             ,
             (
             and
             consequently
             the
             Quintuple
             Arch
             one
             intire
             Revolution
             ;
             )
             the
             Subtense
             of
             that
             Quintuple
             will
             vanish
             ,
             or
             become
             equal
             to
             nothing
             .
          
           
           
             XXVIII
             .
             And
             therefore
             ,
             in
             this
             case
             ,
             
             .
             
             And
             so
             5RqqA
             −
             5RqAc
             +
             Acq
             =
             0
             ;
             and
             5Rqq
             −
             5RqAq
             +
             Aqq
             =
             0
             ;
             or
             ,
             5Rqq
             =
             5RqAq
             −
             Aqq
             ;
             or
             ,
             
             .
             Which
             is
             a
             Quadratick
             Equation
             ,
             whose
             Root
             is
             
             ,
             and
             the
             Co-efficient
             of
             the
             middle
             Term
             5R
             ,
             and
             the
             absolute
             quantity
             5Rq
             .
          
           
             XXIX
             .
             Therefore
             ,
             (
             by
             resolving
             the
             Equation
             )
             :
             
             .
          
           
             XXX
             .
             Of
             which
             ambiguous
             Equation
             ,
             the
             lesser
             Root
             is
             to
             be
             chosen
             ,
             That
             is
             ,
             
             ;
             and
             therefore
             ,
             
             ,
             the
             Subtense
             of
             a
             Quintant
             .
             That
             is
             ,
          
           
             XXXI
             .
             
               The
               Radius
               Multiplied
               into
            
             
             ,
             
               is
               equal
               to
               the
               Subtense
               of
               a
               Quintant
               ,
               or
               of
            
             72
             Degrees
             .
          
           
             XXXII
             .
             The
             same
             may
             be
             otherwise
             thus
             inferred
             :
             If
             ,
             in
             a
             Circle
             ,
             be
             inscribed
             
             a
             Regular
             Pentagon
             ;
             whose
             side
             A
             shall
             be
             reputed
             as
             the
             Subtense
             of
             a
             single
             Arch
             :
             It
             's
             evident
             that
             the
             Subtense
             of
             the
             Duple
             ,
             and
             of
             the
             Triple
             ,
             will
             be
             the
             same
             .
             (
             For
             the
             same
             Chord
             which
             on
             the
             one
             side
             ,
             subtends
             the
             Duple
             ,
             doth
             on
             the
             other
             side
             ,
             subtend
             the
             Triple
             .
             )
             And
             therefore
             ,
             
             .
             And
             
             .
             And
             
             .
             And
             therefore
             ,
             
             .
             And
             therefore
             ,
             (
             as
             before
             )
             
             ;
             and
             so
             onward
             as
             above
             .
          
           
             XXXIII
             .
             Now
             because
             (
             as
             is
             already
             shew'd
             )
             
             :
             This
             therefore
             will
             be
             the
             Subtense
             of
             a
             Sesquiquintant
             (
             or
             one
             Quintant
             and
             an
             half
             ,
             or
             three
             tenth
             parts
             ,
             )
             that
             is
             ,
             of
             108
             Degrees
             :
             As
             being
             that
             Arch
             which
             with
             the
             Quintant
             doth
             complete
             the
             Semicircumference
             .
             That
             is
             ,
          
           
             XXXIV
             .
             
               The
               Difference
               of
               the
               Squares
               of
               the
               Subtenses
               of
               the
               Trient
               ,
               and
               of
               the
               Quintant
               ,
               divided
               by
               the
               Radius
               ;
               is
               equal
               to
               the
               Subtense
               of
               the
               Sesquiquintant
               ,
               or
            
             108
             Degrees
             .
             (
             For
             3Rq
             is
             the
             Square
             of
             the
             Subtense
             of
             the
             Trient
             ;
             and
             Aq
             ,
             of
             the
             Quintant
             ;
             and
             the
             Difference
             of
             these
             3Rq
             −
             Aq
             divided
             by
             the
             Radius
             ,
             is
             the
             Subtense
             .
             )
             Or
             thus
             ,
          
           
             XXXV
             .
             
               If
               from
               the
               Triple
               of
               the
               Radius
            
             3R
             ,
             
               be
               subducted
               the
               Square
               of
               the
               Subtense
               of
               a
               Quintant
               divided
               by
               the
               Radius
               ;
               the
               Remainder
               is
               the
               Subtense
               of
               a
               Sesquiquintant
               ,
               or
            
             108
             Degrees
             .
             
             .
          
           
             XXXVI
             .
             But
             the
             Square
             of
             the
             Subtense
             of
             a
             Quintant
             so
             divided
             ,
             is
             (
             as
             
             before
             )
             
             ;
             which
             therefore
             subtracted
             from
             3R
             ,
             leaves
             
             R
             the
             Subtense
             of
             108
             Degrees
             .
          
           
           
             XXXVII
             .
             Now
             ,
             if
             the
             Radius
             be
             cut
             in
             extream
             and
             mean
             proportion
             ,
             the
             
             greater
             Segment
             thereof
             is
             
             (
             by
             11.
             
             El.
             2.
             )
             to
             which
             if
             1
             R
             be
             added
             ,
             we
             have
             
             (
             the
             Subtense
             of
             108
             Degrees
             as
             before
             ;
             )
             And
             therefore
             ,
          
           
             XXXVIII
             .
             
               If
               the
               Radius
               being
               cut
               in
               extreme
               and
               mean
               proportion
               ,
               the
               greater
               Segment
               thereof
               ,
               be
               added
               to
               the
               whole
               Radius
               ;
               the
               sum
               is
               equal
               to
               the
               Subtense
               of
            
             108
             Degrees
             .
          
           
             XXXIX
             .
             Yet
             again
             ;
             If
             ,
             of
             a
             Pentagone
             so
             inscribed
             ,
             the
             side
             A
             be
             considered
             as
             the
             Subtense
             of
             a
             single
             Arch
             ;
             the
             same
             will
             also
             be
             the
             Subtense
             of
             the
             Quadruple
             .
             (
             For
             the
             same
             Chord
             subtends
             on
             the
             one
             side
             to
             one
             Quintant
             ,
             and
             on
             the
             other
             side
             to
             four
             such
             .
             )
          
           
             XL.
             And
             therefore
             ,
             in
             this
             case
             ,
             
             .
             And
             RcA
             =
             2RqA
             −
             Ac
             into
             
             .
             That
             is
             ,
             Rc
             =
             2Rq
             −
             Aq
             ,
             into
             
             .
             And
             (
             the
             Square
             hereof
             )
             Rcc
             =
             16Rcc
             −
             20RqqAq
             +
             8RqAqq
             −
             Acc
             ;
             or
             15Rcc
             −
             20RqqAq
             +
             8RqAqq
             −
             Acc
             =
             0.
             
          
           
             XLI
             .
             Now
             this
             last
             Equation
             ,
             if
             divided
             by
             3Rq
             −
             Aq
             =
             0
             ,
             will
             afford
             this
             Equation
             ;
             5Rqq
             −
             5RqAq
             +
             Aqq
             =
             0.
             
          
           
           
             XLII
             .
             And
             therefore
             3Rq
             =
             Aq
             ;
             is
             one
             of
             the
             Plain
             Roots
             of
             that
             Equation
             .
             And
             therefore
             ,
             
             ,
             which
             is
             the
             Subtense
             of
             a
             Trient
             .
             (
             Which
             is
             true
             ,
             because
             also
             the
             Quadruple
             of
             a
             Trient
             ,
             hath
             the
             same
             Subtense
             with
             the
             single
             Trient
             .
             )
          
           
             XLIII
             .
             But
             there
             are
             also
             two
             other
             Plain
             Roots
             included
             in
             the
             Resulting
             Equation
             5Rqq
             −
             5RqAq
             +
             Aqq
             =
             0
             ;
             or
             5Rqq
             =
             5RqAq
             −
             Aqq.
             For
             ,
          
           
             XLIV
             .
             The
             lesser
             of
             them
             is
             
             ;
             the
             Square
             of
             the
             Subtense
             of
             a
             Quintant
             .
             As
             before
             ,
          
           
             XLV
             .
             The
             greater
             of
             them
             is
             
             ;
             the
             Square
             of
             the
             Subtense
             of
             a
             double
             Quintant
             ,
             or
             of
             a
             Triple
             ,
             (
             as
             we
             shall
             see
             afterward
             )
             that
             is
             ,
             of
             144
             ,
             or
             of
             216
             Degrees
             .
             For
             the
             Quadruple
             of
             these
             also
             ,
             will
             have
             the
             same
             Subtense
             with
             that
             of
             the
             single
             .
             For
             ⅖
             ×
             4
             =
             ⅗
             =
             1
             +
             ⅗
             .
             And
             ½
             ×
             4
             =
             12
             /
             5
             =
             2
             +
             ⅖
             .
             Where
             the
             Excess
             above
             the
             entire
             Revolutions
             (
             which
             are
             here
             Equivalent
             to
             nothing
             )
             is
             ⅗
             ,
             or
             4
             /
             5
             ,
             both
             which
             have
             the
             same
             Subtense
             (
             as
             at
             §
             32.
             )
             over
             that
             of
             the
             single
             Arch
             ;
             that
             is
             ,
             4
             ⅖
             ,
             or
             ⅗
             .
          
           
           
             XLVI
             .
             Since
             therefore
             (
             as
             is
             shewed
             )
             
             ,
             is
             the
             Square
             of
             
             the
             Subtense
             of
             a
             Quintant
             ;
             the
             Square
             of
             the
             Subtense
             of
             its
             Residue
             to
             the
             Semicircumference
             must
             be
             
             .
             Which
             is
             therefore
             the
             Square
             of
             the
             Subtense
             of
             108
             (
             =
             180
             −
             72.
             )
             And
             the
             Quadratick
             Root
             thereof
             
             as
             was
             also
             before
             shewed
             .
          
           
             XLVII
             .
             And
             for
             as
             much
             as
             
             is
             the
             Subtense
             of
             108
             Degres
             ,
             that
             is
             of
             18
             Degrees
             above
             a
             Quadrant
             ;
             let
             this
             Subtense
             be
             S
             ,
             and
             the
             Subtense
             of
             18
             Degrees
             ,
             (
             which
             is
             the
             Excess
             above
             a
             Quadrant
             )
             E.
             Therefore
             ,
             (
             by
             §
             54
             ,
             
               Chap.
               preced
            
             .
             )
             
             .
             And
             therefore
             ,
             
             .
             And
             (
             by
             resolving
             that
             Equation
             )
             
             .
             The
             lesser
             of
             which
             Roots
             is
             here
             to
             be
             chosen
             ,
             because
             E
             is
             the
             lesser
             of
             the
             two
             S
             ,
             E.
             
          
           
             XLVIII
             .
             But
             (
             as
             is
             shewed
             )
             
             ,
             and
             therefore
             ,
             
             And
             
             ,
             and
             therefore
             ,
             
             ,
             (
             half
             the
             Square
             of
             the
             Subtense
             of
             a
             Quintant
             ,
             )
             whose
             Square
             Root
             is
             
             .
             And
             therefore
             ,
             (
             the
             less
             Root
             being
             here
             of
             use
             )
             
             ,
             the
             Subtense
             of
             18
             Degrees
             .
          
           
             XLIX
             .
             The
             same
             Arch
             of
             18
             Degrees
             ,
             is
             also
             the
             Complement
             of
             a
             Quintant
             to
             a
             Quadrant
             .
             And
             therefore
             if
             the
             Subtense
             of
             a
             Quintant
             (
             or
             72
             Degrees
             ,
             being
             less
             than
             a
             Quadrant
             ,
             )
             be
             called
             
             ;
             and
             the
             Subtense
             of
             its
             Complement
             to
             a
             Quadrant
             (
             or
             of
             18
             Degrees
             )
             E
             :
             Then
             (
             by
             §
             52.
             
             
               Chap.
               preced
            
             .
             )
             
             .
             And
             therefore
             ,
             
             .
             And
             (
             resolving
             the
             Equation
             ,
             )
             
             .
          
           
             L.
             But
             
             ,
             and
             therefore
             ,
             
             (
             half
             the
             Square
             of
             the
             Subtense
             of
             the
             Sesquiquintant
             ,
             or
             108
             Degrees
             ;
             )
             and
             the
             Square
             Root
             thereof
             
             .
             And
             
             .
             And
             therefore
             ,
             
             ,
             the
             Subtense
             of
             18
             Degrees
             ,
             as
             before
             .
             That
             is
             ,
          
           
             LI.
             
               The
               Subtense
               of
               the
               Sesquiquintant
               ,
               or
               of
            
             108
             
               Degrees
               ,
               (
               that
               is
               ,
               the
               greater
               Segment
               of
               the
               Radius
               cut
               in
               extream
               and
               mean
               proportion
               ,
               increased
               by
               the
               entire
               Radius
               ,
               )
               Multiplied
               into
            
             
             (
             for
             
             ,
             )
             
               wanting
               the
               Subtense
               of
               the
               Quintant
               Multiplied
               also
               into
            
             
             (
             for
             
             into
             
             )
             
               is
               equal
               to
               the
               double
               of
               the
               Subtense
               of
            
             18
             Degrees
             .
             (
             And
             half
             thereof
             ,
             equal
             to
             that
             Subtense
             .
             )
             Or
             ,
          
           
             LII
             .
             
               The
               Difference
               of
               the
               Subtenses
               of
               the
               Sesquiquintant
               and
               of
               the
               Quintant
               ,
            
             (
             or
             of
             108
             Degrees
             ,
             and
             of
             72
             Degrees
             )
             
               divided
               by
            
             
             ,
             
               is
               equal
               to
               the
               Subtense
               of
            
             18
             Degrees
             .
             That
             is
             ,
             that
             Difference
             is
             double
             in
             Power
             to
             this
             Subtense
             ,
             (
             
               duplum
               potest
            
             ,
             )
             or
             ,
             the
             Square
             of
             that
             ,
             is
             double
             to
             the
             Square
             of
             this
             .
          
           
           
             LIII
             .
             But
             ,
             
               The
               subtenses
               of
               the
               Quintant
               and
               Sesquiquintant
               ,
            
             (
             that
             is
             ,
             of
             72
             ,
             and
             
             of
             108
             Degrees
             ,
             which
             together
             complete
             the
             Semicircumference
             )
             
               Multiplied
               the
               one
               into
               the
               other
               ,
            
             (
             or
             the
             Rectangle
             of
             them
             ,
             )
             
               divided
               by
               the
               Radius
               ;
               is
               equal
               to
               the
               Subtense
               of
               the
               double
               Arch
               of
               either
               .
            
             For
             ,
             by
             §
             9
             ,
             Chap.
             R
             )
             AE
             (
             B.
             That
             is
             ,
             of
             144
             ,
             or
             of
             216
             Degrees
             .
             That
             is
             ,
             of
             the
             double
             ,
             or
             Triple
             Quintant
             ,
             (
             these
             two
             having
             the
             same
             Subtense
             .
             )
             That
             is
             ,
             
             .
             That
             is
             ,
          
           
             LIV.
             
               The
               Radius
               Multiplied
               into
            
             
             ,
             
               is
               equal
               to
               the
               Subtense
               of
               the
               Biquintant
               ,
               and
               of
               the
               Triquintant
            
             ;
             That
             is
             ,
             
               to
               the
               Subtense
               of
            
             144
             ,
             
               and
               of
            
             216
             Degrees
             .
          
           
             LV.
             And
             the
             Square
             of
             this
             subtracted
             from
             the
             Square
             of
             the
             Diameter
             ,
             leaves
             
             the
             Square
             of
             the
             Subtense
             of
             36
             Degrees
             ;
             (
             as
             being
             what
             144
             Degrees
             wants
             of
             a
             Semicircumference
             ,
             and
             what
             216
             exceeds
             it
             .
             For
             180
             −
             144
             =
             36
             =
             216
             −
             180.
             )
             And
             the
             Square
             Root
             thereof
             is
             that
             Subtense
             ,
             
             .
             That
             is
             ,
          
           
             LVI
             .
             
               The
               greater
               Segment
               of
               the
               Radius
               cut
               in
               extream
               and
               mean
               Proportion
               ,
               is
               the
               Subtense
               of
            
             36
             Degrees
             .
             That
             is
             ,
             of
             half
             a
             Quintant
             ,
             or
             the
             side
             of
             the
             inscribed
             Decagon
             .
          
           
             LVII
             .
             But
             we
             had
             before
             shewn
             (
             at
             §
             38.
             )
             that
             this
             added
             to
             the
             Radius
             (
             which
             is
             the
             Subtense
             of
             60
             Degrees
             ,
             or
             side
             of
             the
             inscribed
             Hexagon
             ,
             )
             is
             equal
             to
             the
             Subtense
             of
             108
             Degrees
             ,
             or
             Sesquiquintant
             :
             Therefore
             ,
          
           
             LVIII
             .
             
               The
               Aggregate
               of
               the
               subtenses
               of
            
             36
             
               Degrees
               ,
               and
               of
            
             60
             Degrees
             ,
             (
             that
             is
             ,
             the
             sides
             of
             the
             inscribed
             Decagon
             and
             Hexagon
             ,
             )
             
               is
               equal
               to
               that
               of
            
             108
             Degrees
             ;
             (
             that
             is
             ,
             of
             the
             Sesquiquintant
             ,
             or
             three
             Tenths
             .
             )
          
           
             LIX
             .
             If
             therefore
             to
             the
             Subtense
             of
             36
             Degrees
             ,
             
             ,
             be
             added
             that
             of
             108
             Degrees
             
             ,
             it
             makes
             
             ,
             or
             
             .
             That
             is
             ,
          
           
             LX.
             
               The
               Subtense
               of
               the
               Semiquintant
            
             (
             or
             36
             Degrees
             )
             
               and
               of
               the
               Sesquiquintant
            
             (
             or
             108
             Degrees
             )
             
               added
               together
               ,
               are
               in
               power
               Quintuple
               to
               the
               Radius
               ,
            
             (
             that
             is
             ,
             the
             Square
             of
             that
             Aggregate
             is
             equal
             to
             five
             Squares
             of
             the
             Radius
             .
             )
             For
             ,
             
             .
          
           
             LXI
             .
             
               And
               their
               Difference
               is
               equal
               to
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXII
             .
             
               And
               the
               Rectangle
               of
               them
               ,
               is
               equal
               to
               the
               Square
               of
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXIII
             .
             
               And
               the
               sum
               of
               their
               Squares
               is
               Triple
               to
               the
               Square
               of
               the
               Radius
               .
            
             (
             Or
             ,
             
               equal
               to
               the
               Square
               of
               the
               side
               of
               the
               inscribed
               Trigone
               .
            
             )
             That
             is
             ,
             
             .
          
           
           
             LXIV
             .
             
               And
               the
               Difference
               of
               their
               Squares
               ,
               is
               in
               Power
               Quintuple
               to
               the
               Square
               of
            
             
             
               the
               Radius
            
             ,
             (
             or
             ,
             
               equal
               to
               five
               squared
               Squares
               of
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXV
             .
             Again
             ,
             
               The
               sum
               of
               the
               Squares
               of
               the
               subtenses
               of
               the
               Quintant
               and
               Biquintant
            
             (
             or
             of
             72
             Degrees
             ,
             and
             of
             144
             Degrees
             ,
             )
             
               is
               Quintuple
               to
               the
               Square
               of
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXVI
             .
             
               And
               the
               Difference
               thereof
               ,
               is
               in
               Power
               Quintuple
               to
               the
               Biquadrate
               of
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXVII
             .
             
               And
               the
               Rectangle
               of
               them
               ,
               is
               Quintuple
               of
               the
               Biquadrate
               of
               the
               Radius
               .
            
             For
             ,
             
             .
          
           
             LXVIII
             .
             We
             have
             therefore
             (
             as
             hath
             been
             severally
             demonstrated
             )
             these
             subtenses
             ,
             answering
             to
             their
             several
             Arches
             ,
             or
             portions
             of
             the
             whole
             Circumferences
             ,
             viz.
             
          
           
             
          
           
             LXIX
             .
             By
             the
             like
             method
             we
             may
             find
             the
             subtenses
             of
             
             ,
             
             ,
             
             ,
             
             ,
             of
             the
             whole
             Circumference
             :
             (
             as
             likewise
             of
             
             ,
             
             ,
             
             ,
             
             ,
             )
             For
             the
             Residue
             of
             ⅕
             to
             a
             Quadrant
             (
             or
             Excess
             of
             
             above
             a
             Quadrant
             )
             is
             
             ;
             and
             therefore
             the
             Subtense
             thereof
             is
             
             ,
             (
             as
             is
             shewed
             before
             at
             §
             48.
             )
             or
             (
             which
             is
             Equivalent
             )
             
             .
             Or
             ,
             
             .
             And
             the
             Residue
             to
             this
             to
             the
             Semicircumference
             is
             
             ;
             whose
             Subtense
             therefore
             is
             
             .
             Or
             ,
             
             .
             Again
             ,
             the
             Residue
             of
             
             to
             a
             Quadrant
             (
             or
             the
             Excess
             of
             ⅖
             above
             a
             Quadrant
             ,
             )
             is
             
             ;
             whose
             Subtense
             therefore
             is
             
             .
             Or
             ,
             
             .
             And
             the
             Residue
             of
             this
             to
             a
             Semicircumference
             is
             
             ;
             whose
             Subtense
             therefore
             is
             
             .
             Or
             ,
             
             .
             For
             ,
             in
             such
             cases
             ,
             the
             same
             value
             may
             be
             expressed
             in
             very
             different
             ways
             .
             (
             All
             which
             may
             be
             easiy
             proved
             by
             computation
             ,
             in
             like
             manner
             as
             those
             before
             going
             ;
             and
             like
             Corollaries
             easily
             deduced
             from
             them
             .
             )
             And
             the
             Remainders
             of
             these
             to
             the
             whole
             Circumferences
             (
             
             ,
             
             ,
             
             ,
             
             ,
             )
             have
             the
             same
             Subtenses
             with
             them
             .
          
           
             LXX
             .
             We
             have
             therefore
             ,
             now
             ,
             these
             Subtenses
             ,
             for
             the
             Arches
             and
             portions
             following
             .
          
           
           
             Degrees
             of
             Arches
             .
             Portions
             of
             the
             whole
             .
             Subtenses
             .
             
          
           
             
          
           
             LXXI
             .
             Now
             if
             all
             these
             Arches
             be
             compared
             with
             the
             Trient
             ;
             and
             the
             Sums
             and
             Differences
             of
             them
             so
             compared
             be
             observed
             :
             We
             shall
             thence
             have
             a
             great
             many
             more
             Subtenses
             ,
             by
             what
             is
             before
             delivered
             ,
             at
             §
             15
             ,
             47
             ,
             Chap.
             30.
             
             As
             for
             Example
             ,
          
           
             LXXII
             .
             Suppose
             the
             Subtense
             of
             72
             Degrees
             to
             be
             
             ,
             and
             the
             Subtense
             of
             48
             (
             =
             120
             −
             72
             )
             to
             be
             E.
             Then
             ,
             (
             by
             §
             15
             ,
             Chap.
             30.
             )
             Aq+AE+Eq
             =
             3Rq
             ;
             and
             therefore
             ,
             AE+Eq
             =
             3Rq
             −
             Aq.
             And
             (
             by
             resolving
             the
             Equation
             )
             
             .
          
           
             LXXIII
             .
             But
             
             ,
             and
             
             .
             Therefore
             ,
             
             the
             Subtense
             of
             48
             Degrees
             ,
             and
             therefore
             also
             of
             312
             Degrees
             .
          
           
             LXXIV
             .
             In
             like
             manner
             :
             Suppose
             (
             as
             before
             )
             A
             the
             Subtense
             of
             72
             Degrees
             ;
             and
             Z
             the
             Subtense
             of
             192
             (
             =
             120+72
             .
             )
             Then
             (
             by
             §
             47
             ,
             Chap.
             30.
             )
             Zq
             −
             AZ+Aq
             =
             3Rq
             ;
             and
             Zq
             −
             AZ
             =
             3Rq
             −
             Aq.
             And
             (
             resolving
             the
             Equation
             )
             
             .
             The
             Subtense
             of
             192
             Degrees
             ,
             and
             therefore
             also
             of
             168
             Degrees
             .
             That
             is
             ,
          
           
             LXXV
             .
             
               If
               the
               Subtense
               of
               a
               Trient
               ,
               be
               increased
               by
               the
               greater
               Segment
               of
               such
               Subtense
               cut
               in
               extream
               and
               mean
               proportion
               ;
               And
               thereunto
               be
               Added
               ,
               or
               taken
               from
               it
               ,
               the
               Subtense
               of
               a
               Quintant
               :
               The
               result
               is
               ,
               in
               case
               of
               Addition
               ,
               the
               double
               Subtense
               of
            
             168
             
               and
               of
            
             192
             
               Degrees
               ;
               in
               case
               of
               Subtraction
               ,
               the
               double
               Subtense
               of
            
             48
             
               and
               of
            
             312
             Degrees
             .
             Or
             thus
             ,
          
           
           
             LXXVI
             .
             
               If
               to
               the
               greater
               Segment
               of
               the
               Subtense
               of
               a
               Trient
               (
               cut
               in
               extream
               and
            
             
             
               mean
               Proportion
               ,
               )
               be
               added
               the
               Sum
               ,
               or
               Difference
               of
               the
               subtenses
               of
               the
               Trient
               and
               of
               the
               Quintant
               :
               The
               Result
               is
               ,
               the
               double
               Subtense
               ,
               in
               the
               first
               case
               ,
               of
            
             168
             
               and
               of
            
             192
             
               Degrees
               ;
               in
               the
               latter
               case
               ,
               of
            
             48
             
               and
               of
            
             312
             Degrees
             .
             For
             ,
             
             ,
             is
             the
             half
             of
             the
             Subtense
             of
             a
             Trient
             (
             
             )
             increased
             by
             its
             greater
             Segment
             is
             so
             cut
             .
             And
             
             ,
             is
             half
             the
             Subtense
             of
             a
             Quintant
             .
          
           
             LXXVII
             .
             And
             the
             Squares
             of
             these
             subtenses
             ,
             subtracted
             from
             4Rq
             ,
             give
             us
             the
             Squares
             of
             the
             subtenses
             of
             their
             Differences
             from
             a
             Semicircumference
             .
             That
             is
             ,
             of
             12
             and
             of
             132
             Degrees
             (
             whereby
             168
             and
             48
             come
             short
             of
             a
             Semicircumference
             ;
             and
             whereby
             192
             and
             312
             exceed
             it
             .
             )
             For
             12
             =
             180
             −
             168
             =
             192
             −
             180
             ;
             and
             132
             =
             180
             −
             48
             =
             312
             −
             180.
             
          
           
             LXXVIII
             .
             Again
             ,
             suppose
             the
             Subtense
             of
             a
             Biquintant
             ,
             or
             144
             Degrees
             ,
             (
             which
             is
             also
             the
             Subtense
             of
             a
             Triquintant
             ,
             or
             216
             Degrees
             )
             being
             greater
             than
             a
             Trient
             ,
             to
             be
             
             And
             the
             Subtense
             of
             96
             =
             216
             −
             120
             =
             240
             −
             144
             ,
             to
             be
             A
             :
             And
             the
             Subtense
             of
             24
             =
             144
             −
             120
             =
             240
             −
             216
             ,
             to
             be
             E.
             Therefore
             ,
             (
             by
             §
             47
             ,
             Chap.
             30.
             )
             Zq
             −
             ZA+Aq
             =
             Zq
             −
             ZE+Eq
             =
             3Rq
             ;
             and
             (
             Zq
             being
             greater
             than
             3Rq
             ,
             )
             Zq
             −
             3Rq
             =
             ZA
             −
             Aq
             =
             ZE
             −
             Eq.
             And
             resolving
             the
             Equation
             )
             
             ,
             the
             Subtense
             of
             96
             Degrees
             if
             connected
             by
             +
             ;
             or
             of
             24
             Degrees
             ,
             if
             by
             −
             .
             (
             That
             is
             ,
             
             ,
             in
             the
             first
             case
             ;
             and
             
             ,
             in
             the
             latter
             .
             )
             That
             is
             ,
          
           
             LXXIX
             .
             
               If
               to
               the
               Subtense
               of
               a
               Biquintant
               be
               Added
               ,
               or
               taken
               from
               it
               ,
               the
               greater
               Segment
               of
               the
               Subtense
               of
               a
               Trient
               (
               cut
               in
               extream
               and
               mean
               Proportion
               ;
               )
               it
               gives
               ,
               in
               the
               first
               case
               ,
               the
               Subtense
               of
            
             96
             ,
             
               and
               of
            
             264
             
               Degrees
               ;
               in
               the
               latter
               ,
               that
               of
            
             24
             ,
             
               and
               of
            
             336
             Degrees
             .
          
           
             LXXX
             .
             And
             by
             these
             again
             (
             by
             subducting
             the
             Squares
             of
             their
             subtenses
             from
             4Rq
             )
             we
             have
             (
             the
             Squares
             of
             )
             the
             subtenses
             of
             their
             Difference
             from
             a
             Semicircumference
             ,
             whether
             in
             Excess
             or
             defect
             .
             As
             of
             84
             =
             180
             −
             96
             =
             264
             −
             180
             ,
             and
             of
             156
             =
             180
             −
             24
             =
             336
             −
             180.
             
          
           
             LXXXI
             .
             And
             if
             in
             like
             manner
             we
             compare
             also
             the
             rest
             of
             those
             at
             §
             70
             ,
             with
             the
             Subtense
             of
             a
             Trient
             ;
             we
             shall
             thence
             have
             the
             subtenses
             of
             these
             Arches
             ,
          
           
             
               Degrees
               of
               Arches
               .
            
             
               120
               −
               108
               =
               12
            
             
               120
               −
               90
               =
               30
            
             
               120
               −
               72
               =
               48
            
             
               120
               −
               54
               =
               66
            
             
               120
               −
               36
               =
               84
            
             
               120
               −
               18
               =
               102
            
             
               120±0
               =
               120
            
             
               120+18
               =
               138
            
             
               120+36
               =
               156
            
             
               120+54
               =
               174
            
          
           
             
               Of
               the
               Residue
               to
               the
               Semicircumference
               .
            
             
               168
            
             
               150
            
             
               132
            
             
               114
            
             
               96
            
             
               78
            
             
               60
            
             
               42
            
             
               24
            
             
               6
            
          
           
             
               Of
               the
               Residues
               to
               the
               whole
               Circumference
               .
            
             
               348.192
            
             
               330.210
            
             
               312.228
            
             
               294.246
            
             
               276.264
            
             
               258.282
            
             
               240.300
            
             
               222.318
            
             
               204.336
            
             
               186.354
            
          
           
           
             LXXXII
             .
             So
             that
             (
             by
             these
             here
             ,
             and
             those
             at
             §
             7●
             .
             )
             we
             have
             subtenses
             for
             
             every
             sixth
             Degree
             of
             the
             whole
             Circumference
             :
             And
             consequently
             the
             Right
             sines
             (
             as
             being
             the
             half
             of
             those
             subtenses
             )
             for
             every
             third
             Degree
             of
             the
             Semicircumference
             .
             And
             this
             by
             the
             Solution
             of
             Quadratick
             Equations
             only
             ,
             without
             the
             help
             of
             Cubicks
             or
             Superior
             Equations
             .
             And
             between
             these
             may
             in
             like
             manner
             ,
             be
             interposed
             as
             many
             more
             as
             we
             please
             ,
             by
             the
             continual
             Bisection
             of
             Arches
             .
          
           
             LXXXIII
             .
             We
             return
             now
             to
             pursue
             the
             former
             Inquisition
             which
             hath
             been
             
             intermitted
             .
             The
             Equation
             formerly
             proposed
             at
             §
             7
             ,
             for
             the
             Quinquisection
             of
             an
             Arch
             ,
             
             ;
             beside
             the
             two
             primary
             Roots
             A
             and
             E
             ,
             contains
             yet
             three
             other
             Roots
             ,
             (
             by
             §
             98
             ,
             99
             ,
             
               Chap.
               preced
            
             .
             )
             answering
             to
             three
             other
             Chords
             drawn
             (
             from
             the
             same
             Point
             with
             A
             and
             E
             )
             to
             three
             other
             Angles
             of
             the
             inscribed
             Pentagon
             :
             Which
             we
             shall
             call
             L
             ,
             M
             ,
             N
             :
             Whereof
             L
             subtends
             a
             Quintant
             increased
             by
             the
             Arch
             of
             A
             ,
             (
             or
             three
             Quintants
             increased
             by
             the
             Arch
             of
             E
             ;
             )
             N
             subtends
             a
             Quintant
             increased
             by
             the
             Arch
             of
             E
             ,
             (
             or
             three
             Quintants
             with
             the
             Arch
             of
             A
             ;
             )
             M
             subtends
             two
             Quintants
             increased
             by
             either
             of
             those
             Arches
             A
             or
             E.
             For
             every
             of
             these
             Arches
             ,
             if
             Quintupled
             ,
             will
             have
             the
             same
             Subtense
             (
             of
             the
             Quintuple
             )
             F
             ,
             as
             well
             as
             the
             Quintuple
             of
             the
             Arches
             A
             or
             E.
             
          
           
             LXXXIV
             .
             Of
             these
             three
             ,
             (
             in
             case
             A
             and
             E
             be
             supposed
             Affirmative
             Roots
             ,
             )
             L
             and
             N
             will
             be
             Negative
             ;
             but
             M
             ,
             Affirmative
             .
             But
             contrarywise
             ,
             in
             case
             A
             and
             E
             be
             supposed
             Negative
             :
             For
             then
             L
             ,
             N
             ,
             will
             be
             Affirmative
             ,
             and
             M
             Negative
             .
             For
             ,
          
           
             LXXXV
             .
             
               When
               the
               single
               Arch
               is
               less
               than
               a
               Quintant
               (
               or
               greater
               than
               four
            
             
             
               Quintants
               ;
               )
               or
               when
               it
               is
               greater
               than
               Two
               ,
               but
               less
               than
               Three
               ;
               the
               Subtense
               of
               the
               Triple
               Arch
               will
               be
               greater
               than
               that
               of
               the
               double
               .
            
             (
             As
             is
             easie
             to
             apprehend
             ,
             or
             may
             be
             proved
             if
             need
             be
             ,
             in
             like
             manner
             as
             we
             have
             formerly
             done
             in
             like
             cases
             ;
             as
             is
             after
             shewed
             at
             §
             93
             ,
             &c.
             )
          
           
             LXXXVI
             .
             And
             therefore
             ,
             if
             A
             (
             or
             E
             ,
             )
             be
             the
             Subtense
             of
             the
             single
             Arch
             ;
             then
             Cq
             −
             Bq
             =
             AF
             ,
             (
             or
             cq
             −
             bq
             =
             EF
             ,
             )
             will
             be
             an
             Affirmative
             quantity
             ,
             (
             as
             at
             §
             2.
             )
          
           
             LXXXVII
             .
             And
             ,
             in
             like
             manner
             ,
             if
             M
             be
             the
             Subtense
             of
             the
             single
             ,
             (
             greater
             
             than
             Two
             Quintants
             ,
             but
             less
             than
             Three
             ,
             )
             Cq
             −
             Bq
             =
             MF
             ,
             will
             be
             also
             Affirmative
             .
          
           
             LXXXVIII
             .
             
               But
               when
               the
               single
               Arch
               is
               greater
               than
               a
               Quintant
               ,
               but
               less
               than
               Two
               ,
               or
               greater
               than
               Three
               ,
               but
               less
               than
               Four
               :
               The
               Subtense
               of
               the
               double
               will
               be
               greater
               than
               that
               of
               the
               Triple
               .
            
          
           
             LXXXIX
             .
             And
             therefore
             ,
             if
             the
             Subtense
             of
             the
             single
             be
             N
             ,
             then
             Cq
             −
             Bq
             =
             NF
             ,
             
             will
             be
             Negative
             .
          
           
             XC
             .
             And
             in
             like
             manner
             ,
             if
             it
             be
             L
             ;
             then
             Cq
             −
             Bq
             =
             LF
             will
             be
             also
             
             Negative
             .
          
           
             XCI
             .
             That
             therefore
             ,
             L
             ,
             N
             ,
             may
             have
             Affirmative
             values
             (
             as
             well
             as
             F
             ,
             )
             we
             must
             put
             the
             Equations
             thus
             ,
             Bq
             −
             Cq
             =
             NF
             ;
             and
             Bq
             −
             Cq
             =
             LF
             .
             And
             ,
             if
             so
             ;
             then
             the
             value
             of
             the
             other
             three
             Roots
             will
             be
             Negative
             .
          
           
             XCII
             .
             
               But
               if
               the
               single
               Arch
               be
               just
               a
               Quintant
            
             (
             or
             two
             ,
             three
             or
             more
             Quintants
             )
             
             
               the
               Subtense
               of
               the
               double
               will
               be
               equal
               to
               that
               of
               the
               Triple
               .
            
             And
             therefore
             ,
             (
             putting
             V
             or
             X
             ,
             for
             the
             Subtense
             of
             the
             single
             ,
             )
             Cq
             ∼
             Bq
             =
             VF
             =
             o
             ;
             or
             Cq
             ∼
             Bq
             =
             XF
             =
             o.
             The
             Subtense
             of
             the
             Quintuple
             (
             in
             this
             case
             )
             vanishing
             to
             nothing
             .
          
           
           
             XCIII
             .
             Now
             that
             ,
             for
             the
             Arches
             A
             ,
             E
             ,
             M
             ,
             the
             Subtense
             of
             the
             Triple
             is
             
             greater
             (
             or
             at
             least
             not
             less
             )
             than
             that
             of
             the
             Duple
             ;
             but
             contrarywise
             for
             the
             Arches
             L
             ,
             N
             ;
             is
             easie
             to
             apprehend
             upon
             a
             little
             consideration
             .
             For
             if
             the
             Arch
             A
             ,
             or
             E
             ,
             be
             to
             Degrees
             ;
             B
             is
             20
             ;
             C
             ,
             30
             :
             If
             that
             be
             20
             ;
             B
             is
             40
             ;
             C
             ,
             60
             :
             If
             A
             be
             40
             ;
             B
             is
             80
             ;
             C
             ,
             120
             :
             If
             A
             be
             60
             ;
             B
             is
             120
             ;
             C
             ,
             180.
             
             (
             And
             hitherto
             is
             no
             doubt
             ,
             because
             we
             are
             not
             yet
             past
             a
             Semicircumference
             ;
             and
             ,
             till
             then
             ,
             as
             the
             Arches
             increase
             ,
             the
             Chords
             increase
             also
             ;
             though
             not
             when
             we
             are
             past
             180
             Degrees
             .
             )
             If
             A
             be
             ●0
             ;
             B
             is
             140
             =
             180
             −
             40
             ;
             C
             ,
             2●0
             =
             180
             +
             30.
             
             So
             that
             yet
             the
             Chord
             of
             C
             ,
             though
             past
             a
             Semicircumference
             ,
             is
             greater
             than
             that
             of
             B
             ,
             because
             nearer
             to
             a
             Semicircumference
             ,
             or
             180
             Degrees
             ;
             (
             for
             this
             doth
             less
             exceed
             it
             ,
             than
             that
             wants
             of
             it
             .
             )
             And
             so
             'till
             we
             come
             to
             72
             Degrees
             ,
             (
             or
             1
             /
             ●
             of
             the
             whole
             )
             for
             then
             B
             is
             144
             =
             180
             −
             36
             ,
             and
             C
             ,
             216
             =
             180
             +
             36
             ;
             where
             the
             distance
             is
             equal
             ,
             and
             accordingly
             the
             Chord
             of
             the
             Triple
             equal
             to
             that
             of
             the
             double
             .
             But
             when
             we
             be
             past
             a
             Quintant
             ,
             that
             of
             the
             Triple
             becomes
             less
             ;
             for
             if
             the
             single
             Arch
             N
             =
             ●
             /
             5
             +
             E
             be
             73
             ;
             B
             is
             146
             =
             180
             −
             34
             ;
             C
             ,
             219
             =
             180
             +
             39
             ;
             and
             this
             doth
             therefore
             more
             exceed
             180
             ,
             than
             the
             other
             comes
             short
             of
             it
             ;
             and
             hath
             therefore
             the
             shorter
             Chord
             .
             So
             likewise
             ,
             if
             N
             be
             80
             ;
             B
             is
             160
             =
             180
             −
             20
             ;
             C
             ,
             240
             =
             180
             +
             60
             :
             If
             N
             be
             90
             ,
             B
             is
             180
             ;
             C
             ,
             270
             =
             180
             +
             90
             :
             If
             N
             be
             100
             ;
             B
             is
             200
             =
             180
             +
             20
             =
             360
             −
             160
             ;
             C
             ,
             300
             =
             180
             +
             120
             =
             360
             −
             60
             :
             Where
             the
             Triple
             is
             farther
             from
             a
             Semicircumference
             ,
             as
             more
             exceeding
             it
             ;
             and
             nearer
             to
             a
             whole
             Revolution
             (
             which
             is
             Equivalent
             to
             nothing
             )
             as
             approaching
             nearer
             to
             it
             ;
             and
             therefore
             the
             Chord
             of
             the
             Triple
             ,
             less
             than
             that
             of
             the
             double
             :
             So
             ,
             if
             N
             ,
             or
             L
             be
             108
             ;
             B
             is
             216
             =
             180
             +
             36
             =
             360
             −
             144
             ;
             C
             ,
             324
             =
             360
             −
             36
             =
             180
             +
             144.
             
             If
             L
             =
             ⅕
             +
             A
             be
             120
             ;
             B
             is
             243
             =
             180
             +
             60
             =
             360
             −
             120
             ;
             C
             ,
             360.
             
             And
             therefore
             that
             B
             ,
             the
             greater
             Chord
             :
             And
             so
             it
             will
             be
             'till
             we
             come
             to
             144
             Degrees
             (
             or
             ⅖
             )
             when
             again
             they
             will
             become
             equal
             ;
             for
             then
             B
             will
             be
             288
             =
             180
             +
             108
             =
             360
             −
             72
             ;
             and
             C
             =
             432
             =
             360
             +
             72
             =
             540
             −
             108
             ;
             which
             doth
             as
             much
             surpass
             a
             whole
             Revolution
             as
             the
             other
             wants
             of
             it
             ;
             and
             doth
             as
             much
             want
             of
             a
             third
             Semicircumference
             as
             the
             other
             exceeds
             the
             first
             ;
             and
             therefore
             their
             Chords
             become
             equal
             .
             But
             after
             this
             ,
             the
             Chord
             of
             the
             Triple
             doth
             again
             become
             the
             greater
             :
             For
             if
             M
             the
             single
             Arch
             be
             145
             ;
             B
             will
             be
             290
             =
             180
             +
             110
             =
             360
             −
             70
             ;
             C
             ,
             435
             =
             360
             +
             75
             =
             540
             −
             105
             :
             If
             M
             be
             150
             ;
             B
             is
             300
             =
             360
             −
             60
             ;
             C
             ,
             450
             =
             360
             +
             90
             :
             If
             M
             be
             180
             ;
             B
             is
             360
             ;
             C
             ,
             is
             540
             =
             360
             +
             180
             :
             If
             M
             be
             200
             ;
             B
             is
             400
             =
             360
             +
             40
             ;
             C
             ,
             600
             =
             360
             +
             240
             ;
             where
             the
             Arch
             C
             (
             as
             farther
             remote
             from
             an
             intire
             Revolution
             )
             requires
             the
             greater
             Chord
             .
             And
             so
             onward
             'till
             we
             come
             to
             216
             ,
             (
             or
             ⅗
             )
             where
             the
             Chords
             of
             B
             and
             C
             do
             again
             become
             equal
             ,
             for
             B
             will
             be
             432
             =
             360
             +
             72
             ;
             C
             ,
             648
             =
             720
             −
             72
             ;
             where
             the
             Arch
             B
             doth
             as
             much
             exceed
             one
             Revolution
             ,
             as
             C
             wants
             of
             two
             ;
             and
             therefore
             require
             equal
             Chords
             .
             After
             this
             ,
             the
             Arches
             L
             ,
             N
             ,
             from
             216
             to
             288
             ,
             have
             the
             same
             Chords
             with
             those
             of
             L
             ,
             N
             ,
             from
             144
             backward
             to
             72
             ,
             (
             as
             being
             their
             Complements
             to
             a
             whole
             Revolution
             ,
             )
             and
             the
             same
             Chords
             of
             their
             Doubles
             and
             Triples
             ,
             with
             the
             Doubles
             and
             Triples
             of
             those
             ;
             and
             therefore
             (
             as
             there
             )
             the
             Chords
             of
             the
             Duple
             greater
             than
             those
             of
             the
             Triple
             .
             And
             from
             thence
             to
             360
             (
             which
             is
             an
             entire
             Revolution
             )
             the
             Chords
             are
             the
             same
             with
             those
             of
             A
             and
             E
             ,
             (
             as
             being
             the
             Remainders
             of
             these
             to
             an
             intire
             Revolution
             )
             and
             therefore
             here
             also
             ,
             the
             Chord
             of
             the
             Triple
             is
             greater
             than
             that
             of
             the
             Duple
             .
          
           
             XCIV
             .
             All
             which
             depends
             on
             this
             General
             Consideration
             ;
             (
             which
             equally
             serves
             for
             all
             such
             Comparings
             of
             Arches
             and
             their
             Subtenses
             ;
             and
             is
             therefore
             to
             be
             taken
             notice
             of
             ,
             once
             for
             all
             .
             )
             That
             is
             ,
          
           
             XCV
             .
             
               Arches
               equally
               distant
               from
               the
               beginning
               or
               end
               of
            
             (
             one
             or
             more
             )
             
               entire
               Revolutions
               ,
               have
               equal
               Subtenses
               ,
            
             (
             for
             the
             same
             Chord
             doth
             indifferently
             subtend
             both
             or
             all
             of
             them
             ;
             )
             
               But
               those
               which
               are
               less
               distant
               from
               such
               beginning
               or
               end
               ,
               have
               the
               lesser
               subtenses
            
             ;
             (
             as
             nearest
             approaching
             to
             nothing
             .
             )
          
           
           
             XCVI
             .
             Again
             ,
             
               Arches
               equally
               distant
            
             (
             whether
             in
             Excess
             or
             defect
             )
             from
             
             1
             ,
             2
             ,
             3
             ,
             (
             or
             any
             odd
             number
             of
             )
             
               Semicircumferences
               ,
               have
               equal
               subtenses
            
             ,
             (
             for
             here
             also
             the
             same
             Chord
             subtends
             both
             or
             all
             ;
             )
             
               but
               those
               which
               are
               less
               distant
               from
               such
               Semicircumferences
               ,
               have
               the
               greater
               Subtense
               ,
            
             (
             as
             nearest
             approaching
             to
             that
             of
             a
             Semicircumference
             ,
             or
             180
             Degrees
             ,
             the
             greatest
             Chord
             of
             all
             .
             )
          
           
             XCVII
             .
             'T
             is
             manifest
             therefore
             ,
             that
             ,
             if
             the
             Arch
             E
             or
             A
             be
             not
             greater
             than
             60
             Degrees
             ,
             and
             consequently
             the
             Triple
             Arch
             do
             not
             exceed
             one
             Semicircumference
             ,
             That
             of
             the
             Treble
             (
             as
             nearest
             approaching
             to
             it
             )
             will
             be
             greater
             than
             that
             of
             the
             Double
             .
             And
             though
             A
             be
             greater
             than
             60
             Degrees
             ;
             that
             of
             the
             Triple
             will
             yet
             be
             the
             greater
             ,
             'till
             this
             do
             as
             much
             exceed
             a
             Semicircumference
             as
             the
             Double
             comes
             short
             of
             it
             :
             That
             is
             ,
             'till
             2
             ½
             A
             =
             180
             Deg.
             or
             ½
             of
             the
             whole
             Circumference
             ;
             that
             is
             ,
             'till
             A
             =
             ⅕
             ,
             or
             72
             Degrees
             .
             And
             what
             is
             said
             of
             E
             and
             A
             less
             than
             ●
             /
             5
             ,
             doth
             equally
             hold
             of
             ⅘
             +
             A
             =
             1
             −
             E
             ,
             and
             ⅘
             +
             E
             =
             1
             −
             A
             ,
             which
             have
             the
             same
             Chords
             with
             E
             and
             A
             ;
             and
             their
             Double
             and
             Treble
             ,
             the
             same
             with
             the
             Double
             and
             Treble
             of
             E
             and
             A.
             
          
           
             XCVIII
             .
             But
             if
             N
             ,
             or
             L
             ,
             the
             single
             Arch
             exceed
             ⅕
             ,
             suppose
             ⅕
             +
             E
             or
             ⅕
             +
             A
             ;
             the
             Subtense
             of
             the
             double
             will
             be
             the
             longer
             .
             For
             the
             Subtense
             of
             ●
             /
             5
             being
             the
             same
             with
             that
             of
             ⅗
             =
             1
             −
             ⅖
             ;
             that
             of
             2
             N
             =
             ⅖
             +
             2
             E
             will
             be
             longer
             than
             it
             ,
             as
             nearer
             approaching
             to
             ½
             ;
             (
             'till
             2
             N
             or
             2
             L
             =
             ½
             ,
             that
             is
             ,
             N
             or
             L
             =
             ¼
             or
             90
             Degrees
             ;
             )
             but
             that
             of
             3
             N
             =
             ●
             /
             5
             +
             3
             E
             less
             than
             it
             ,
             as
             nearer
             approaching
             to
             1
             intire
             Revolution
             .
             And
             even
             when
             2
             L
             exceeds
             ½
             ,
             yet
             3
             L
             will
             have
             the
             less
             Chord
             ,
             as
             nearer
             approaching
             to
             1
             intire
             Revolution
             ;
             'till
             it
             become
             equal
             to
             it
             ;
             that
             is
             ,
             3
             L
             =
             1
             ,
             and
             L
             =
             ⅓
             ,
             or
             120
             Deg.
             And
             even
             after
             this
             ,
             'till
             3
             L
             do
             as
             much
             exceed
             1
             ,
             as
             2
             L
             comes
             short
             of
             it
             ;
             that
             is
             ,
             'till
             2
             ½
             L
             =
             1
             ;
             or
             L
             =
             ⅖
             or
             144
             Degrees
             .
             But
             then
             (
             as
             before
             at
             A
             or
             N
             =
             ⅕
             )
             the
             Chords
             will
             be
             equal
             ;
             for
             then
             the
             double
             is
             ⅘
             =
             1
             −
             ●
             /
             5
             ;
             the
             Treble
             6
             /
             5
             =
             1
             +
             ⅕
             .
             And
             what
             is
             said
             of
             N
             =
             ⅕
             +
             E
             ,
             or
             L
             =
             ●
             /
             5
             +
             A
             ;
             holds
             equally
             true
             of
             N
             =
             ●
             /
             5
             +
             A
             ,
             or
             L
             =
             ●
             /
             5
             +
             E
             ;
             (
             that
             is
             of
             1
             −
             N
             ,
             or
             1
             −
             L
             ;
             )
             as
             having
             the
             same
             Chords
             with
             those
             .
          
           
             XCIX
             .
             But
             when
             M
             the
             single
             Arch
             exceeds
             ●
             /
             5
             ,
             suppose
             ⅖
             +
             E
             ;
             the
             Chord
             of
             the
             Treble
             will
             again
             be
             longer
             than
             that
             of
             the
             Double
             .
             For
             the
             Treble
             of
             ⅖
             as
             much
             exceeding
             ,
             as
             the
             Double
             of
             it
             comes
             short
             of
             ,
             1
             Revolution
             ;
             the
             Treble
             of
             ⅖
             +
             E
             will
             more
             exceed
             it
             ,
             (
             approaching
             nearer
             to
             the
             third
             Semicircumference
             )
             and
             the
             Double
             want
             less
             of
             it
             ,
             (
             approaching
             nearer
             to
             1
             Revolution
             ,
             )
             'till
             3
             M
             =
             ●
             /
             2
             ;
             that
             is
             ,
             M
             =
             ½
             or
             180
             Degrees
             .
             And
             what
             is
             said
             of
             M
             =
             ⅖
             +
             E
             less
             than
             ½
             ;
             holds
             also
             of
             M
             =
             ⅖
             +
             A
             =
             ●
             /
             5
             −
             E
             :
             Which
             doth
             as
             much
             exceed
             a
             Semicircumference
             ,
             as
             the
             other
             comes
             short
             of
             it
             .
          
           
             C.
             'T
             is
             manifest
             therefore
             ,
             that
             for
             the
             Arches
             A
             ,
             E
             ,
             less
             than
             ⅕
             ,
             or
             more
             than
             ⅘
             (
             but
             less
             than
             1
             Revolution
             ;
             )
             and
             again
             for
             the
             Arch
             M
             ,
             more
             than
             ⅖
             but
             less
             than
             ●
             /
             5
             ;
             the
             Chord
             of
             the
             Triple
             is
             greater
             than
             that
             of
             the
             Double
             ;
             by
             §
             97
             ,
             99.
             
             But
             ,
             for
             the
             Arches
             L
             or
             M
             ,
             more
             than
             ⅕
             but
             less
             than
             ⅖
             ;
             or
             more
             than
             ⅕
             ,
             but
             less
             than
             ⅘
             ;
             the
             Chord
             of
             the
             Double
             is
             greater
             than
             that
             of
             the
             Treble
             ;
             by
             §
             98.
             
             But
             in
             case
             the
             single
             Arch
             be
             ⅕
             ,
             ⅖
             ,
             ●
             /
             5
             ,
             ⅘
             ;
             (
             or
             any
             number
             of
             Quintants
             ,
             )
             the
             Chords
             of
             the
             Double
             and
             Treble
             are
             equal
             .
             And
             the
             same
             method
             may
             be
             pursued
             in
             other
             like
             Comparisons
             of
             Arches
             and
             Chords
             .
          
           
             CI.
             Now
             ,
             (
             to
             return
             where
             we
             left
             off
             at
             §
             92.
             )
             what
             hath
             been
             particularly
             delivered
             ,
             may
             be
             Collected
             into
             this
             General
             .
             Namely
             ,
             (
             putting
             O
             for
             the
             Subtense
             of
             the
             single
             Arch
             )
             C
             q
             ∼
             B
             q
             =
             O
             F
             (
             by
             §
             85
             ,
             88
             ,
             92.
             )
             And
             
             .
             And
             
             .
             That
             is
             ,
          
           
           
             CII
             .
             
               The
               Difference
               of
               the
               Squares
               of
               the
               subtenses
               of
               the
               Triple
               and
               double
               Arches
               ,
            
             
             
               is
               equal
               to
               the
               Rect-angle
               of
               the
               subtenses
               of
               the
               single
               and
               Quintuple
               .
               And
               that
               Difference
               applied
               to
               either
               of
               these
               ,
               gives
               the
               other
               .
            
             (
             Which
             is
             a
             General
             to
             that
             of
             §
             3.
             )
             Namely
             ,
             if
             O
             be
             interpreted
             of
             A
             ,
             E
             ,
             M
             ;
             then
             C
             q
             −
             B
             q
             =
             O
             F
             ,
             and
             
             .
             If
             ,
             of
             L
             ,
             N
             ;
             then
             B
             q
             −
             C
             q
             =
             O
             F
             ,
             and
             
             .
             If
             ,
             of
             V
             ,
             X
             ;
             then
             B
             q
             ∼
             C
             q
             =
             V
             F
             =
             0.
             And
             
             .
             Or
             ,
             B
             q
             ∼
             C
             q
             =
             X
             F
             =
             0
             ,
             and
             
             .
          
           
             CIII
             .
             Again
             ,
             because
             C
             q
             ∼
             B
             q
             =
             C
             +
             B
             into
             C
             ∼
             B
             ;
             therefore
             ,
             O.
             B
             +
             C
             :
             :
             B
             ∼
             C.
             (
             interpreting
             C
             ∼
             B
             ,
             of
             C
             −
             B
             ,
             for
             A
             ,
             E
             ,
             M
             ;
             but
             of
             B
             −
             C
             ,
             for
             L
             ,
             N.
             )
             That
             is
             ,
          
           
             CIV
             .
             
               As
               the
               Subtense
               of
               the
               single
               Arch
               ,
               to
               the
               Aggregate
               of
               the
               subtenses
               of
               the
               Double
               and
               Triple
               ;
               so
               is
               the
               Difference
               of
               these
               ,
               to
               that
               of
               the
               Quintuple
               .
            
             (
             Which
             is
             a
             General
             to
             that
             of
             §
             6.
             )
          
           
             CV
             .
             But
             (
             by
             §
             45
             ,
             Chap.
             30.
             )
             
             ;
             and
             therefore
             ,
             
             :
             And
             (
             by
             §
             7
             ,
             Chap.
             29.
             )
             
             ;
             and
             therefore
             
             :
             From
             hence
             therefore
             ,
             we
             may
             have
             the
             value
             of
             C
             q
             ∼
             B
             q
             =
             O
             F
             ,
             and
             of
             
             ,
             sutable
             to
             each
             case
             .
             Namely
             ,
          
           
             CVI.
             If
             the
             Arch
             O
             be
             less
             than
             ⅕
             ,
             or
             more
             than
             ⅘
             ;
             (
             that
             is
             ,
             from
             0
             ,
             to
             72°
             and
             from
             288
             ,
             to
             360.
             )
             Or
             more
             than
             ⅖
             but
             less
             than
             ⅗
             ,
             (
             that
             is
             ,
             from
             144
             ,
             to
             216°
             )
             there
             is
             
             :
             And
             
             .
             And
             O
             to
             be
             understood
             of
             A
             ,
             E
             ,
             and
             M.
             
          
           
             CVII
             .
             But
             if
             the
             Arch
             O
             be
             more
             than
             ⅕
             but
             less
             than
             ⅖
             ;
             or
             more
             than
             ⅗
             but
             less
             than
             ⅘
             ;
             (
             that
             is
             ,
             from
             72
             Degrees
             to
             144
             ,
             and
             from
             216
             to
             288
             :
             )
             Then
             is
             ,
             
             :
             And
             
             .
             And
             O
             to
             be
             interpreted
             of
             L
             ,
             N.
             
          
           
             CVIII
             .
             That
             is
             ,
             (
             to
             reduce
             all
             to
             a
             brief
             Synopsis
             )
             
             And
             ,
             in
             the
             common
             term
             (
             or
             Point
             of
             connexion
             )
             of
             these
             Intervals
             ,
             it
             is
             indifferent
             to
             whether
             of
             the
             two
             to
             refer
             them
             :
             As
             at
             36
             Degrees
             ,
             to
             E
             or
             A
             ;
             at
             72°
             to
             A
             or
             N
             ;
             and
             so
             of
             the
             rest
             .
          
           
           
             CIX
             .
             Hence
             follows
             this
             Five-fold
             Equation
             ;
             containing
             five
             Roots
             .
             5
             R
             q
             q
             E
             
             −
             5
             R
             q
             E
             c
             +
             E
             q
             c
             =
             5
             R
             q
             q
             A
             −
             5
             R
             q
             A
             c
             +
             A
             q
             c
             =
             (
             R
             q
             q
             F
             =
             )
             −
             5
             R
             q
             q
             N
             +
             5
             R
             q
             N
             c
             −
             N
             q
             c
             =
             −
             5
             R
             q
             q
             L
             +
             5
             R
             q
             L
             c
             −
             L
             q
             c
             =
             +
             5
             R
             q
             q
             M
             −
             5
             R
             q
             M
             c
             +
             M
             q
             c.
             
          
           
             CX
             .
             Now
             because
             5
             R
             q
             q
             A
             −
             5
             R
             q
             A
             c
             +
             A
             q
             c
             =
             5
             R
             q
             q
             E
             −
             5
             R
             q
             E
             c
             +
             E
             q
             c
             ;
             therefore
             ,
             (
             by
             transposition
             )
             5
             R
             q
             q
             A
             −
             5
             R
             q
             q
             E
             =
             5
             R
             q
             A
             c
             −
             5
             R
             q
             E
             c
             −
             A
             q
             c
             +
             E
             q
             c
             :
             And
             (
             dividing
             all
             by
             A
             −
             E
             ,
             )
             
             .
          
           
             CXI
             .
             But
             (
             by
             §
             13
             ,
             14
             ,
             15
             ,
             Chap.
             30.
             )
             
             ,
             and
             therefore
             ,
             
             :
             Therefore
             ,
             
             :
             That
             is
             ,
             
             .
          
           
             CXII
             .
             And
             again
             ,
             because
             (
             as
             will
             appear
             by
             Division
             )
             
             :
             Therefore
             ,
             A
             q
             q
             +
             A
             c
             E
             +
             A
             q
             E
             q
             +
             A
             E
             c
             +
             E
             q
             q
             =
             10
             R
             q
             q.
             
          
           
             CXIII
             .
             But
             the
             Angle
             contained
             by
             A
             ,
             E
             ,
             is
             of
             144
             Degrees
             .
             (
             As
             being
             an
             Angle
             in
             the
             Circumference
             insisting
             on
             an
             Arch
             of
             288
             Degrees
             ,
             or
             ⅘
             of
             the
             whole
             .
             )
             Therefore
             ,
          
           
             CXIV
             .
             
               The
               Difference
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             144
             
               Degrees
               ,
               or
               divided
               by
               the
               Difference
               of
               those
               Legs
               ,
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circles
               .
            
             (
             by
             §
             111.
             )
             That
             is
             ,
             (
             by
             §
             112.
             )
          
           
             CXV
             .
             
               The
               Biquadrates
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             144
             
               Degrees
               ,
               together
               with
               the
               three
               means
               proportional
               between
               these
               Biquadrates
               ,
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               .
            
          
           
             CXVI
             .
             But
             now
             the
             Base
             of
             this
             Triangle
             ,
             being
             the
             side
             of
             an
             Inscribed
             Equilater
             Pentagon
             ,
             or
             Subtense
             of
             72
             Degrees
             ;
             is
             ,
             (
             by
             §
             31.
             )
             
             ;
             and
             therefore
             the
             Square
             of
             this
             
             ;
             and
             ,
             its
             Biquadrate
             ,
             
             :
             Which
             is
             ,
             to
             10
             R
             q
             q
             ,
             as
             
             to
             10
             ;
             or
             as
             
             .
             Therefore
             ,
          
           
             CXVII
             .
             
               The
               Difference
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             144
             
               Degrees
               ,
               divided
               by
               the
               Difference
               of
               those
               Legs
               ,
               or
               the
               Biquadrates
               of
               the
               Legs
               containing
               such
               Angle
               ,
               together
               with
               the
               three
               means
               Proportional
               between
               these
               Biquadrates
               ;
               is
               ,
               to
               the
               Biquadrate
               of
               the
               Base
               subtending
               that
               Angle
               ;
               as
            
             
             .
          
           
             CXVIII
             .
             Again
             ,
             because
             (
             by
             §
             109.
             )
             5
             R
             q
             q
             M
             −
             5
             R
             q
             M
             c
             +
             M
             q
             c
             =
             (
             R
             q
             q
             F
             =
             )
             5
             R
             q
             q
             A
             −
             5
             R
             q
             A
             c
             +
             A
             q
             c
             (
             M
             being
             greater
             than
             A
             ;
             )
             therefore
             (
             as
             at
             §
             110
             ,
             111
             ,
             112
             ,
             )
             
             .
             And
             
             :
          
           
           
             CXIX
             .
             And
             (
             by
             the
             same
             reason
             )
             
             .
             
             And
             
             .
          
           
             CXX
             .
             And
             also
             ,
             because
             (
             by
             §
             109.
             )
             −
             5RqqL
             +
             5RqLc
             −
             Lqq
             =
             (
             RqqF
             =
             )
             −
             5RqqN
             +
             5RqNc
             −
             Nqc
             ;
             and
             (
             changing
             all
             the
             signs
             )
             5RqqL
             −
             5RqLc
             +
             Lqc
             =
             5RqqN
             −
             5RqNc
             +
             Nqc
             (
             L
             being
             greater
             than
             N
             :
             )
             Therefore
             ,
             (
             as
             at
             §
             118.
             )
             
             .
             And
             
             .
          
           
             CXXI
             .
             But
             the
             Angles
             contained
             by
             M
             ,
             A
             ;
             and
             M
             ,
             E
             ;
             and
             L
             ,
             N
             ;
             are
             of
             72
             Degrees
             ;
             (
             as
             being
             Angles
             in
             the
             Circumference
             insisting
             on
             an
             Arch
             of
             144
             Degrees
             ,
             or
             ⅖
             of
             the
             whole
             :
             )
             And
             ,
             as
             to
             M
             ,
             E
             ;
             one
             of
             the
             Angles
             at
             the
             Base
             ,
             obtuse
             ;
             but
             ,
             as
             to
             M
             ,
             A
             ;
             all
             acute
             :
             (
             This
             being
             an
             Angle
             in
             a
             greater
             Segment
             ;
             that
             ,
             in
             a
             less
             ,
             than
             a
             Semicircle
             ;
             )
             and
             likewise
             ,
             as
             to
             L
             ,
             N
             ,
             all
             acute
             .
             Therefore
             ,
             (
             as
             at
             §
             114
             ,
             115.
             )
          
           
             CXXII
             .
             
               The
               Difference
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             72
             
               Degrees
               ,
               divided
               by
               the
               Difference
               of
               those
               Legs
               ;
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               .
            
             And
             ,
          
           
             CXXIII
             .
             
               The
               Biquadrates
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             72
             
               Degrees
               ,
               together
               with
               the
               three
               means
               Proportional
               between
               those
               Biquadrates
               ,
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               .
            
          
           
             CXXIV
             .
             But
             here
             the
             Base
             of
             this
             Triangle
             (
             subtended
             to
             that
             Angle
             of
             72
             Degrees
             ,
             )
             is
             the
             Subtense
             of
             a
             Biquintant
             ,
             or
             Triquintant
             ;
             that
             is
             ,
             of
             ⅖
             =
             144
             Degrees
             ;
             or
             of
             ⅗
             =
             216
             Degrees
             ;
             which
             is
             (
             by
             §
             54.
             )
             
             .
             And
             the
             Square
             of
             this
             
             And
             its
             Biquadrate
             
             .
             Which
             is
             ,
             to
             10Rqq
             ,
             as
             
             to
             10
             ;
             or
             ,
             as
             
             to
             4.
             
             Therefore
             ,
          
           
             CXXV
             .
             
               The
               Difference
               of
               the
               Quadricubes
               of
               the
               Legs
               contanining
               an
               Angle
               of
            
             72
             
               Degrees
               divided
               by
               the
               Difference
               of
               those
               Legs
            
             ;
             or
             ,
             
               the
               Biquadrates
               of
               the
               Legs
               containing
               such
               Angle
               ,
               together
               with
               the
               three
               means
               Proportional
               between
               those
               Biquadrates
               ;
               is
               ,
               to
               the
               Biquadrate
               of
               the
               Base
               subtending
               that
               Angle
               ;
               as
            
             4
             to
             
             .
          
           
             CXXVI
             .
             Again
             ,
             because
             (
             by
             §
             109
             ,
             )
             5RqqA
             −
             5RqAc
             +
             Aqc
             =
             (
             RqqF
             =
             )
             −
             5RqqL
             +
             5RqLc
             −
             Lqc
             (
             L
             being
             greater
             than
             A
             :
             )
             Therefore
             ,
             (
             by
             transposition
             )
             5RqqL
             +
             5RqqA
             =
             5RqLc
             +
             5RqAc
             −
             Lqc
             −
             Aqc.
             And
             (
             dividing
             all
             by
             L
             +
             A
             ,
             )
             
             .
          
           
             CXXVII
             .
             But
             (
             by
             §
             46
             ,
             47
             ,
             Chap.
             30.
             )
             
             ,
             and
             therefore
             ,
             
             Therefore
             ,
             
             .
             That
             is
             ,
             
             .
          
           
           
             CXXVIII
             .
             And
             again
             ,
             because
             (
             as
             will
             appear
             by
             Division
             )
             
             :
             
             Therefore
             ,
             Lqq
             −
             LcA
             +
             LqAq
             −
             LAc
             +
             Aqq
             =
             10Rqq
             .
          
           
             CXXIX
             .
             But
             the
             Angle
             contained
             by
             L
             ,
             A
             ,
             is
             of
             36
             Degrees
             (
             as
             being
             an
             Angle
             at
             the
             Circumfererence
             insisting
             on
             an
             Arch
             of
             72
             Degrees
             ,
             or
             ⅗
             of
             the
             whole
             ,
             )
             and
             one
             of
             the
             other
             ,
             obtuse
             .
          
           
             CXXX
             .
             And
             the
             same
             is
             to
             be
             said
             (
             for
             the
             same
             reasons
             )
             of
             N
             ,
             E
             ,
             as
             of
             L
             ,
             A.
             
          
           
             CXXXI
             .
             And
             also
             ,
             because
             in
             like
             manner
             (
             by
             §
             109
             ,
             )
             5RqqM
             −
             5RqMc
             +
             Mqc
             =
             (
             RqqF
             =
             )
             −
             5RqqN
             +
             5RqNc
             −
             Nqc
             ;
             (
             M
             being
             greater
             than
             N
             :
             )
             Therefore
             ,
             (
             by
             the
             same
             methods
             ,
             )
             
             .
             And
             the
             Angle
             contained
             by
             M
             ,
             N
             ,
             is
             of
             36
             Degrees
             ;
             and
             one
             of
             the
             other
             ,
             obtuse
             .
          
           
             CXXXII
             .
             And
             just
             the
             same
             (
             for
             the
             same
             reasons
             )
             of
             M
             ,
             L
             ;
             save
             that
             here
             the
             Angles
             be
             all
             acute
             .
          
           
             CXXXIII
             .
             And
             these
             are
             all
             the
             cases
             that
             can
             happen
             ,
             the
             Angle
             at
             the
             Vertex
             being
             36
             Degrees
             ;
             for
             that
             of
             the
             Legs
             V
             ,
             X
             ;
             is
             to
             be
             reduced
             to
             that
             of
             A
             ,
             L
             ;
             and
             that
             of
             X
             ,
             X
             ;
             to
             that
             of
             L
             ,
             M
             ;
             (
             and
             the
             like
             is
             to
             be
             understood
             of
             other
             like
             cases
             ,
             where
             A
             is
             extended
             to
             the
             whole
             Quintant
             ,
             and
             E
             vanisheth
             into
             nothing
             .
             )
             Therefore
             ,
          
           
             CXXXIV
             .
             
               The
               Sum
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             36
             
               Degrees
               ,
               divided
               by
               the
               Sum
               of
               those
               Legs
               ,
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               .
            
             (
             By
             §
             127
             ,
             130
             ,
             131
             ,
             132.
             )
             And
             ,
          
           
             CXXXV
             .
             
               The
               Biquadrates
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             36
             
               Degrees
               ,
               with
               a
               mean
               Proportional
               between
               those
               Biquadrates
               ,
               wanting
               the
               first
               and
               third
               of
               three
               means
               Proportional
               between
               them
               ;
               are
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               .
            
          
           
             CXXXVI
             .
             But
             the
             Base
             subtended
             to
             this
             Angle
             of
             36
             Degrees
             ,
             being
             the
             side
             of
             an
             inscribed
             Equilater
             Pentagon
             ;
             (
             as
             at
             §
             116
             ,
             )
             the
             Biquadrate
             hereof
             is
             to
             10Rqq
             as
             
             to
             4.
             
             And
             therefore
             ,
          
           
             CXXXVII
             .
             
               The
               Sum
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             36
             
               Degrees
               ,
               divided
               by
               the
               Sum
               of
               those
               Legs
               :
            
             Or
             ,
             
               the
               Biquadrates
               of
               the
               Legs
               containing
               such
               Angle
               ,
               with
               a
               mean
               Proportional
               between
               those
               Biquadrates
               ,
               wanting
               the
               first
               and
               third
               of
               three
               mean
               Proportionals
               between
               them
               ;
               is
               ,
               to
               the
               Biquadrate
               of
               the
               Base
               subtending
               that
               Angle
               ;
               as
            
             4
             ,
             to
             
             .
          
           
             CXXXVIII
             .
             Again
             ,
             because
             (
             by
             §
             109.
             )
             5RqqA
             −
             5RqAc
             +
             Aqc
             =
             (
             RqqF
             =
             )
             −
             5RqqN
             +
             5RqNc
             −
             Nqc
             ;
             (
             N
             being
             greater
             than
             A
             ;
             )
             Therefore
             ,
             (
             as
             at
             §
             126
             ,
             &c.
             )
             
             .
          
           
             CXXXIX
             .
             And
             ,
             in
             like
             manner
             ,
             because
             5RqqE
             −
             5RqEc
             +
             Eqc
             =
             (
             RqqF
             =
             )
             −
             5RqqL
             +
             5RqLc
             −
             Lqc
             ,
             (
             L
             being
             greater
             than
             E
             :
             )
             Therefore
             ,
             
             .
          
           
           
             CXL
             .
             But
             the
             Angles
             contained
             by
             N
             ,
             A
             ;
             or
             by
             L
             E
             ;
             are
             Angles
             of
             108
             
             Degrees
             ,
             (
             as
             being
             Angles
             at
             the
             Circumference
             ,
             insisting
             on
             an
             Arch
             of
             216
             Degrees
             ,
             or
             ⅗
             of
             the
             whole
             .
             )
             Therefore
             ,
          
           
             CXLI
             .
             
               The
               Sum
               of
               the
               Quadricubes
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             108
             
               Degrees
               ,
               divided
               by
               the
               Sum
               of
               those
               Legs
               ,
               is
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               a
               Circumscribed
               Circle
               .
            
             And
             ,
          
           
             CXLII
             .
             
               The
               Biquadrates
               of
               the
               Legs
               containing
               an
               Angle
               of
            
             108
             
               Degrees
               ,
               with
               a
               mean
               Proportional
               between
               those
               Biquadrates
               ,
               wanting
               the
               first
               and
               third
               of
               three
               means
               Proportional
               between
               them
               ;
               are
               equal
               to
               Ten
               Biquadrates
               of
               the
               Radius
               of
               a
               Circumscribed
               Circle
               .
            
          
           
             CXLIII
             .
             But
             the
             Base
             subtended
             to
             this
             Angle
             of
             108
             Degrees
             ,
             is
             the
             Subtense
             of
             a
             Biquintant
             ,
             or
             (
             which
             is
             the
             same
             )
             of
             a
             Triquintant
             ;
             that
             is
             ,
             of
             ⅖
             or
             ⅗
             of
             the
             whole
             Circumference
             :
             And
             therefore
             ,
             (
             as
             at
             §
             124
             )
             is
             to
             10Rqq
             ,
             as
             
             to
             4.
             
             Therefore
             ,
          
           
             CXLIV
             .
             
               The
               Sum
               of
               the
               Quadricubes
               of
               Legs
               containing
               an
               Angle
               of
            
             108
             
               Degrees
               ,
               divided
               by
               the
               Sum
               of
               those
               Legs
               :
            
             Or
             ,
             
               The
               Biquadrates
               of
               the
               Legs
               containing
               such
               Angle
               ,
               with
               a
               mean
               Proportional
               between
               those
               Biquadrates
               ,
               wanting
               the
               first
               and
               third
               of
               three
               means
               Proportional
               between
               them
               ;
               is
               ,
               to
               the
               Biquadrate
               of
               the
               Base
               subtending
               that
               Angle
               ;
               as
            
             4
             to
             
             .
          
           
             CXLV
             .
             Now
             these
             several
             Theorems
             thus
             delivered
             in
             particular
             ,
             may
             be
             Collected
             into
             these
             Generals
             following
             .
             Namely
             ,
          
           
             CXLVI
             .
             
               The
               Difference
               of
               the
               Quadricubes
               of
               Legs
               containing
               an
               Angle
               of
            
             144
             
               or
               of
            
             72
             
               Degrees
               ,
               divided
               by
               the
               Differences
               of
               those
               Legs
               :
            
             Or
             ,
             
               The
               Sum
               of
               the
               Quadricubes
               of
               Legs
               containing
               an
               Angle
               of
            
             36
             
               Degrees
               ,
               or
               of
            
             108
             
               Degrees
               ,
               divided
               by
               the
               Sum
               of
               those
               Legs
               :
            
             Or
             ,
             (
             which
             is
             Equivalent
             to
             those
             )
             
               The
               Biquadrates
               of
               the
               Legs
            
             (
             in
             the
             former
             case
             )
             
               with
               the
               three
               means
               Proportional
               between
               them
            
             ;
             Or
             ,
             (
             in
             the
             latter
             case
             )
             
               The
               Biquadrates
               of
               the
               Legs
               ,
               with
               a
               mean
               Proportional
               between
               them
               ,
               wanting
               the
               first
               and
               third
               of
               three
               means
               Proportionals
               :
            
             Are
             equal
             to
             
               Ten
               Biquadrates
               of
               the
               Radius
               of
               a
               Circumscribed
               Circle
               .
            
             And
             ,
             
               These
               ,
               to
               the
               Biquadrates
               of
               their
               respective
               Bases
               subtending
               such
               Angle
               of
            
             144
             
               Degrees
               ,
               or
               of
            
             36
             Degrees
             ;
             are
             as
             4
             to
             
             ;
             
               but
               ,
               of
               the
               Bases
               subtending
               such
               Angle
               of
            
             72
             
               Degrees
               ,
               or
               of
            
             108
             
               Degrees
               ;
               as
            
             4
             to
             
             :
             Or
             ,
             as
             8
             to
             
             ,
             and
             8
             to
             
             .
             That
             is
             ,
             in
             the
             Duplicate
             proportion
             
             to
             
             ;
             and
             of
             
             to
             
             .
          
           
             CXLVII
             .
             And
             those
             sides
             ,
             contain
             these
             following
             Angles
             .
          
           
             
               
                 Sides
                 .
              
               
                 Deg.
                 
              
            
             
               
                 A
                 ,
                 E.
                 
              
               
                 144
              
            
             
               
                 
                   
                     M
                     ,
                     A.
                     
                  
                   
                     M
                     ,
                     E.
                     
                  
                   
                     L
                     ,
                     N.
                     
                  
                
              
               
                 72
              
            
             
               
                 
                   
                     L
                     ,
                     A.
                     
                  
                   
                     N
                     ,
                     E.
                     
                  
                   
                     M
                     ,
                     L.
                     
                  
                   
                     M
                     ,
                     N.
                     
                  
                
              
               
                 36
              
            
             
               
                 
                   
                     N
                     ,
                     A.
                     
                  
                   
                     L
                     ,
                     E.
                     
                  
                
              
               
                 108.
                 
              
            
          
           
             whereof
             the
             Four
             first
             Couple
             ,
             are
             sides
             of
             like
             signs
             ;
             the
             six
             latter
             ,
             of
             unlike
             .
          
           
             CXLVIII
             .
             The
             same
             Equations
             may
             be
             thus
             also
             considered
             .
             Because
             (
             by
             §
             110
             )
             5RqqA
             −
             5RqqE
             =
             5RqAc
             −
             5RqEc
             −
             Aqc
             +
             Eqc
             :
             Therefore
             ,
             (
             dividing
             all
             by
             A
             −
             E
             ,
             and
             again
             by
             5Rq
             ,
             )
             
             And
             (
             by
             transposition
             )
          
           
             Aq
             +
             AE
             +
             Eq
             −
             Rq
             ,
             into
             5Rq
             ,
             =
             Aqq
             +
             AcE
             +
             AqEq
             +
             AEc
             +
             Eqq.
             
          
           
           
             CXLIX
             .
             And
             in
             like
             manner
             (
             because
             M
             ,
             A
             ,
             and
             M
             ,
             E
             ,
             and
             L
             ,
             N
             ,
             have
             
             also
             like
             signs
             .
             )
          
           
             
               Mq+MA+Aq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Mqq+McA+MqAq+MAc+Aqq
               .
            
             
               Mq+ME+Eq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Mqq+McE+MqEq+MEc+Eqq
               .
            
             
               Lq+LN+Nq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Lqq+LcN+LqNq+LNc+Nqq
               .
            
          
           
             CL.
             And
             therefore
             ,
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Legs
               contain
               an
               Angle
               of
            
             144
             Degrees
             ,
             (
             as
             A
             ,
             E
             ,
             )
             or
             72
             Degrees
             ,
             (
             as
             M
             ,
             A
             ,
             or
             M
             ,
             E
             ,
             or
             L
             ,
             N
             ,
             )
             
               if
               the
               Squares
               of
               the
               Legs
               ,
               with
               the
               Rectangle
               of
               them
               ,
               wanting
               the
               Square
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               ,
               be
               all
               Multiplied
               into
               five
               times
               the
               Square
               of
               that
               Radius
               :
               The
               Product
               is
               equal
               to
               the
               Biquadrates
               of
               the
               Legs
               ,
               with
               three
               means
               Proportional
               between
               those
               Biquadrates
               .
            
          
           
             CLI
             .
             In
             like
             manner
             may
             be
             shewed
             ,
             (
             where
             the
             signs
             of
             the
             Legs
             be
             unlike
             ,
             )
             That
             ,
          
           
             
               Lq
               −
               LA+Aq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Lqq
               −
               LcA+LqAq
               −
               LAc+Aqq
               .
            
             
               Nq
               −
               NE+Eq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Nqq
               −
               NcE+NqEq
               −
               NEc+Eqq
               .
            
             
               Mq
               −
               ML+Lq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Mqq
               −
               McL+MqLq
               −
               MLc+Lqq
               .
            
             
               Mq
               −
               MN+Nq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Mqq
               −
               McN+MqNq
               −
               MNc+Nqq
               .
            
             
               Nq
               −
               NA+Aq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Nqq
               −
               NcA+NqAq
               −
               NAc+Aqq
               .
            
             
               Lq
               −
               LE+Eq
               −
               Rq
               ,
               into
               5Rq
               ,
               =
               Lqq
               −
               LcE+LqEq
               −
               LEc+Eqq
               .
            
          
           
             CLII.
             And
             therefore
             ,
             
               In
               a
               Right-lined
               Triangle
               ,
               whose
               Legs
               contain
               an
               Angle
               of
            
             36
             Degrees
             ,
             (
             as
             L
             ,
             A
             ,
             or
             N
             ,
             E
             ,
             or
             M
             ,
             L
             ,
             or
             M
             ,
             N
             ;
             )
             or
             108
             Degrees
             ,
             (
             as
             N
             ,
             A
             ,
             or
             L
             ,
             E
             ;
             )
             
               if
               the
               Squares
               of
               the
               Legs
               ;
               wanting
               the
               Rectangle
               of
               them
               ,
               and
               the
               Square
               of
               the
               Radius
               of
               the
               Circumscribed
               Circle
               ,
               be
               all
               Multiplied
               into
               five
               times
               the
               Square
               of
               that
               Radius
               ;
               the
               Product
               is
               equal
               to
               the
               Biquadrates
               of
               the
               Legs
               ,
               and
               a
               mean
               Proportional
               between
               those
               Biquadrates
               ,
               wanting
               the
               first
               and
               third
               of
               three
               means
               Proportional
               between
               them
               .
            
          
           
             CLIII
             .
             Now
             all
             this
             variety
             of
             cases
             ,
             and
             Deductions
             from
             them
             ;
             from
             
             §
             83
             ,
             hitherto
             ,
             ariseth
             from
             the
             first
             Construction
             ,
             at
             §
             1.
             and
             what
             is
             Analogous
             thereunto
             :
             Where
             the
             six
             Lines
             ,
             for
             the
             four
             sides
             and
             two
             Diagonals
             of
             the
             Quadrilater
             ,
             are
             F
             ,
             A
             ;
             B
             ,
             B
             ;
             C
             ,
             C.
             And
             the
             variety
             ariseth
             from
             hence
             ,
             that
             sometimes
             C
             ,
             C
             ,
             are
             the
             Diagonals
             ;
             and
             B
             ,
             B
             ,
             opposite
             sides
             ;
             sometimes
             C
             ,
             C
             ,
             are
             opposite
             sides
             ;
             and
             B
             ,
             B
             ,
             Diagonals
             ;
             according
             as
             C
             or
             B
             happens
             to
             be
             greater
             .
          
           
             CLIV.
             But
             ,
             by
             a
             like
             method
             ,
             with
             some
             little
             alteration
             ,
             we
             may
             infer
             most
             
             of
             the
             same
             things
             ;
             (
             and
             observe
             thence
             like
             Deductions
             ,
             or
             others
             Analogous
             thereunto
             ;
             with
             like
             variety
             of
             cases
             ;
             )
             from
             the
             second
             Construction
             ,
             at
             §
             9
             ;
             where
             the
             six
             Lines
             are
             ,
             F
             ,
             C
             ;
             A
             ,
             A
             ;
             D
             ,
             D.
             Where
             the
             variety
             of
             cases
             proceedeth
             from
             hence
             ,
             that
             sometimes
             D
             ,
             D
             ,
             are
             Diagonals
             ,
             and
             A
             ,
             A
             ,
             opposite
             sides
             ;
             sometimes
             D
             ,
             D
             ,
             are
             opposite
             sides
             ;
             and
             A
             ,
             A
             ,
             (
             or
             what
             answers
             to
             them
             )
             Diagonals
             ;
             according
             as
             D
             ,
             or
             A
             ,
             (
             or
             what
             answers
             to
             this
             ,
             E
             ,
             L
             ,
             M
             ,
             N
             ,
             )
             are
             greater
             .
          
           
             CLV
             .
             And
             accordingly
             ,
             the
             propositions
             at
             §
             10
             ,
             and
             12
             ,
             may
             be
             delivered
             more
             generally
             .
             Namely
             ,
          
           
           
             CLVI
             .
             
               The
               Difference
               of
               the
               Squares
               of
               the
               subtenses
               of
               the
               Quadruple
               and
               of
               the
            
             
             
               single
               Arch
               ;
               is
               equal
               to
               the
               Rectangle
               of
               the
               subtenses
               of
               the
               Triple
               and
               Quintuple
               .
            
             And
             ,
             being
             divided
             by
             either
             of
             these
             ,
             it
             gives
             the
             other
             .
             And
             ,
          
           
             CLVII
             .
             
               As
               the
               Subtense
               of
               the
               Triple
               Arch
               ,
               to
               the
               Sum
               of
               the
               subtenses
               of
               the
               Quadruple
               and
               of
               the
               single
               ;
               so
               is
               the
               Differences
               of
               these
               ,
               to
               the
               Subtense
               of
               the
               Quintuple
               .
            
             Whether
             such
             single
             Arch
             be
             lesser
             ,
             or
             greater
             ,
             or
             equal
             to
             a
             Quintant
             .
          
           
             CLVIII
             .
             And
             in
             like
             manner
             ,
             from
             the
             Third
             Construction
             ,
             at
             §
             14.
             where
             
             the
             six
             Lines
             are
             ,
             F
             ,
             A
             ;
             C
             ,
             A
             ;
             B
             ,
             D.
             And
             what
             is
             there
             delivered
             (
             at
             §
             15
             ,
             17.
             )
             of
             an
             Arch
             less
             than
             a
             Quintant
             ,
             may
             be
             more
             generally
             delivered
             ,
             thus
             ,
          
           
             CLIX.
             
               The
               Difference
               of
               the
               Rectangles
               of
               the
               subtenses
               of
               the
               double
               and
               of
               the
               Quadruple
               Arch
               ;
               and
               ,
               of
               the
               single
               and
               Triple
               ;
               is
               equal
               to
               that
               of
               the
               Subtenses
               of
               the
               single
               and
               Quintuple
               .
            
             And
             ,
             being
             divided
             by
             either
             of
             these
             ,
             it
             gives
             the
             other
             .
             And
             ,
          
           
             CLX
             .
             
               As
               the
               Subtense
               of
               the
               single
               Arch
               ,
               to
               that
               of
               the
               double
               ;
               so
               is
               that
               of
               the
               Quadruple
               ,
               to
               the
               Sum
               or
               Difference
               of
               the
               subtenses
               of
               the
               Triple
               and
               Quintuple
               ;
               according
               as
            
             B
             ,
             D
             ,
             
               happen
               to
               be
               Diagonals
               or
               opposite
               sides
            
          
           
             CLXI
             .
             And
             in
             like
             manner
             ,
             from
             the
             Fourth
             Construction
             ,
             at
             §
             20
             ;
             where
             
             the
             six
             Lines
             are
             ,
             F
             ,
             B
             ;
             A
             ,
             B
             ;
             C
             ,
             D.
             And
             what
             is
             there
             delivered
             at
             §
             21
             ,
             23
             ,
             may
             be
             more
             generally
             delivered
             ;
             thus
             ,
          
           
             CLXII
             .
             
               The
               Difference
               of
               the
               Rectangles
               ,
               of
               the
               subtenses
               of
               the
               Triple
               and
               Quadruple
               Arches
               ;
               and
               ,
               of
               the
               single
               and
               double
               ;
               is
               equal
               to
               that
               of
               the
               subtenses
               of
               the
               double
               and
               Quintuple
               .
            
             And
             ,
             being
             divided
             by
             either
             ,
             gives
             the
             other
             of
             them
             .
             And
             ,
          
           
             CLXIII
             .
             
               As
               the
               Subtense
               of
               the
               double
               Arch
               ,
               is
               to
               that
               of
               the
               Triple
               ;
               so
               is
               that
               of
               the
               Quadruple
               ,
               to
               the
               Sum
               or
               Difference
               of
               the
               subtenses
               of
               the
               Quintuple
               ,
               and
               single
               ;
               according
               as
            
             C
             ,
             D
             ,
             
               happen
               to
               be
               Diagonals
               ,
               or
               opposite
               sides
               .
            
          
           
             CLXIV
             .
             And
             from
             every
             of
             these
             Constructions
             ,
             may
             be
             derived
             like
             varietie
             of
             cases
             and
             Consequences
             ,
             (
             with
             Figures
             suited
             to
             those
             cases
             :
             )
             as
             (
             at
             §
             83
             ,
             &c.
             )
             is
             done
             from
             the
             first
             Construction
             .
             But
             I
             forbear
             to
             pursue
             these
             any
             farther
             ;
             and
             leave
             it
             to
             any
             who
             shall
             think
             fit
             ,
             (
             for
             their
             own
             Exercise
             ,
             )
             to
             pursue
             these
             as
             I
             have
             done
             the
             first
             .
          
        
         
           
           
             
               CHAP.
               V.
            
             Of
             the
             Sextuplation
             ,
             and
             Sextisection
             of
             an
             ARCH
             or
             ANGLE
             :
             And
             other
             following
             Multiplications
             and
             Sections
             .
          
           
             I.
             ACCORDING
             to
             the
             same
             methods
             may
             be
             had
             ,
             the
             Sextuplation
             ,
             Septuplation
             ,
             and
             other
             consequent
             Multiplications
             ;
             as
             also
             the
             Sextisection
             ,
             Septisection
             ,
             and
             other
             consequent
             Sections
             ,
             of
             an
             Arch
             or
             Angle
             .
             Of
             which
             I
             shall
             briefly
             touch
             at
             some
             .
          
           
             II.
             The
             Sextuplation
             ,
             may
             be
             had
             ,
             by
             Tripling
             the
             Double
             ,
             or
             Doubling
             the
             Triple
             Arch.
             And
             ,
             accordingly
             ,
             the
             Sextisection
             ,
             by
             Bisecting
             the
             Subtriple
             ,
             or
             Trisecting
             the
             Subduple
             .
             (
             as
             is
             of
             it self
             manifest
             .
             )
             And
             the
             same
             holds
             ,
             in
             like
             manner
             ,
             for
             Multiplications
             and
             Sections
             which
             take
             their
             Denomination
             from
             a
             Compound
             number
             .
             For
             Multiplications
             and
             Sections
             successively
             made
             ,
             according
             to
             the
             Components
             of
             such
             Compound
             number
             ,
             amount
             to
             the
             same
             as
             one
             by
             such
             Compound
             number
             .
          
           
             III.
             But
             though
             Six
             were
             not
             a
             Compound
             number
             ,
             or
             be
             not
             considered
             as
             such
             ;
             yet
             may
             such
             Sextuplation
             and
             Sextisection
             be
             had
             in
             like
             manner
             as
             those
             before
             .
             Namely
             ,
          
           
             IV.
             If
             in
             a
             Circle
             be
             inscribed
             a
             Quadrilater
             ,
             whose
             opposite
             sides
             are
             B
             ,
             B
             ,
             
             subtenses
             of
             the
             Duple
             ;
             and
             B
             ,
             G
             ,
             subtenses
             of
             the
             Duple
             and
             Sextuple
             ;
             and
             the
             Diagonals
             D
             ,
             D
             ,
             subtenses
             of
             the
             Quadruple
             .
             Then
             is
             ,
             Dq
             −
             Bq
             =
             BG
             ;
             and
             B
             )
             Dq
             −
             Bq
             (
             G.
             
          
           
             V.
             Or
             ,
             Let
             the
             opposite
             sides
             be
             A
             ,
             A
             ,
             and
             D
             ,
             G
             ;
             and
             the
             Diagonals
             F
             ,
             F.
             
             Then
             is
             ,
             Fq
             −
             Aq
             =
             DG
             ;
             and
             D
             )
             Fq
             −
             Aq
             (
             G.
             
          
           
             VI.
             Or
             ,
             Let
             the
             opposite
             sides
             be
             A
             ,
             B
             ,
             and
             C
             ,
             G
             ;
             and
             the
             Diagonals
             D
             ,
             F.
             
             Then
             is
             ,
             DF
             −
             AB
             =
             CG
             ;
             and
             C
             )
             DF
             −
             AB
             (
             G.
             
          
           
             VII
             .
             Or
             ,
             Let
             the
             opposite
             sides
             be
             A
             ,
             C
             ,
             and
             B
             ,
             G
             ;
             the
             Diagonals
             C
             ,
             F.
             
             Then
             CF
             −
             AC
             =
             BG
             ;
             and
             B
             )
             CF
             −
             AC
             (
             G.
             
          
           
             VIII
             .
             And
             therefore
             ,
             Dq
             −
             Bq
             =
             CF
             −
             CA.
             
          
           
             IX
             .
             Or
             ,
             Let
             the
             opposite
             sides
             be
             A
             ,
             G
             ,
             and
             A
             ,
             D
             ;
             the
             Diagonals
             B
             ,
             F.
             
             Then
             BF
             −
             AD
             =
             AG
             ;
             and
             A
             )
             BF
             −
             AD
             (
             G.
             
          
           
             X.
             Or
             ,
             Let
             the
             opposite
             sides
             be
             B
             ,
             C
             ,
             and
             A
             ,
             G
             ;
             the
             Diagonals
             C
             ,
             D.
             
             Then
             CD
             −
             BC
             −
             AG
             ;
             and
             A
             )
             CD
             −
             BC
             (
             G.
             
          
           
             XI
             .
             And
             therefore
             ,
             BF
             −
             AD
             =
             CD
             −
             BC.
             
          
           
             XII
             .
             It
             is
             manifest
             that
             from
             hence
             may
             be
             deduced
             a
             great
             number
             of
             Equations
             ,
             and
             Analogies
             ,
             and
             great
             variety
             of
             Theorems
             ,
             in
             like
             manner
             ,
             as
             is
             done
             in
             the
             Chapters
             foregoing
             .
             But
             I
             forbear
             here
             to
             pursue
             them
             in
             particular
             as
             is
             there
             done
             .
          
           
           
             XIII
             .
             But
             from
             every
             of
             those
             Constructions
             ,
             (
             the
             values
             of
             B
             ,
             C
             ,
             D
             ,
             F
             ,
             
             being
             known
             as
             is
             above
             declared
             ,
             )
             we
             have
             (
             by
             ordering
             the
             Equations
             in
             due
             manner
             ,
             )
             
             .
             Or
             ,
             
             .
             And
             (
             taking
             the
             Squares
             of
             these
             ,
             )
             4GqRcccc
             −
             GqRqqccAq
             =
             144RccccAq
             −
             456RqqccAqq
             +
             553RqccAcc
             −
             328RccAqcc
             +
             102RqqAqqcc
             −
             16RqAcccc
             +
             Aqcccc.
             
          
           
             XIV
             .
             That
             is
             ,
             (
             dividing
             all
             by
             4Rq
             −
             Aq
             ,
             )
             RqqccGq
             =
             36RqqccAq
             −
             105RqccAqq
             +
             112RccAcc
             −
             54RqqAqcc
             +
             12RqAqqcc
             −
             Acccc.
             
          
           
             XV.
             Of
             this
             Equation
             there
             be
             Six
             plain
             Roots
             ,
             answering
             to
             Aq
             ;
             the
             
             Square
             Roots
             of
             which
             ,
             are
             A.
             Which
             are
             so
             many
             streight
             Lines
             from
             some
             one
             Point
             of
             the
             Circumference
             ,
             to
             the
             Six
             Angles
             of
             an
             inscribed
             regular
             Hexagon
             .
             (
             So
             that
             ,
             any
             one
             of
             them
             being
             known
             ,
             the
             rest
             are
             known
             also
             .
             And
             the
             like
             in
             all
             such
             Equations
             .
             )
          
           
             XVI
             .
             Of
             these
             ,
             the
             Two
             least
             ,
             A
             ,
             E
             ,
             (
             which
             subtend
             ,
             on
             the
             one
             side
             ,
             to
             Arches
             less
             than
             a
             Sextant
             ;
             and
             ,
             on
             the
             other
             side
             to
             more
             than
             five
             Sextants
             ;
             )
             And
             the
             Two
             greatest
             ,
             x
             ,
             y
             ,
             (
             which
             subtend
             to
             Arches
             greater
             than
             two
             Sextants
             ,
             but
             less
             than
             four
             ;
             )
             are
             Affirmative
             Roots
             ;
             (
             because
             the
             Subtendent
             of
             the
             double
             Arch
             is
             less
             than
             that
             of
             the
             Quadruple
             ;
             and
             therefore
             Dq
             −
             Bq
             an
             Affirmative
             Quantity
             :
             )
             But
             the
             Two
             betwen
             them
             I
             ,
             K
             ,
             (
             which
             subtend
             on
             the
             one
             side
             ,
             to
             Arches
             greater
             than
             one
             Sextant
             but
             less
             than
             two
             ;
             and
             on
             the
             other
             side
             ,
             to
             Arches
             greater
             than
             four
             Sextants
             but
             less
             than
             five
             ;
             )
             are
             Negatives
             ,
             (
             because
             of
             D
             less
             than
             B
             ;
             and
             therefore
             Dq
             −
             Bq
             a
             Negative
             Quantity
             ;
             )
             G
             being
             in
             all
             ,
             reputed
             Affirmative
             .
          
           
             XVII
             .
             If
             a
             Chord
             be
             subtendent
             to
             just
             a
             Sextant
             ,
             or
             two
             or
             more
             Sextants
             ;
             it
             is
             indifferent
             to
             whether
             of
             the
             two
             cases
             on
             either
             side
             it
             be
             referred
             ;
             suppose
             
             .
             (
             which
             is
             to
             be
             understood
             in
             all
             cases
             of
             like
             nature
             .
             )
             And
             when
             ever
             this
             happens
             ,
             one
             of
             the
             Roots
             vanish
             ,
             or
             become
             equal
             to
             nothing
             .
          
           
             XVIII
             .
             For
             the
             Septuplation
             or
             Septisection
             of
             an
             Arch
             or
             Angle
             ;
             we
             shall
             have
             ,
             according
             as
             the
             Quadrilater
             may
             be
             differently
             inscribed
             ,
             the
             Subtense
             of
             the
             Septuple
             Arch
             ,
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             ,
             or
             
             .
          
           
             XIX
             .
             From
             every
             of
             which
             Equations
             ,
             (
             having
             the
             values
             of
             B
             ,
             C
             ,
             D
             ,
             F
             ,
             G
             ,
             known
             as
             before
             ,
             )
             we
             shall
             have
             (
             by
             due
             ordering
             such
             Equation
             )
             
             :
             Or
             ,
             RccH
             =
             7RccA
             −
             14RqqAc
             +
             7RqAqc
             −
             Aqqc.
             
          
           
           
             XX.
             The
             Seven
             Roots
             of
             this
             Equation
             ;
             are
             ,
             so
             many
             streight
             Lines
             from
             some
             one
             Point
             of
             the
             Circumference
             ,
             to
             the
             Seven
             Angles
             of
             an
             inscribed
             Regular
             Heptagon
             .
          
           
             XXI
             .
             Of
             these
             Roots
             (
             putting
             H
             Affirmative
             ,
             )
             the
             two
             least
             are
             Affirmative
             ;
             the
             two
             next
             ,
             are
             Negative
             ;
             the
             two
             next
             to
             these
             ,
             are
             again
             Affirmative
             ;
             and
             ,
             the
             greatest
             Negative
             .
          
           
             XXII
             .
             And
             after
             the
             same
             manner
             we
             may
             proceed
             as
             far
             as
             we
             please
             :
             Collecting
             the
             consequent
             Multiplications
             and
             Sections
             ,
             by
             the
             help
             of
             those
             Antecedent
             .
          
           
             XXIII
             .
             And
             all
             such
             as
             are
             denominated
             by
             a
             Compound
             number
             (
             as
             4
             =
             2
             ×
             2
             ,
             6
             =
             2
             ×
             3
             ,
             8
             =
             2
             ×
             4
             =
             2
             ×
             2
             ×
             2
             ,
             9
             =
             3
             ×
             3
             ,
             &c.
             )
             may
             ,
             with
             more
             convenience
             ,
             (
             at
             lest
             ,
             as
             to
             the
             Section
             ,
             if
             not
             as
             to
             the
             Multiplication
             also
             ,
             )
             be
             performed
             by
             two
             or
             more
             operations
             ,
             according
             to
             the
             Components
             of
             such
             Compound
             number
             .
          
           
             XXIV
             .
             But
             ,
             both
             these
             ,
             and
             those
             which
             are
             Denominated
             from
             Prime
             numbers
             ,
             (
             as
             3
             ,
             5
             ,
             7
             ,
             11
             ,
             &c.
             )
             may
             (
             by
             such
             inscription
             of
             Quadrilaters
             )
             be
             Reduced
             to
             such
             Equations
             ,
             as
             will
             contain
             as
             many
             Roots
             as
             is
             the
             number
             from
             which
             such
             Multiplication
             or
             Section
             takes
             its
             Denomination
             .
          
           
             XXV
             .
             And
             ,
             of
             these
             ,
             those
             which
             are
             Denominated
             by
             an
             Even
             number
             ,
             will
             afford
             Equations
             having
             Plain
             Roots
             ;
             the
             Square
             Root
             of
             which
             Plains
             ,
             are
             the
             subtenses
             of
             the
             Arches
             .
          
           
             XXVI
             .
             But
             those
             which
             are
             Denominated
             by
             Odd
             numbers
             ,
             afford
             Equations
             whose
             Roots
             are
             those
             subtenses
             .
          
           
             XXVII
             .
             And
             ,
             of
             these
             subtenses
             (
             as
             well
             in
             the
             one
             case
             as
             in
             the
             other
             ,
             )
             the
             two
             least
             (
             which
             I
             look
             upon
             as
             the
             Principal
             Roots
             of
             the
             Equation
             ,
             )
             are
             Affirmatives
             (
             supposing
             the
             Subtense
             of
             the
             Multiple
             Arch
             to
             be
             always
             put
             Affirmative
             ;
             )
             the
             two
             next
             greater
             than
             these
             ,
             Negatives
             ;
             the
             two
             next
             Affirmatives
             ;
             and
             so
             onward
             ,
             Alternately
             ,
             as
             long
             as
             there
             be
             Roots
             remaining
             :
             save
             that
             ,
             when
             the
             number
             is
             Odd
             ,
             the
             greatest
             of
             all
             will
             be
             singular
             ,
             whereas
             the
             rest
             go
             by
             Couples
             .
          
        
         
           
           
             
               CHAP.
               VI.
            
             Of
             the
             Proportion
             of
             the
             Base
             to
             the
             Legs
             of
             a
             Triangle
             ,
             according
             as
             is
             the
             Angle
             at
             the
             Top
             of
             it
             .
          
           
             I.
             THE
             noted
             Proposition
             of
             Pythagoras
             ,
             (
             which
             is
             in
             Euclid
             ,
             47
             è
             1.
             )
             concerning
             the
             Square
             of
             the
             Base
             equal
             to
             the
             Squares
             of
             the
             two
             Legs
             containing
             a
             Right-angle
             :
             And
             two
             more
             in
             
               Euclid
               (
               Pr.
            
             12
             ,
             13
             ,
             è
             2.
             )
             concerning
             the
             Excess
             ,
             (
             in
             case
             the
             Angle
             at
             the
             Top
             be
             Obtuse
             ;
             )
             or
             the
             Defect
             ,
             (
             in
             case
             it
             be
             Acute
             ;
             )
             of
             the
             Square
             of
             the
             Base
             ,
             compared
             with
             the
             Squares
             of
             the
             two
             Legs
             :
             And
             some
             other
             Propositions
             in
             the
             foregoing
             Chapters
             ,
             shewing
             what
             Proportion
             that
             Excess
             or
             Defect
             bears
             to
             a
             Rectangle
             of
             the
             Legs
             ,
             in
             divers
             cases
             :
             Gave
             me
             occasion
             to
             pursue
             that
             Speculation
             a
             little
             further
             ;
             according
             to
             the
             following
             Propositions
             .
          
           
           
             II.
             If
             by
             the
             Legs
             of
             a
             Triangle
             C
             ,
             D
             ,
             the
             Angle
             at
             the
             Top
             contained
             A
             ,
             be
             a
             Right-angle
             ,
             (
             or
             of
             90
             Degrees
             ;
             )
             and
             from
             thence
             a
             Perpendicular
             G
             ,
             on
             the
             Base
             ,
             cut
             this
             into
             two
             Segments
             χ
             ,
             δ
             :
             The
             two
             Triangles
             hence
             arising
             ,
             χ
             G
             C
             ,
             G
             δ
             D
             ,
             are
             like
             to
             the
             whole
             CDB
             ;
             (
             because
             of
             one
             Angle
             common
             ,
             the
             other
             a
             Right-angle
             ,
             and
             therefore
             the
             third
             equal
             to
             the
             third
             .
             )
             And
             therefore
             (
             the
             Triangles
             being
             here
             designed
             by
             their
             sides
             .
             )
          
           
             
             and
             therefore
             
             and
             therefore
             ,
             Cq+Dq
             =
             (
             Bχ+Bδ
             =
             B
             :
             into
             χ+δ
             =
             B
             :
             =
             )
             Bq.
             That
             is
             ,
          
           
             
               The
               Square
               of
               the
               Base
               is
               equal
               to
               the
               two
               Squares
               of
               the
               Legs
               containing
               a
               Right-angle
               .
            
          
           
           
             III.
             If
             the
             Angle
             A
             be
             120
             Deg.
             or
             4
             /
             3
             of
             a
             Right-angle
             ;
             and
             thence
             be
             drawn
             the
             Base
             two
             streight
             Lines
             G
             ,
             Γ
             ;
             making
             ,
             with
             it
             ,
             Angles
             equal
             to
             that
             at
             the
             Top
             :
             The
             Two
             Triangles
             χ
             GC
             ,
             ΓδD
             ,
             are
             like
             to
             the
             whole
             CDB
             ,
             (
             because
             of
             one
             Angle
             common
             ,
             another
             equal
             to
             A
             ,
             and
             therefore
             the
             third
             equal
             to
             the
             third
             ;
             )
             with
             a
             Triangle
             between
             GΓμ
             Equilater
             (
             because
             each
             of
             the
             Angles
             at
             the
             Base
             ,
             and
             therefore
             that
             at
             the
             Top
             ,
             are
             of
             60
             Degrees
             :
             )
             And
             the
             Base
             B
             =
             χ+δ+μ
             .
             And
             therefore
             ,
             
             and
             
             Therefore
             ,
             Cq+Dq+CD
             =
             (
             Bχ+Bδ+Bμ
             =
             B
             :
             into
             χ+δ+μ
             =
             )
             B
             :
             =
             Bq.
             That
             is
             ,
          
           
             
               The
               Square
               of
               the
               Base
               (
               of
               an
               Angle
               of
            
             120
             
               Degrees
               )
               is
               equal
               to
               the
               Squares
               of
               the
               Legs
               and
               a
               Rectangle
               of
               them
               .
            
          
           
           
           
             
               The
               Square
               of
               the
               Base
               (
               of
               an
               Angle
               of
            
             60
             
               Degrees
               )
               is
               equal
               to
               the
               Squares
               of
               the
               Legs
               ,
               wanting
               the
               Rectangle
               of
               them
               .
            
          
           
             (
             Note
             here
             ,
             that
             ,
             by
             χ
             ,
             I
             understand
             the
             Base
             to
             the
             Legs
             CG
             ;
             by
             δ
             ,
             that
             of
             the
             Legs
             DΓ
             ;
             by
             μ
             ,
             that
             of
             GΓ
             ;
             and
             ,
             by
             B
             ,
             that
             of
             CD
             ,
             which
             is
             ever
             equal
             to
             χ+δ±μ
             ,
             however
             these
             parts
             be
             intermingled
             .
             Which
             where
             it
             is
             +μ
             ,
             is
             commonly
             more
             obvious
             to
             the
             Eye
             ;
             but
             where
             it
             is
             −
             μ
             ,
             is
             more
             perplex
             ,
             and
             will
             need
             more
             consideration
             to
             discern
             ;
             but
             it
             is
             equally
             true
             in
             both
             cases
             .
             )
          
           
           
             
               The
               Square
               of
               the
               Base
               (
               of
               an
               Angle
               of
            
             135
             
               Degrees
               )
               is
               equal
               to
               the
               Squares
               of
               the
               Legs
               ,
               with
               a
               Rectangle
               of
               them
               Multiplied
               into
            
             
          
           
           
             VI.
             If
             A
             be
             45
             Degrees
             :
             It
             will
             in
             like
             manner
             be
             shewed
             ,
             that
             (
             because
             of
             B
             =
             χ+δ
             −
             μ.
             )
             
             into
             χ+δ
             −
             μ
             =
             B
             :
             =
             )
             Bq.
             That
             is
             ,
          
           
             
               The
               Square
               of
               the
               Base
               (
               of
               an
               Angle
               of
            
             45
             
               Degrees
               )
               is
               equal
               to
               the
               Squares
               of
               the
               Legs
               ,
               wanting
               a
               Rectangle
               of
               them
               Multiplied
               into
            
             
          
           
           
             VII
             .
             And
             ,
             universally
             ,
             what
             ever
             be
             the
             Angle
             A
             ;
             it
             will
             (
             by
             like
             process
             )
             be
             shewed
             :
             That
             ,
          
           
             
             .
             That
             is
             ,
             
               The
               Square
               of
               the
               Base
               (
               whatever
               be
               the
               Angle
               at
               the
               Vertex
               )
               is
               equal
               to
               the
               Squares
               of
               the
               Legs
               ,
               together
               with
               (
               if
               it
               be
               greater
               than
               a
               Right-angle
               )
               or
               wanting
               (
               if
               less
               than
               such
               )
               a
               Plain
               ,
               which
               shall
               be
               ,
               to
               the
               Rect-angle
               of
               the
               Legs
               ,
               as
               a
               Portion
               in
               the
               Base-line
               ,
               intercepted
               between
               two
               Lines
               from
               the
               Vertex
               ,
               making
               at
               the
               Base
               a
               like
               Angle
               with
               that
               of
               the
               Vertex
               ,
               to
               one
               of
               those
               two
               Lines
               so
               drawn
               .
            
          
           
             VIII
             .
             Of
             this
             we
             are
             to
             give
             great
             variety
             of
             Examples
             in
             the
             following
             Chapter
             ,
             where
             this
             General
             Theorem
             is
             applied
             to
             particular
             cases
             :
             And
             which
             is
             further
             improved
             by
             these
             two
             ensuing
             Propositions
             .
          
           
           
             IX
             .
             
               The
               Radius
               of
               a
               Circle
               ,
               with
               the
               subtenses
               of
               two
               Arches
               ,
               being
               given
               ;
               the
               Subtense
               of
               their
               Aggregate
               is
               also
               given
               .
            
             For
             ,
             supposing
             the
             subtenses
             of
             the
             given
             Arches
             to
             be
             A
             ,
             E
             :
             The
             subtenses
             of
             their
             Remainders
             to
             a
             Semicircle
             ,
             are
             also
             had
             :
             Suppose
             
             And
             
             .
             And
             therefore
             ,
             inscribing
             a
             Quadrilater
             whose
             opposite
             sides
             are
             A
             ,
             ε
             ;
             and
             E
             ,
             α
             ;
             one
             of
             the
             Diagonals
             is
             the
             Diameter
             =
             2R
             ;
             the
             other
             the
             Subtense
             of
             the
             Sum
             or
             Aggregate
             of
             those
             Arches
             ,
             suppose
             
             .
          
           
           
             X.
             
               The
               same
               being
               given
               ;
               the
               Subtense
               of
               the
               Difference
               of
               those
               Arches
               is
               also
               given
               .
            
             For
             ,
             having
             (
             as
             before
             )
             A
             ,
             α
             ;
             E
             ,
             ε
             ;
             2R
             :
             we
             have
             (
             by
             a
             Quadrilater
             duly
             inscribed
             )
             the
             Subtense
             of
             the
             Difference
             ,
             
             .
          
           
             XI
             .
             It
             is
             manifest
             also
             ,
             (
             from
             what
             is
             before
             delivered
             ,
             )
             that
             the
             same
             Triangle
             GΓμ
             ,
             doth
             indifferently
             serve
             for
             the
             Angle
             of
             120
             Degrees
             and
             of
             60
             Degrees
             :
             And
             ,
             in
             like
             manner
             ,
             for
             135
             ,
             and
             45
             :
             And
             so
             ,
             for
             any
             two
             Arches
             whereof
             one
             doth
             as
             much
             exceed
             as
             the
             other
             wants
             of
             a
             Quadrant
             .
             For
             ,
             the
             Angle
             V
             is
             in
             both
             the
             same
             ;
             and
             the
             Angles
             at
             the
             Base
             differ
             only
             in
             this
             :
             That
             ,
             in
             one
             ,
             the
             External
             Angle
             ;
             in
             the
             other
             ,
             the
             Internal
             ,
             (
             which
             is
             the
             others
             Complement
             to
             two
             Right-angles
             ;
             )
             is
             equal
             to
             the
             Angle
             of
             CD
             at
             the
             Vertex
             .
          
           
             XII
             .
             Hence
             it
             follows
             :
             That
             ,
             
               Of
               two
               Angles
               ,
               where
               the
               Legs
               of
               the
               one
               are
               respectively
               equal
               to
               those
               of
               the
               other
               ;
               the
               one
               as
               much
               exceeding
               a
               Right-angle
               ,
               as
               the
               other
               wants
               of
               it
               :
               The
               Square
               of
               the
               Base
               in
               the
               one
               ,
               doth
               as
               much
               exceed
               the
               two
               Squares
               of
               the
               Legs
               ;
               as
               ,
               in
               the
               other
               ,
               it
               wants
               thereof
               .
            
          
           
           
             XIII
             .
             And
             consequently
             ,
             
               In
               any
               Right-lined
               Triangle
               ,
               (
               however
               inclined
               ,
               )
               the
               Squares
               of
               the
               Axis
               or
               Diameter
               ,
               and
               of
               the
               half
               Bases
               twice
               taken
               ;
               are
               equal
               to
               the
               Squares
               of
               the
               Legs
               .
            
             For
             ,
             supposing
             C
             ,
             C
             ,
             the
             two
             halfs
             of
             the
             Base
             ;
             and
             B
             ,
             the
             Diameter
             or
             Axis
             of
             the
             Triangle
             ,
             (
             meaning
             thereby
             a
             streight
             Line
             from
             the
             Vertex
             to
             the
             middle
             of
             the
             Base
             ;
             )
             and
             B
             ,
             β
             ,
             the
             two
             Legs
             :
             It
             is
             manifest
             ,
             that
             ,
             of
             the
             two
             Angles
             at
             the
             Base
             (
             which
             are
             each
             others
             Complement
             to
             two
             Right-angles
             ;
             )
             the
             one
             doth
             as
             much
             exceed
             ,
             as
             the
             other
             wants
             of
             ,
             a
             Right-angle
             :
             And
             therefore
             the
             Square
             of
             one
             of
             the
             Legs
             ,
             as
             Bq
             ,
             doth
             as
             much
             exceed
             ;
             as
             the
             other
             ,
             βq
             ,
             doth
             come
             short
             of
             ;
             Dq+Cq
             .
             And
             therefore
             ,
             both
             together
             ,
             Bq+βq
             =
             2Dq+2Cq
             .
          
           
           
             XIV
             .
             And
             therefore
             ,
             
               The
               Base
               ,
               and
               Axis
               (
               or
               Diameter
               )
               of
               a
               Triangle
               remaining
               the
               same
               ;
               (
               however
               differently
               inclined
               :
               )
               the
               Aggregate
               of
               the
               Squares
               of
               the
               two
               Legs
               ,
               remains
               the
               same
               .
            
          
           
             XV.
             And
             the
             same
             is
             to
             be
             understood
             of
             the
             
               Squares
               of
               Tangents
            
             ,
             of
             a
             Parabola
             ,
             Hyperbola
             ,
             Elipsis
             ,
             (
             or
             other
             Curve
             Line
             having
             Diameter
             and
             Ordinates
             ,
             )
             
               from
               the
               two
               ends
               of
               an
               Inscribed
               Ordinate
               ,
               to
               the
               Point
               of
               the
               Diameter
               (
               produced
               if
               need
               be
               )
               wherein
               those
               Tangents
               meet
               .
            
          
           
             XVI
             .
             The
             same
             may
             be
             likewise
             accommodated
             ,
             to
             the
             Segments
             (
             of
             such
             Legs
             ,
             or
             Tangents
             ,
             )
             
               Cut
               off
               by
               Lines
               Parallel
               to
               the
               Base
               .
            
             Namely
             ,
             
               The
               Squares
               of
               such
               Segments
            
             (
             intercepted
             by
             those
             Parallels
             )
             
               together
               taken
               ,
               (
               the
               Axe
               of
               such
               Trapezium
               remaining
               the
               same
               ,
               )
               are
               the
               same
               :
            
             Whether
             such
             Trapezium
             be
             Erect
             ,
             or
             however
             inclined
             .
             For
             such
             Segments
             ,
             are
             still
             Proportional
             to
             their
             Wholes
             .
          
        
         
           
             
               CHAP.
               VII
            
             .
             Application
             thereof
             to
             particular
             cases
             .
          
           
             I.
             IF
             A
             be
             a
             Right-angle
             ,
             (
             or
             of
             90
             Degrees
             ,
             )
             GΓ
             are
             Co-incident
             ,
             and
             μ
             =
             0.
             and
             therefore
             ,
             
             .
             And
             consequently
             (
             by
             §
             7
             ,
             
               Chap.
               preced
            
             .
             )
             
             .
          
           
             II.
             If
             A
             =
             120
             Degrees
             ;
             then
             is
             V
             (
             that
             is
             ,
             the
             Angle
             contained
             of
             GΓ
             )
             =
             60
             Degrees
             :
             (
             as
             being
             always
             the
             Difference
             of
             2
             A
             from
             two
             Right-angles
             :
             )
             And
             consequently
             GΓμ
             an
             Equilater
             Triangle
             ,
             (
             for
             such
             also
             are
             the
             Angles
             at
             the
             Base
             ;
             each
             of
             which
             is
             the
             Complement
             of
             A
             to
             two
             Right-angles
             :
             )
             And
             therefore
             ,
             μ
             =
             G
             ;
             and
             Bq
             =
             Cq+Dq+CD
             .
          
           
             III.
             If
             A
             =
             60
             Degrees
             :
             Then
             also
             is
             V
             =
             60
             Degrees
             ,
             and
             μ
             =
             G
             ,
             as
             before
             .
             And
             therefore
             ,
             Bq
             =
             Cq+Dq
             −
             CD
             .
          
           
             IV.
             If
             A
             =
             135.
             
             Then
             V
             =
             90
             :
             And
             therefore
             ,
             (
             by
             §
             1.
             )
             μq
             =
             Gq+Γq
             ,
             that
             is
             (
             because
             G
             =
             Γ
             ,
             )
             μq
             =
             2Gq
             ;
             and
             
             And
             therefore
             ,
             
             .
          
           
             V.
             If
             A
             =
             45.
             
             Then
             also
             ,
             V
             =
             90
             :
             And
             therefore
             ,
             (
             as
             before
             )
             
             ;
             and
             consequently
             ,
             
             .
          
           
             VI.
             If
             A
             =
             150
             Then
             V
             =
             120.
             
             And
             therefore
             ,
             (
             by
             §
             2.
             )
             μq
             =
             Gq+Γq+GΓ
             ;
             that
             is
             (
             because
             G
             =
             Γ
             ,
             )
             μq
             =
             3Gq
             ,
             and
             
             .
             And
             
             .
          
           
             VII
             .
             If
             A
             =
             30
             Then
             V
             =
             120.
             
             And
             therefore
             ,
             (
             by
             §
             2.
             )
             μq
             =
             Gq+Γq+GΓ
             ;
             that
             is
             (
             because
             G
             =
             Γ
             ,
             )
             μq
             =
             3Gq
             ,
             and
             
             .
             And
             
             .
          
           
             VIII
             .
             If
             A
             =
             157½
             Then
             V
             =
             135.
             And
             
             (
             by
             §
             4.
             )
             And
             therefore
             ,
             
             .
          
           
             IX
             .
             If
             A
             =
             22½
             Then
             V
             =
             135.
             And
             
             (
             by
             §
             4.
             )
             And
             therefore
             ,
             
             .
          
           
             X.
             If
             A
             =
             112½
             Then
             V
             =
             45.
             And
             
             (
             by
             §
             5.
             )
             And
             therefore
             ,
             
             .
          
           
             XI
             .
             If
             A
             =
             6
             −
             ½
             Then
             V
             =
             45.
             And
             
             (
             by
             §
             5.
             )
             And
             therefore
             ,
             
             .
          
           
           
             XII
             .
             If
             A
             =
             165
             Then
             V
             =
             150.
             And
             
             (
             by
             §
             6.
             )
             And
             therefore
             ,
             
             .
          
           
             XIII
             .
             If
             A
             =
             15
             Then
             V
             =
             150.
             And
             
             (
             by
             §
             6.
             )
             And
             therefore
             ,
             
             .
          
           
             XIV
             .
             If
             A
             =
             105
             Then
             V
             =
             30.
             And
             
             (
             by
             §
             7.
             )
             And
             therefore
             ,
             
             .
          
           
             XV.
             If
             A
             =
             75
             Then
             V
             =
             30.
             And
             
             (
             by
             §
             7.
             )
             And
             therefore
             ,
             
             .
          
           
             XVI
             .
             If
             A
             =
             172½
             Then
             V
             =
             165.
             
             And
             (
             by
             §
             12.
             )
             
             .
             And
             therefore
             ,
             
             .
          
           
             XVII
             .
             If
             A
             =
             7½
             Then
             V
             =
             165.
             
             And
             (
             by
             §
             12.
             )
             
             .
             And
             therefore
             ,
             
             .
          
           
             XVIII
             .
             If
             A
             =
             97½
             Then
             V
             =
             15.
             
             And
             (
             by
             §
             13.
             )
             
             .
             And
             
             .
          
           
             XIX
             .
             If
             A
             =
             82½
             Then
             V
             =
             15.
             
             And
             (
             by
             §
             13.
             )
             
             .
             And
             
             .
          
           
             XX.
             If
             A
             =
             142½
             Then
             V
             =
             105.
             
             And
             (
             by
             §
             14.
             )
             
             .
             And
             
             .
          
           
             XXI
             .
             If
             A
             =
             37½
             Then
             V
             =
             105.
             
             And
             (
             by
             §
             14.
             )
             
             .
             And
             
             .
          
           
             XXII
             .
             If
             A
             =
             127½
             Then
             V
             =
             75.
             
             And
             (
             by
             §
             15.
             )
             
             .
             And
             
             .
          
           
             XXIII
             .
             If
             A
             =
             52½
             Then
             V
             =
             75.
             
             And
             (
             by
             §
             15.
             )
             
             .
             And
             
             .
          
           
             And
             ,
             in
             like
             manner
             ,
             we
             may
             proceed
             to
             lesser
             Arches
             ,
             determined
             by
             quarters
             of
             Degrees
             .
             For
             like
             as
             here
             ,
             by
             help
             of
             §
             4
             ,
             5
             ,
             12
             ,
             13
             ,
             14
             ,
             15.
             we
             have
             performed
             §
             8
             ,
             9
             ,
             10
             ,
             11
             ,
             16
             ,
             17
             ,
             18
             ,
             19
             ,
             20
             ,
             21
             ,
             22
             ,
             23.
             which
             proceed
             to
             half
             Degrees
             :
             So
             by
             the
             help
             of
             these
             ,
             we
             may
             proceed
             to
             Quarters
             of
             Degrees
             .
             And
             farther
             if
             we
             please
             :
             But
             I
             shall
             at
             present
             rest
             at
             half
             Degrees
             .
          
           
             Moreover
             ,
             assuming
             (
             as
             elsewhere
             proved
             )
             the
             Subtense
             of
             36
             Degrees
             ,
             or
             the
             side
             of
             the
             inscribed
             Decagon
             ;
             Namely
             ,
             
             .
             (
             by
             9
             El.
             13.
             and
             4
             El.
             14.
             
             Or
             ,
             55
             ,
             56
             ,
             Chap.
             32.
             )
             we
             may
             ,
             from
             thence
             ,
             thus
             proceed
             .
          
           
             XXIV
             .
             If
             A
             =
             108
             Then
             V
             =
             36.
             And
             
             .
             And
             
             .
          
           
             XXV
             .
             If
             A
             =
             72
             Then
             V
             =
             36.
             And
             
             .
             And
             
             .
          
           
             XXVI
             .
             If
             A
             =
             144
             Then
             V
             =
             108.
             
             And
             μ
             =
             (
             by
             §
             24.
             )
             
             .
             And
             
             .
          
           
             XXVII
             .
             If
             A
             =
             36
             Then
             V
             =
             108.
             
             And
             μ
             =
             (
             by
             §
             24.
             )
             
             .
             And
             
             .
          
           
           
             XXVIII
             .
             If
             A
             =
             126
             Then
             V
             =
             72.
             
             And
             (
             by
             §
             25.
             )
             
             .
             And
             
             .
          
           
             XXIX
             .
             If
             A
             =
             54
             Then
             V
             =
             72.
             
             And
             (
             by
             §
             25.
             )
             
             .
             And
             
             .
          
           
             XXX
             .
             If
             A
             =
             144
             Then
             V
             =
             144.
             
             And
             (
             by
             §
             26.
             )
             
             And
             
             .
          
           
             XXXI
             .
             If
             A
             =
             18
             Then
             V
             =
             144.
             
             And
             (
             by
             §
             26.
             )
             
             .
             And
             
             .
          
           
             XXXII
             .
             If
             A
             =
             153
             Then
             V
             =
             126.
             
             And
             (
             by
             §
             28.
             )
             
             .
             And
             
             .
          
           
             XXXIII
             .
             If
             A
             =
             27
             Then
             V
             =
             126.
             
             And
             (
             by
             §
             28.
             )
             
             .
             And
             
             .
          
           
             XXXIV
             .
             If
             A
             =
             117
             Then
             V
             =
             54.
             
             And
             (
             by
             §
             29.
             )
             
             .
             And
             
             .
          
           
             XXXV
             .
             If
             A
             =
             63
             Then
             V
             =
             54.
             
             And
             (
             by
             §
             29.
             )
             
             .
             And
             
             .
          
           
             XXXVI
             .
             If
             A
             =
             171
             Then
             V
             =
             162.
             
             And
             (
             by
             §
             30.
             )
             
             .
             And
             
             .
          
           
             XXXVII
             .
             If
             A
             =
             9
             Then
             V
             =
             162.
             
             And
             (
             by
             §
             30.
             )
             
             .
             And
             
             .
          
           
             XXXVIII
             .
             If
             A
             =
             99
             Then
             V
             =
             18.
             
             And
             (
             by
             §
             31.
             )
             
             .
             And
             
             .
          
           
             XXXIX
             .
             If
             A
             =
             81
             Then
             V
             =
             18.
             
             And
             (
             by
             §
             31.
             )
             
             .
             And
             
             .
          
           
             XL.
             If
             A
             =
             166½
             Then
             V
             =
             153.
             
             And
             (
             by
             §
             32.
             )
             
             .
             And
             
             .
          
           
             XLI
             .
             If
             A
             =
             13½
             Then
             V
             =
             153.
             
             And
             (
             by
             §
             32.
             )
             
             .
             And
             
             .
          
           
             XLII
             .
             If
             A
             =
             103½
             Then
             V
             =
             27.
             
             And
             (
             by
             §
             33.
             )
             
             .
             And
             
             .
          
           
             XLIII
             .
             If
             A
             =
             76½
             Then
             V
             =
             27.
             
             And
             (
             by
             §
             33.
             )
             
             .
             And
             
             .
          
           
             XLIV
             .
             If
             A
             =
             148½
             Then
             V
             =
             117.
             
             And
             consequently
             ,
             (
             by
             §
             34.
             )
             
             .
          
           
             XLV
             .
             If
             A
             =
             31½
             Then
             V
             =
             117.
             
             And
             consequently
             ,
             (
             by
             §
             34.
             )
             
             .
          
           
             XLVI
             .
             If
             A
             =
             121½
             Then
             V
             =
             63.
             
             And
             (
             by
             §
             35.
             )
             
             .
          
           
             XLVII
             .
             If
             A
             =
             58½
             Then
             V
             =
             63.
             
             And
             (
             by
             §
             35.
             )
             
             .
          
           
           
             XLVIII
             .
             If
             A
             =
             175½
             Then
             V
             =
             171.
             
             And
             (
             by
             §
             36.
             )
             
             .
          
           
             XLIX
             .
             If
             A
             =
             4½
             Then
             V
             =
             171.
             
             And
             (
             by
             §
             36.
             )
             
             .
          
           
             L.
             If
             A
             =
             94½
             Then
             V
             =
             9.
             
             And
             (
             by
             §
             37.
             )
             
             .
          
           
             LI.
             If
             A
             =
             85½
             Then
             V
             =
             9.
             
             And
             (
             by
             §
             37.
             )
             
             .
          
           
             LII
             .
             If
             A
             =
             139½
             Then
             V
             =
             99.
             
             And
             (
             by
             §
             38.
             )
             
             .
          
           
             LIII
             .
             If
             A
             =
             40½
             Then
             V
             =
             99.
             
             And
             (
             by
             §
             38.
             )
             
             .
          
           
             LIV.
             If
             A
             =
             130½
             Then
             V
             =
             81.
             
             And
             (
             by
             §
             39.
             )
             
             .
          
           
             LV.
             If
             A
             =
             49½
             Then
             V
             =
             81.
             
             And
             (
             by
             §
             39.
             )
             
             .
          
           
             And
             ,
             in
             like
             manner
             ,
             by
             help
             of
             §
             40
             ,
             41
             ,
             &c.
             
             We
             may
             proceed
             to
             Arches
             determined
             by
             Quarters
             of
             Degrees
             ;
             and
             further
             if
             need
             be
             .
          
           
             Again
             ,
             because
             the
             Subtense
             of
             72
             Degrees
             ,
             is
             
             and
             the
             Subtense
             of
             60
             Degrees
             is
             R
             :
             We
             may
             thence
             Collect
             the
             Subtense
             of
             their
             Difference
             ,
             which
             is
             that
             of
             12
             Degrees
             ;
             namely
             ,
             R
             into
             
             ;
             or
             
             .
             And
             thence
             proceed
             thus
             ,
          
           
             LVI
             .
             If
             A
             =
             96
             Then
             V
             =
             12.
             
             And
             therefore
             ,
             
             .
          
           
             LVII
             .
             If
             A
             =
             84
             Then
             V
             =
             12.
             
             And
             therefore
             ,
             
             .
          
           
             LVIII
             .
             If
             A
             =
             138
             Then
             V
             =
             96.
             
             And
             (
             by
             §
             56.
             )
             
             .
          
           
             LIX
             .
             If
             A
             =
             42
             Then
             V
             =
             96.
             
             And
             (
             by
             §
             56.
             )
             
             .
          
           
             LX.
             If
             A
             =
             132
             Then
             V
             =
             84.
             
             And
             (
             by
             §
             57.
             )
             
             .
          
           
             LXI
             .
             If
             A
             =
             48
             Then
             V
             =
             84.
             
             And
             (
             by
             §
             57.
             )
             
             .
          
           
             LXII
             .
             If
             A
             =
             159
             Then
             V
             =
             138.
             
             And
             (
             by
             §
             58.
             )
             
          
           
             LXIII
             .
             If
             A
             =
             21
             Then
             V
             =
             138.
             
             And
             (
             by
             §
             58.
             )
             
          
           
             LXIV
             .
             If
             A
             =
             111
             Then
             V
             =
             42.
             
             And
             (
             by
             §
             59.
             )
             
          
           
             LXV
             .
             If
             A
             =
             69
             Then
             V
             =
             42.
             
             And
             (
             by
             §
             59.
             )
             
          
           
           
             LXVI
             .
             If
             A
             =
             156
             Then
             V
             =
             132.
             
             And
             (
             by
             §
             60.
             )
             
             .
          
           
             LXVII
             .
             If
             A
             =
             24
             Then
             V
             =
             132.
             
             And
             (
             by
             §
             60.
             )
             
             .
          
           
             LXVIII
             .
             If
             A
             =
             114
             Then
             V
             =
             48.
             
             And
             (
             by
             §
             61.
             )
             
             .
          
           
             LXIX
             .
             If
             A
             =
             66
             Then
             V
             =
             48.
             
             And
             (
             by
             §
             61.
             )
             
             .
          
           
             LXX
             .
             If
             A
             =
             169½
             Then
             V
             =
             159.
             
             And
             (
             by
             §
             62.
             )
             
             .
          
           
             LXXI
             .
             If
             A
             =
             10½
             Then
             V
             =
             159.
             
             And
             (
             by
             §
             62.
             )
             
             .
          
           
             LXXII
             .
             If
             A
             =
             100½
             Then
             V
             =
             21.
             
             And
             (
             by
             §
             63.
             )
             
             .
          
           
             LXXIII
             .
             If
             A
             =
             79½
             Then
             V
             =
             21.
             
             And
             (
             by
             §
             63.
             )
             
             .
          
           
             LXXIV
             .
             If
             A
             =
             145½
             Then
             V
             =
             111.
             
             And
             (
             by
             §
             64.
             )
             
             .
          
           
             LXXV
             .
             If
             A
             =
             34½
             Then
             V
             =
             111.
             
             And
             (
             by
             §
             64.
             )
             
             .
          
           
             LXXVI
             .
             If
             A
             =
             124½
             Then
             V
             =
             69.
             
             And
             (
             by
             §
             65.
             )
             
             .
          
           
             LXXVII
             .
             If
             A
             =
             55
             Then
             V
             =
             69.
             
             And
             (
             by
             §
             65.
             )
             
             .
          
           
             LXXVIII
             .
             If
             A
             =
             168
             Then
             V
             =
             156.
             
             And
             (
             by
             §
             66.
             )
             
             .
          
           
             LXXIX
             .
             If
             A
             =
             12
             Then
             V
             =
             156.
             
             And
             (
             by
             §
             66.
             )
             
             .
          
           
             LXXX
             .
             If
             A
             =
             102
             Then
             V
             =
             24.
             
             And
             (
             by
             §
             67.
             )
             
             .
          
           
             LXXXI
             .
             If
             A
             =
             78
             Then
             V
             =
             24.
             
             And
             (
             by
             §
             67.
             )
             
             .
          
           
             LXXXII
             .
             If
             A
             =
             147
             Then
             V
             =
             114.
             
             And
             (
             by
             §
             68.
             )
             
             .
          
           
             LXXXIII
             .
             If
             A
             =
             33
             Then
             V
             =
             114.
             
             And
             (
             by
             §
             68.
             )
             
             .
          
           
           
             LXXXIV
             .
             If
             A
             =
             123
             Then
             V
             =
             66.
             
             And
             (
             by
             §
             69.
             )
             
             .
          
           
             LXXXV
             .
             If
             A
             =
             57
             Then
             V
             =
             66.
             
             And
             (
             by
             §
             69.
             )
             
             .
          
           
             LXXXVI
             .
             If
             A
             =
             174
             Then
             V
             =
             168.
             
             And
             (
             by
             §
             78.
             )
             
             .
          
           
             LXXXVII
             .
             If
             A
             =
             6
             Then
             V
             =
             168.
             
             And
             (
             by
             §
             78.
             )
             
             .
          
           
             LXXXVIII
             .
             If
             A
             =
             141
             Then
             V
             =
             102.
             
             And
             (
             by
             §
             80.
             )
             
             .
          
           
             LXXXIX
             .
             If
             A
             =
             39
             Then
             V
             =
             102.
             
             And
             (
             by
             §
             80.
             )
             
             .
          
           
             XC
             .
             If
             A
             =
             129.
             
             Then
             V
             =
             78.
             
             And
             (
             by
             §
             81.
             )
             
             .
          
           
             XCI
             .
             If
             A
             =
             51
             Then
             V
             =
             78.
             
             And
             (
             by
             §
             81.
             )
             
             .
          
           
             XCII
             .
             If
             A
             =
             163
             ½
             Then
             V
             =
             147.
             
             And
             (
             by
             §
             82.
             )
             
             .
          
           
             XCIII
             .
             If
             A
             =
             16
             ½
             Then
             V
             =
             147.
             
             And
             (
             by
             §
             82.
             )
             
             .
          
           
             XCIV
             .
             If
             A
             =
             106
             ½
             Then
             V
             =
             33.
             
             And
             (
             by
             §
             83.
             )
             
             .
          
           
             XCV
             .
             If
             A
             =
             73
             ½
             Then
             V
             =
             33.
             
             And
             (
             by
             §
             83.
             )
             
             .
          
           
             XCVI
             .
             If
             A
             =
             151
             ½
             Then
             V
             =
             123.
             
             And
             (
             by
             §
             84.
             )
             
             .
          
           
             XCVII
             .
             If
             A
             =
             28
             ½
             Then
             V
             =
             123.
             
             And
             (
             by
             §
             84.
             )
             
             .
          
           
             XCVIII
             .
             If
             A
             =
             118
             ½
             Then
             V
             =
             57.
             
             And
             (
             by
             §
             85.
             )
             
             .
          
           
             XCIX
             .
             If
             A
             =
             61
             ½
             Then
             V
             =
             57.
             
             And
             (
             by
             §
             85.
             )
             
             .
          
           
             C.
             If
             A
             =
             177
             Then
             V
             =
             174.
             
             And
             (
             by
             §
             86.
             )
             
             .
          
           
             CI.
             If
             A
             =
             3
             Then
             V
             =
             174.
             
             And
             (
             by
             §
             86.
             )
             
             .
          
           
             CII
             .
             If
             A
             =
             93
             Then
             V
             =
             6.
             
             And
             (
             by
             §
             87.
             )
             
             .
          
           
             CIII
             .
             If
             A
             =
             87
             Then
             V
             =
             6.
             
             And
             (
             by
             §
             87.
             )
             
             .
          
           
             CIV
             .
             If
             A
             =
             160
             ½
             Then
             V
             =
             141.
             
             And
             (
             by
             §
             88.
             )
             
             .
          
           
             CV
             .
             If
             A
             =
             19
             ½
             Then
             V
             =
             141.
             
             And
             (
             by
             §
             88.
             )
             
             .
          
           
           
             CVI.
             If
             A
             =
             109
             ½
             Then
             V
             =
             39.
             
             And
             (
             by
             §
             89.
             )
             
             .
          
           
             CVII
             .
             If
             A
             =
             70
             ½
             Then
             V
             =
             39.
             
             And
             (
             by
             §
             89.
             )
             
             .
          
           
             CVIII
             .
             If
             A
             =
             154
             ½
             Then
             V
             =
             129.
             
             And
             (
             by
             §
             90.
             )
             
             .
          
           
             CIX
             .
             If
             A
             =
             25
             ½
             Then
             V
             =
             129.
             
             And
             (
             by
             §
             90.
             )
             
             .
          
           
             CX
             .
             If
             A
             =
             115
             ½
             Then
             V
             =
             51.
             
             And
             (
             by
             §
             91.
             )
             
             .
          
           
             CXI
             .
             If
             A
             =
             64
             ½
             Then
             V
             =
             51.
             
             And
             (
             by
             §
             91.
             )
             
             .
          
           
             CXII
             .
             If
             A
             =
             178
             ½
             Then
             V
             =
             177.
             
             And
             (
             by
             §
             100.
             )
             
             .
          
           
             CXIII
             .
             If
             A
             =
             1
             ½
             Then
             V
             =
             177.
             
             And
             (
             by
             §
             100.
             )
             
             .
          
           
             CXIV
             .
             If
             A
             =
             91
             ½
             Then
             V
             =
             3.
             
             And
             (
             by
             §
             101.
             )
             
             .
          
           
             CXV
             .
             If
             A
             =
             88
             ½
             Then
             V
             =
             3.
             
             And
             (
             by
             §
             101.
             )
             
             .
          
           
             CXVI
             .
             If
             A
             =
             136
             ½
             Then
             V
             =
             93.
             
             And
             (
             by
             §
             102.
             )
             
             .
          
           
             CVII
             .
             If
             A
             =
             43
             ½
             Then
             V
             =
             93.
             
             And
             (
             by
             §
             102.
             )
             
             .
          
           
             CXVIII
             .
             If
             A
             =
             133
             ½
             Then
             V
             =
             87.
             
             And
             (
             by
             §
             107.
             )
             
             .
          
           
             CXIX
             .
             If
             A
             =
             46
             ½
             Then
             V
             =
             87.
             
             And
             (
             by
             §
             107.
             )
             
             .
          
           
             And
             ,
             in
             like
             manner
             ,
             (
             by
             help
             of
             §
             70
             ,
             71
             ,
             &c.
             92
             ,
             93
             ,
             &c.
             104
             ,
             105
             ,
             &c.
             as
             was
             shewed
             at
             §
             23.
             )
             we
             may
             proceed
             to
             Arches
             determined
             by
             Quarters
             of
             Degrees
             ;
             or
             yet
             further
             ,
             if
             there
             be
             occasion
             .
          
           
             But
             we
             content
             our selves
             at
             present
             to
             rest
             at
             half
             Degrees
             .
             Having
             hereby
             sitted
             subtenses
             to
             every
             three
             halves
             of
             a
             Degree
             throughout
             the
             Semicircle
             .
          
        
         
           
           
             
               CHAP.
               VIII
            
             .
             Of
             the
             Canon
             of
             Subtenses
             ,
             and
             Sines
             ;
             Of
             Tangents
             also
             and
             of
             Secants
             .
          
           
             FROM
             what
             is
             delivered
             in
             the
             foregoing
             Chapter
             ;
             it
             is
             easie
             to
             construct
             a
             Canon
             of
             Subtenses
             or
             Chords
             ,
             in
             Surd
             Roots
             ,
             to
             every
             Threehalves
             of
             a
             Degree
             throughout
             the
             Semicircle
             .
             The
             halves
             of
             which
             Subtenses
             ,
             are
             the
             Right-sines
             for
             every
             Three-quarters
             of
             a
             Degree
             throughout
             the
             Quadrant
             .
          
           
             (
             And
             thence
             ,
             if
             need
             be
             ,
             many
             Canons
             of
             Tangents
             and
             Secants
             ,
             be
             deduced
             ,
             in
             Surds
             Roots
             .
             )
          
           
             And
             hereby
             ,
             any
             who
             please
             ,
             may
             either
             make
             new
             Tables
             ,
             in
             Numbers
             ,
             (
             to
             what
             accuracy
             he
             please
             ,
             )
             or
             examin
             those
             already
             made
             .
          
           
             For
             to
             every
             Subtense
             ,
             to
             be
             successively
             sought
             ,
             there
             will
             need
             but
             one
             extraction
             of
             the
             Square
             Root
             ;
             (
             and
             ,
             sometimes
             ,
             not
             this
             ;
             )
             the
             rest
             of
             the
             work
             being
             dispatched
             by
             only
             Addition
             and
             Subtraction
             ;
             or
             ,
             at
             most
             ,
             Division
             also
             by
             2
             or
             4.
             
          
           
             As
             ,
             for
             Example
             :
             Supposing
             the
             Radius
             of
             a
             Circle
             R
             =
             1.
             
             Then
             (
             because
             these
             ,
             in
             the
             same
             Circle
             be
             all
             equal
             ,
             )
             C
             =
             D
             =
             1.
             
             And
             likewise
             Cq
             =
             Dq
             =
             CD
             =
             1.
             
             And
             B
             will
             be
             the
             Subtense
             of
             the
             Angle
             proposed
             .
          
           
             Therefore
             ,
             (
             by
             §
             1.
             )
             the
             Square
             of
             the
             Subtense
             of
             90
             Degrees
             ,
             Bq
             =
             Cq
             +
             Dq
             =
             1
             +
             1
             =
             2.
             
             And
             the
             Subtense
             it self
             
             :
             Which
             is
             had
             by
             one
             extraction
             of
             the
             Square
             Root
             of
             the
             number
             2
             ;
             continued
             in
             Decimal
             Parts
             to
             to
             what
             accuracy
             we
             please
             .
             Suppose
             
             proximè
             .
          
           
             Again
             ,
             (
             by
             §
             2.
             )
             the
             Square
             of
             the
             Subtense
             of
             120
             Degrees
             ,
             is
             Bq
             =
             Cq
             +
             Dq
             +
             CD
             =
             1
             +
             1
             +
             1
             =
             3.
             
             And
             the
             Subtense
             it self
             
             :
             Which
             is
             likewise
             had
             by
             one
             extraction
             of
             the
             Square
             Root
             of
             3.
             Suppose
             
             −
             proximè
             .
          
           
             The
             Square
             of
             the
             Subtense
             of
             60
             Degrees
             ,
             is
             (
             by
             §
             3.
             )
             Bq
             =
             Cq
             +
             Dq
             −
             CD
             =
             1
             +
             1
             −
             1
             =
             1.
             
             And
             therefore
             the
             Subtense
             B
             =
             1.
             
          
           
             The
             Square
             of
             the
             Subtense
             of
             135
             Degrees
             ,
             is
             (
             by
             §
             4.
             )
             
             .
             Which
             is
             had
             by
             adding
             2
             ,
             to
             the
             value
             of
             
             already
             found
             at
             §
             1.
             
             That
             is
             ,
             
             .
             And
             so
             by
             one
             extraction
             of
             the
             Square
             Root
             of
             this
             number
             ,
             we
             have
             the
             Subtense
             of
             135
             Degrees
             :
             Namely
             ,
             
             proximè
             .
          
           
             So
             (
             by
             §
             5.
             )
             the
             Square
             of
             the
             Subtense
             of
             45
             Degrees
             ,
             is
             
             .
             That
             is
             (
             by
             subtracting
             from
             2
             the
             value
             of
             
             already
             found
             ,
             )
             Bq
             =
             0.58578643
             ½
             proximè
             .
             The
             Square
             Root
             of
             which
             ,
             now
             to
             be
             extracted
             ,
             is
             
             proximè
             .
          
           
           
             And
             (
             by
             §
             6.
             )
             the
             Square
             of
             the
             Subtense
             of
             150
             Degrees
             ,
             is
             
             .
             Which
             is
             had
             ,
             by
             adding
             2
             to
             the
             value
             of
             
             all
             ready
             found
             .
             That
             is
             ,
             
             .
             The
             Root
             of
             which
             (
             now
             to
             be
             extracted
             )
             is
             
             proximè
             .
          
           
             Or
             thus
             :
             Because
             
             ,
             (
             as
             will
             appear
             ,
             either
             by
             the
             Squaring
             of
             this
             ,
             or
             by
             extracting
             the
             Square
             Root
             of
             the
             Binomial
             
             )
             Having
             ,
             as
             before
             ,
             the
             value
             of
             
             ;
             and
             (
             by
             one
             extraction
             now
             to
             be
             made
             )
             the
             value
             of
             
             (
             or
             ,
             it
             may
             be
             had
             by
             Multiplying
             the
             value
             of
             
             ,
             by
             that
             of
             
             ,
             already
             known
             ;
             because
             
             )
             we
             have
             thence
             
             ;
             and
             the
             half
             thereof
             
             .
             As
             before
             ,
          
           
             So
             (
             by
             §
             7.
             )
             the
             Square
             of
             the
             Subtense
             of
             30
             Degrees
             ,
             is
             
             ,
             (
             which
             is
             had
             by
             Subduction
             only
             ,
             the
             value
             of
             
             being
             found
             before
             .
             )
             The
             Square
             Root
             of
             which
             (
             now
             to
             be
             extracted
             )
             is
             the
             Subtense
             
             proximè
             .
          
           
             Or
             thus
             ,
             (
             without
             extracting
             a
             Root
             ;
             )
             because
             
             :
             Therefore
             ,
             (
             the
             values
             of
             
             and
             
             being
             had
             before
             :
             )
             by
             Subduction
             only
             we
             have
             
             proximè
             ;
             and
             (
             the
             half
             of
             this
             )
             
             proximè
             :
             As
             before
             ,
          
           
             And
             in
             the
             rest
             ,
             (
             taking
             the
             Propositions
             or
             Paragraphs
             as
             they
             are
             before
             set
             down
             in
             the
             former
             Chapter
             ,
             )
             there
             is
             need
             but
             of
             one
             extraction
             of
             the
             Square
             Root
             (
             and
             oft-times
             not
             of
             one
             ,
             )
             for
             finding
             of
             each
             Subtense
             .
          
           
             These
             Subtenses
             being
             thus
             had
             ;
             the
             halves
             thereof
             are
             the
             Right-sines
             of
             the
             half
             Arch.
             As
             for
             Example
             .
          
           
             
               
                 
                   Arches
                   .
                
                 
                   Subtenses
                   .
                
                 
                   Sines
                   .
                
                 
                   Arches
                   .
                
              
               
                 
                   Degrees
                   .
                   90
                
                 
                   1.41421356
                   +
                
                 
                   0.70710678
                   +
                
                 
                   45
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   120
                
                 
                   1.73205081
                   −
                
                 
                   0.86602540
                   ½
                
                 
                   60
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   60
                
                 
                   1.00000000
                
                 
                   0.50000000
                
                 
                   30
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   135
                
                 
                   1.84775906
                   ½
                
                 
                   0.92387953
                   ¼
                
                 
                   67
                   ½
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   45
                
                 
                   0.76536687
                   −
                
                 
                   0.38268343
                   ½
                
                 
                   22
                   ½
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   150
                
                 
                   1.93185165
                   +
                
                 
                   0.96592582
                   ½
                
                 
                   75
                   Degrees
                   .
                
              
               
                 
                   Degrees
                   .
                   30
                
                 
                   0.51763809
                
                 
                   0.25881904
                   ½
                
                 
                   15
                   Degrees
                   .
                
              
            
          
           
             Now
             follows
             the
             Table
             of
             Subtenses
             in
             Surd
             Roots
             ,
             answering
             to
             each
             three
             halves
             of
             a
             Degree
             throughout
             the
             whole
             Semicircle
             ;
             (
             and
             consequently
             ,
             of
             their
             Residuals
             to
             a
             whole
             Circle
             ,
             whose
             Subtenses
             are
             the
             same
             with
             these
             :
             )
             Putting
             the
             Radius
             R
             =
             1
             ,
             and
             therefore
             C
             =
             D
             =
             1
             ,
             and
             likewise
             Cq
             =
             Dq
             =
             CD
             =
             1
             :
             With
             references
             in
             the
             Margin
             to
             the
             Paragraphs
             of
             the
             former
             Chapter
             from
             whence
             they
             are
             derived
             .
          
           
           
             
             
             
             
             
             
             
             
             And
             ,
             in
             like
             manner
             ,
             we
             may
             proceed
             to
             design
             ,
             by
             Surd
             Roots
             ,
             the
             Subtenses
             of
             Arches
             as
             small
             as
             we
             please
             ,
             by
             a
             continual
             Bisection
             of
             these
             Arches
             .
             The
             halves
             of
             which
             Subtenses
             ,
             are
             the
             Right
             Sines
             of
             the
             Half-Arches
             .
          
           
             But
             ,
             to
             design
             an
             intire
             Canon
             of
             Subtenses
             and
             Sines
             ,
             answering
             to
             each
             single
             Degree
             ,
             and
             the
             Sexagesims
             or
             first
             Minutes
             of
             such
             Degrees
             :
             Will
             (
             beside
             the
             extracting
             the
             Square
             Roots
             ,
             of
             such
             Surds
             ,
             in
             Numbers
             ,
             )
             require
             also
             the
             Analysis
             (
             in
             Numbers
             )
             of
             Two
             Trisections
             ,
             and
             of
             one
             Quinquisection
             of
             an
             Arch.
             
          
           
             For
             ,
             the
             former
             process
             reaching
             no
             farther
             than
             to
             the
             Subtense
             of
             1½
             Degree
             ;
             and
             consequently
             to
             the
             sine
             of
             ¾
             of
             a
             Degree
             ,
             or
             of
             Min.
             45
             =
             3
             ×
             3
             ×
             5
             :
             We
             may
             thence
             ,
             by
             a
             Trisection
             twice
             performed
             ;
             and
             a
             Quinquisection
             once
             ,
             proceed
             to
             the
             sine
             of
             1
             Minute
             .
             But
             not
             by
             Bisections
             only
             ,
             or
             operations
             thence
             deduced
             .
          
           
             But
             ,
             these
             operations
             being
             so
             (
             as
             is
             said
             )
             performed
             ;
             the
             rest
             of
             the
             work
             is
             easily
             dispatched
             by
             help
             of
             §
             9
             ,
             10.
             
             Chap.
             6.
             for
             finding
             the
             Subtense
             of
             the
             Sum
             or
             Difference
             of
             those
             Arches
             whose
             Subtenses
             are
             already
             known
             .
          
        
         
           
             
               CHAP.
               IX
            
             .
             Of
             Angles
             compared
             with
             they
             Arches
             on
             which
             the
             stand
             .
          
           
             I.
             THAT
             
               The
               Angle
               of
               a
               Sector
               ,
               is
               proportional
               to
               the
               Arch
               on
               which
               it
               doth
               insist
               ;
               whether
               such
               Angle
               be
               at
               the
               Center
               ,
               or
               at
               the
               Circumference
               :
            
             And
             ,
             that
             
               such
               Angle
               at
               the
               Center
               ,
               is
               double
               to
               that
               at
               the
               Circumference
               :
            
             Is
             shewed
             by
             Euclide
             long
             since
             ;
             and
             is
             generally
             known
             .
             But
             not
             so
             ,
             in
             case
             such
             Angle
             be
             any
             where
             else
             ,
             whether
             within
             or
             without
             the
             Circle
             .
             It
             will
             therefore
             be
             not
             amiss
             ,
             to
             pursue
             that
             Notion
             a
             little
             farther
             ;
             as
             here
             followeth
             .
          
           
           
             II.
             If
             on
             the
             Diameter
             of
             a
             Circle
             D
             F
             ,
             be
             formed
             ,
             at
             the
             Center
             C
             ,
             an
             Angle
             B
             A
             D
             ;
             the
             intercepted
             Arch
             B
             D
             ,
             is
             Proportional
             to
             the
             Angle
             .
             (
             by
             26
             ,
             27.
             
             El.
             3.
             and
             33
             ,
             El.
             6
             ,
             of
             Euclid
             .
             )
             That
             is
             ,
             the
             Arch
             intercepted
             B
             D
             ,
             is
             such
             a
             Part
             of
             the
             whole
             Circumference
             ;
             as
             is
             the
             Angle
             A
             ,
             of
             Four
             Right-angles
             .
             And
             accordingly
             the
             Angle
             A
             ,
             is
             said
             to
             be
             ,
             of
             so
             many
             Degrees
             ,
             as
             is
             the
             Arch
             B
             D.
             
          
           
             III.
             If
             B
             A
             ,
             D
             A
             ,
             the
             Legs
             containing
             such
             Angle
             ,
             after
             a
             Decussation
             at
             C
             ,
             forming
             the
             Vertical
             Angle
             E
             ,
             be
             continued
             ,
             on
             the
             other
             side
             ,
             to
             the
             Circumference
             ;
             the
             intercepted
             Arch
             G
             F
             ,
             will
             ,
             (
             by
             §
             2.
             )
             be
             equal
             to
             B
             D
             ,
             (
             because
             of
             the
             Vertical
             Angle
             E
             ,
             equal
             to
             A
             :
             )
             And
             ,
             consequently
             ,
             the
             Aggregate
             of
             both
             B
             D
             +
             G
             F
             ,
             is
             double
             to
             B
             D.
             That
             is
             ,
             B
             D
             +
             G
             F
             =
             2
             B
             D.
             
          
           
             IV.
             If
             at
             F
             the
             end
             of
             the
             Diameter
             ,
             be
             formed
             a
             like
             Angle
             A
             ;
             the
             intercepted
             Arch
             H
             D
             ,
             is
             likewise
             double
             to
             B
             D.
             That
             is
             ,
             H
             D
             =
             B
             D
             +
             G
             F
             =
             2
             B
             D.
             Because
             (
             by
             20.
             
             El.
             3.
             )
             the
             Angle
             at
             the
             Center
             ,
             is
             double
             to
             that
             at
             the
             Circumference
             ,
             on
             the
             same
             Arch.
             
          
           
           
             Or
             thus
             ,
             because
             (
             by
             Construction
             )
             F
             H
             ,
             G
             C
             B
             ,
             are
             Parallels
             :
             (
             as
             making
             equal
             Angles
             with
             F
             D
             :
             )
             Therefore
             ,
             H
             B
             =
             F
             G
             =
             B
             D.
             And
             ,
             consequently
             ,
             H
             D
             =
             H
             B
             +
             B
             D
             =
             F
             G
             +
             B
             D
             =
             2
             B
             D.
             (
             By
             §
             3.
             )
          
           
             V.
             If
             at
             K
             ,
             (
             any
             other
             Point
             of
             the
             Diameter
             within
             the
             Circle
             ,
             )
             such
             Angle
             A
             be
             made
             ,
             (
             with
             its
             Vertical
             E
             :
             )
             the
             Aggregate
             of
             the
             two
             Arches
             intercepted
             ,
             F
             L
             +
             M
             D
             ,
             is
             Double
             to
             B
             D.
             That
             is
             ,
             F
             L
             +
             M
             D
             =
             F
             G
             +
             B
             D
             =
             2
             B
             D.
             For
             ,
             (
             drawing
             the
             streight
             Line
             A
             M
             ,
             )
             the
             two
             Internal
             Angles
             F
             M
             L
             +
             M
             F
             D
             ,
             are
             equal
             to
             the
             External
             M
             K
             D
             =
             A
             =
             H
             F
             D
             (
             which
             is
             an
             Angle
             at
             the
             Circumference
             .
             )
             And
             therefore
             ,
             the
             Arches
             opposite
             to
             those
             F
             L
             +
             M
             D
             ,
             equal
             to
             the
             Arch
             opposite
             to
             this
             H
             D
             =
             2
             B
             D.
             (
             By
             §
             4.
             )
          
           
             Or
             thus
             ,
             because
             F
             H
             ,
             L
             M
             ,
             be
             Parallels
             (
             as
             making
             like
             Angles
             with
             F
             D
             ,
             )
             therefore
             ,
             F
             L
             =
             H
             M
             ,
             and
             F
             L
             +
             M
             D
             =
             H
             M
             +
             M
             D
             =
             H
             D
             =
             2
             B
             D.
             (
             By
             §
             4.
             )
          
           
             VI.
             If
             at
             N
             ,
             a
             Point
             of
             the
             Diameter
             produced
             ,
             without
             the
             Circle
             ;
             be
             formed
             a
             like
             Angle
             A
             ;
             the
             Difference
             of
             the
             two
             intercepted
             Arches
             ,
             Q
             D
             −
             P
             F
             ,
             is
             equal
             to
             the
             same
             H
             D
             ,
             or
             the
             Double
             of
             B
             D.
             For
             ,
             P
             Q
             ,
             F
             H
             ,
             being
             Parallels
             ,
             (
             as
             making
             like
             Angles
             with
             D
             F
             produced
             ,
             )
             and
             therefore
             ,
             P
             F
             =
             Q
             H
             :
             Therefore
             ,
             Q
             D
             −
             P
             F
             =
             Q
             D
             −
             Q
             H
             =
             H
             D
             =
             2
             B
             D.
             (
             By
             §
             4.
             )
          
           
             VII
             .
             The
             same
             will
             hold
             ,
             though
             neither
             of
             the
             Legs
             containing
             the
             Angle
             do
             pass
             through
             the
             Center
             ,
             (
             and
             therefore
             lie
             not
             upon
             a
             Diameter
             ;
             )
             as
             I
             shall
             now
             shew
             in
             the
             several
             Cases
             .
          
           
             VIII
             .
             If
             M
             L
             ,
             μ
             λ
             ,
             make
             M
             K
             μ
             an
             Angle
             at
             K
             any
             where
             ,
             within
             the
             Circle
             ,
             Let
             B
             c
             β
             be
             a
             like
             Angle
             at
             the
             Center
             ,
             and
             the
             Legs
             of
             this
             Parallel
             to
             those
             of
             that
             .
             And
             ,
             by
             the
             Angular
             Point
             K
             ,
             draw
             the
             Diameter
             F
             K
             D.
             Then
             is
             (
             by
             §
             5.
             )
             M
             D
             +
             L
             F
             =
             2
             B
             D
             :
             And
             μ
             D
             +
             λ
             F
             =
             2
             β
             D.
             Therefore
             ,
             (
             the
             Sum
             or
             Difference
             )
             M
             μ
             (
             =
             M
             D
             ±
             μ
             D
             )
             +
             L
             λ
             (
             =
             L
             F
             ±
             λ
             F
             )
             =
             (
             2
             B
             D
             ±
             2
             β
             D
             =
             )
             2
             B
             β.
             
          
           
           
             IX
             .
             In
             like
             manner
             :
             If
             H
             F
             η
             be
             an
             Angle
             at
             the
             Circumference
             :
             And
             ,
             at
             the
             Center
             ,
             B
             C
             β
             like
             to
             it
             ,
             and
             with
             Legs
             Parallel
             to
             those
             :
             And
             F
             D
             a
             Diameter
             .
             Then
             (
             by
             §
             4.
             )
             H
             D
             =
             2
             B
             D
             ,
             and
             η
             D
             =
             2
             β
             D.
             Therefore
             ,
             H
             η
             (
             =
             H
             D
             ±
             η
             D
             )
             =
             (
             2
             B
             D
             ±
             2
             β
             D
             =
             )
             2
             B
             β.
             
          
           
           
           
             X.
             The
             same
             is
             to
             be
             understood
             ,
             in
             case
             one
             of
             the
             Legs
             touch
             the
             Circumference
             at
             F
             ,
             (
             the
             Points
             F
             ,
             H
             ,
             being
             in
             this
             case
             Co-incident
             ;
             the
             Arch
             F
             H
             vanishing
             to
             nothing
             ,
             and
             the
             Arch
             intercepted
             H
             η
             ,
             the
             same
             with
             F
             η.
             )
             For
             here
             also
             ,
             H
             F
             D
             =
             2
             B
             D
             ,
             and
             η
             F
             D
             =
             2
             β
             D
             ;
             and
             therefore
             ,
             H
             F
             η
             (
             =
             H
             F
             D
             ±
             η
             F
             D
             )
             =
             (
             2
             B
             D
             ±
             2
             β
             D
             =
             )
             2
             B
             β.
             
          
           
           
             XI
             .
             In
             like
             manner
             :
             If
             Q
             N
             χ
             be
             an
             Angle
             without
             the
             Circle
             ,
             whose
             Legs
             cut
             it
             in
             P
             ,
             π
             :
             And
             ,
             at
             the
             Center
             ,
             a
             like
             Angle
             and
             like
             sited
             B
             C
             :
             And
             N
             F
             D
             a
             Diameter
             produced
             .
             Then
             (
             by
             §
             6.
             )
             Q
             D
             −
             P
             F
             =
             2
             B
             D
             ,
             and
             χ
             D
             −
             π
             F
             =
             2
             β
             D.
             And
             therefore
             Q
             D
             ±
             χ
             D
             wanting
             P
             F
             ±
             π
             F
             ,
             that
             is
             Q
             χ
             −
             P
             π
             ,
             is
             equal
             to
             2
             B
             D
             ±
             2
             β
             D
             =
             2
             B
             β.
             
          
           
           
             XII
             .
             The
             same
             is
             to
             be
             understood
             ,
             in
             case
             one
             or
             both
             of
             the
             Legs
             do
             not
             cut
             ,
             but
             only
             touch
             the
             Circle
             .
             For
             then
             the
             Points
             P
             ,
             Q
             ;
             or
             π
             ,
             χ
             ;
             (
             or
             both
             ,
             )
             being
             Co-incident
             ;
             the
             rest
             proceeds
             as
             before
             .
             For
             still
             Q
             D
             −
             P
             F
             =
             2
             B
             D
             ,
             and
             χ
             D
             −
             π
             F
             =
             2
             β
             D
             ;
             and
             therefore
             ,
             (
             the
             Sum
             or
             Difference
             )
             Q
             χ
             −
             P
             π
             =
             2
             B
             β.
             
          
           
           
             XIII
             .
             But
             in
             case
             one
             or
             both
             of
             the
             Legs
             pass
             by
             the
             Circle
             ,
             and
             neither
             cut
             ,
             nor
             so
             much
             as
             touch
             it
             :
             It
             doth
             not
             concern
             the
             present
             business
             ;
             for
             such
             Angle
             doth
             not
             insist
             on
             a
             Circular
             Arch.
             The
             whole
             therefore
             (
             thus
             demonstrated
             )
             amounts
             to
             this
             General
             .
          
           
             XIV
             .
             
               If
               a
               Circle
               be
               cut
            
             (
             or
             at
             least
             touched
             )
             
               by
               two
               streight
               Lines
               ,
               making
               an
               Angle
               :
            
             (
             and
             so
             ,
             when
             continued
             ,
             intersecting
             each
             other
             :
             )
             
               The
               Sum
            
             (
             if
             their
             intersection
             be
             within
             the
             Circle
             ,
             )
             
               or
               Difference
            
             (
             if
             without
             ,
             )
             
               of
               the
               two
               Arches
               intercepted
               by
               them
            
             (
             produced
             ,
             if
             need
             be
             ,
             )
             or
             (
             if
             their
             intersection
             be
             at
             the
             Circumference
             )
             
               the
               single
               Arch
               by
               them
               intercepted
               ;
               is
               Double
               to
               the
               Arch
               of
               a
               like
               Angle
               at
               the
               Center
               .
            
          
        
         
           FINIS
           .
        
      
    
     
       
         
         
           
             
               I
            
          
           
             
               II
            
          
           
             
               III
            
          
           
             
               IV
            
          
           
             
               V
            
          
           
             
               VI
            
          
           
             
               VII
            
          
           
             
               VIII
            
          
           
             
               IX
            
          
           
             
               X
            
          
           
             
               XI
            
          
           
             
               XII
            
          
           
             
               XIII
            
          
           
             
               XIV
            
          
           
             
               XV
            
          
           
             
               XVI
            
          
           
             
               XVII
            
          
           
             
               XVIII
            
          
           
             
               XIX
            
          
           
             
               XX
            
          
           
             
               XXI
            
          
           
             
               XXII
            
          
           
             
               XXIII
            
          
           
             
             
               XXIV
            
          
           
             
               XXV
            
          
           
             
               XXVI
            
          
           
             
               XXVII
            
          
           
             
               XXVIII
            
          
           
             
               XXIX
            
          
           
             
               XXX
            
          
           
             
               XXXI
            
          
           
             
               XXXII
            
          
           
             
               XXXIII
            
          
           
             
               XXXIV
            
          
           
             
               XXXV
            
          
           
             
               XXXVI
            
          
           
             
               XXXVII
            
          
           
             
               XXXVIII
            
          
           
             
               XXXIX
            
          
        
      
       
         Notes, typically marginal, from the original text
         
           Notes for div A67419-e150
           
             Fig.
             I.
             
          
           
             Fig.
             II.
             
          
           
             Fig.
             III.
             
          
           
             Fig.
             IV.
             V
             
          
           
             Fig.
             VI.
             
          
           
             Fig.
             VII
             .
          
           
             Fig.
             VIII
             .
          
           
             Fig.
             IX
             .
          
           
             Fig.
             IX
             .
          
           
             Fig.
             X.
             
          
           
             Fig.
             XI
             .
          
           
             Fig.
             XI
             .
          
           
             Fig.
             XII
             .
          
           
             Fig.
             XII
             .
          
           
             Fig.
             XII
             .
          
           
             Fig.
             XIII
             .
          
           
             Fig.
             XIII
             
          
           
             Fig.
             XIII
             
          
           
             Fig.
             XIV
             .
          
           
             Fig.
             XIV
             .
          
           
             Fig.
             XV.
             
          
           
             Fig.
             XV.
             
          
           
             Fig.
             XVI
             .
          
           
             Fig.
             XV.
             
          
           
             Fig.
             XVII
             .
          
           
             Fig.
             XVII
             .
          
           
             Fig.
             XVII
             .
          
           
             Fig.
             XVIII
             .
          
           
             Fig.
             XVIII
             .
          
           
             Fig.
             XVIII
             .
          
           
             Fig.
             XIX
             
          
           
             Fig.
             XIX
             
          
           
             Fig.
             XIX
             
          
           
             Fig.
             XIX
             .
          
           
             Fig.
             XX.
             
          
           
             Fig.
             XVI
             
          
           
             Fig.
             XXI
             
          
           
             Fig.
             XXII
             .
          
           
             Fig.
             I.
             
          
           
             Fig.
             XI
             .
          
           
             Fig.
             XXIII
             .
          
           
             Fig.
             XXIV
             .
          
           
             Fig.
             XXV
             .
          
           
             Fig.
             XXV
             .
          
           
             Fig.
             XXVI
             .
          
           
             Fig.
             XXVII
             .
          
           
             Fig.
             XXVII
             .
          
           
             Fig.
             XXVII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXVIII
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIV
             .
          
           
             Fig.
             XXX
             .
          
           
             Fig.
             XXXI
             .
          
           
             Fig.
             XXXII
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIX
             .
          
           
             Fig.
             XXIV
             .
          
           
             Fig.
             XXV
             .
          
           
             Fig.
             XXV
             .
          
           
             Fig.
             XXVI
             .
          
           
             Fig.
             XXVII
             .
          
           
             Fig.
             XXXIII
             
          
           
             Fig.
             XXXIV
             .
          
           
             Fig.
             XXXV
             .
          
           
             Fig.
             XXXVI
             .
          
           
             Fig.
             XXXVII
             
          
           
             Fig.
             XXXViii
             
          
           
             Fig.
             XXXViii
             
          
           
             Fig.
             XXXIX
             .